Back to GEOMETRY GARRET

Alan H. Schoen

Comments are welcome.

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Triply-periodic minimal surfaces (TPMS)

This is an illustrated account of my amateur study of TPMS, aimed at both beginner and specialist. It contains  
links to the contemporary mathematical and scientific literature. I describe some of the chance events in 1966  
and 1967 that led to my three-year immersion in this study, in which I was guided by both mathematics and    
              physical experimentation. I benefited greatly from discussions with several eminent mathematicians, some of whom     
appear in photos below, but especially useful in my study of the geometry of periodic structures were the two  
  books 'Third Dimension in Chemistry', by A. F. Wells and 'Regular Polytopes', by H.S.M. Coxeter. I owe           
      special thanks to the architect Peter Pearce, who in 1966 demonstrated for me his concept of saddle polyhedron.   
     Two months later, I had the good luck to be visited by the geometer Norman Johnson, who had just completed his
mathematics PhD under Prof. Coxeter at the University of Toronto. Norman told me about Coxeter's paper on
regular skew polytopes (Proc. Lond. Math. Soc. (2), 43 (1937) 33), which eventually proved to be critically    
useful to me in my pursuit of the elusive gyroid minimal surface (cf. §1 below).                                                   

    In 1967, I accepted the position of senior scientist at NASA's Electronics Research Center ('ERC') in Cambridge,
      Massachusetts. I had been invited to join ERC by the physicist Lester C. Van Atta, who was the associate director.
               During my three years at ERC, the sculptor Harald Robinson provided invaluable technical support. Fashioning special
               tools for the fabrication of plastic models of minimal surfaces was just one of several tasks he performed with unfailing
         skill and ingenuity. Others who helped me in various ways during my 1967 to 1970 stint at ERC include Bob Davis,
    Jay Epstein, Rick Kondrat, Randall Lundberg, Kenneth Paciulan, Charlie Strauss, Jim Wixson, Norman Johnson,
           Blaine Lawson, John Milnor, and Bob Osserman. From 1970 to 1973, my student collaborators at California Institute
         of the Arts were John Brennan and Bob Fuller. At Southern Illinois University/Carbondale (1973-1983), my student
            collaborators included Bob Gray, Thad Heckman, and several others. I am enormously indebted to all of these people!

      In September 1999, I began a collaboration with Ken Brakke, who makes precise mathematical models of minimal
  surfaces with Surface Evolver, his powerful interactive program. Some results of our joint work are displayed at
Triply periodic minimal surfaces, which is one of Ken's many fascinating web sites.                                             

  I pay special attention here to the gyroid minimal surface, G. My 1970 NASA Technical Note, which is entitled
                      Infinite Periodic Minimal Surfaces Without Self-Intersections, contains a detailed description of the gyroid, which I claimed
                         is the only surface free of transverse self-intersections (i.e., embedded) in the family of surfaces that are associates of the two
                      minimal surfaces D and P first described in 1866 by H.A. Schwarz. The evidence for my claim that the gyroid is embedded
                     included a computer-generated movie of Bonnet bending of the surface and also a physical demonstration of such bending,
using thin plastic models of the surface. The stereoscopic version of the movie was subsequently lost, but a    
                    non-stereoscopic version that did survive is included in this video, starting at about 3^m35^s after the beginning. A sequence
                      of eleven frames from the lost stereoscopic version is shown in §16 (below). If you view these frames stereoscopically, you
             may see at least a suggestion of the self-intersections that occur at bending angles different from those for D, G, and P.

Eventually the gyroid was rigorously proved to be free of self-intersections by Karsten Grosse-Brauckmann   
and Meinhard Wohlgemuth, in their article, 'The gyroid is embedded and has constant mean curvature             
companions', Calc. Var. Partial Differential Equations 4 (1996), no. 6, 499-523.                                                  

        The Wikipedia entry for the gyroid mentions some of its currently recognized 'applications'. One of the first            
            published examples of such an application describes how the gyroid serves as a template for self-assembled periodic
        surfaces separating two interpenetrating regions of matter. Depending on the constutuent materials, the unit cell in
such cases ranges in size over several orders of magnitude — according to the case. Additional examples      
of applications continue to be reported, and in the future I expect to add links here to some of them.               

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§0. Eversion of the Laves graph

The Laves graph is triply-periodic (on a bcc lattice) and chiral. It is of interest for a    
variety of reasons, not least because a left- and right-handed pair of these graphs (an  
enantiomorphic pair) are the skeletal graphs of the two intertwined labyrinths of the  
              gyroid, a triply-periodic minimal surface or TPMS (cf. gyroid history, images, analysis,           
below in §1). Because the Laves graph is one of the rare examples of a triply-periodic
graph on a cubic lattice that is symmetric, any finite piece of it can be everted by a     
continuous transformation in which its edges remain unchanged in length.                  

    The video below is a [cross-eyed] stereoscopic animation of the eversion of a
        finite piece of the Laves graph. It is just 'eye candy' and has no direct bearing   
    on the relation between the Laves graph and the gyroid. It was the result of a
            short excursion in the summer of 1969 into the design of expandable spaceframes,
            after I was informed that some official at NASA headquarters was less than happy
           to hear that I was 'playing with soap bubbles' (they were not soap bubbles — they
were soap films!).                                                                                            

      If you choose an arbitrary vertex P₀ somewhere in this infinite graph, you will
  find that during the eversion transformation, every other vertex in the graph
            travels on an elliptical trajectory that is centered on P₀ . For the three vertices that
         are nearest neighbors of P₀, the ellipse is a circle. In this video, there is a certain
       vertex P_green (it happens to be green, but it could equally well have been a red,
           yellow, or blue one) that is stationary with respect to you, the observer. After you
    find P_green, hold your finger on it during the animation, and observe the path
followed by any one of its three nearest neighbors — P_red, P_yellow, P_blue.   

      You will see that for each of these three neighbor vertices, the ellipse is indeed
     a circle centered on P_green. Exactly halfway through the video, the graph has
        again become fully 'deployed', but now it is the enantiomorphic [mirror] image
        of its initial state. At the end of the video, it has been restored to its initial state.

    P_yellow travels in a horizontal plane, P_blue in a plane parallel to your monitor
screen, and P_red in a plane orthogonal to the other two.                                

       I made the original b&w movie of Laves graph eversion in 1969 with the help
                  of a team headed by the computer scientist Charlie Strauss. (The other team members
             were Bob Davis, Ken Pacuilan, and Jay Epstein.) The movie can be viewed in this
      video, starting about 7^m4^s after the beginning. Five stereoscopic snapshots of
the eversion are copied from pp. 86-88 of my 1970 NASA Technical Note.

     The animation shows a single period of the eversion. After mid-period, the
     handedness of each regular helical polygon in the graph has changed from
   CW to CCW. At the end, the initial CW handedness has been restored.     

     If you look carefully, you'll be able to identify the stationary green vertex.
  At mid-period, all of the other vertices have been inverted in this vertex.
(Hint: Consider how the color sequence and positions of the nearest     
        neighbors of this fixed vertex have changed by the end of every half-period.)

For additional details concerning the eversion of the Laves graph, see   
Reflections Concerning Triply-Periodic Minimal Surfaces, pp.663-665,
     a summary of my minimal surface research that was published in October,
        2012 in Interface Focus, a journal of The Royal Society.                                 

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§1. The gyroid

All paired images — both computer drawings and photos — like those just
below are stereoscopic. They are arranged to be viewed with eyes crossed.

  You can run this time-lapse sequence by using the Page Down and Page Up keys.   
  It shows how a sequence of rotations and rotatory reflections transforms a curved   
triangular Flächenstück ( 'asymmetric unit') of the gyroid into this skew hexagon,  
which corresponds to a face of the Coxeter-Petrie map {6,4|4}. The infinite gyroid
could in principle be constructed by first attaching a second hexagon to any open   
edge of an initial hexagon and then repeating the attachment process at every open
edge of the emerging assembly. Each attachment is implemented by applying a      
symmetry of the gyroid: a half-turn about an axis normal to the surface through the
midpoint of each edge. Each of these 'C2 axes' is parallel to a line joining the          
  midpoints of opposite edges of a cube like the cube shown in the illustration above.   

A more economical way to describe the construction of the entire infinite surface is
based on the translation symmetries of the surface. A connected assembly of eight  
suitably located hexagons — no two oriented alike — defines a lattice fundamental
domain. The infinite surface is generated by appropriate translations of this domain.
(It is not essential that the domain be connected, but it is conventional for it to be.)  

_{An example of a lattice fundamental domain of the gyroid.
The lattice is bcc.}

_{The first physical model (1968) approximating the gyroid.
It is composed of several lattice fundamental domains.}

_{If sheets of paper are rolled into cylinders and inserted into the open
  tunnels that are visible here and also into the tunnels visible from the
    side and from above, the curved edges of the faces are found to define
     cylindrical helices centered on lines parallel to the x, y, and z axes. The
     helices in nearest-neighbor parallel tunnels are of opposite handedness.}

_{In 1969, I took this model to a math conference in Tbilisi, Georgia.
Before packing it in a wooden box, I inserted rolled-up tubes of     
   newspaper into its three orthogonal sets of tunnels in order to protect
    it from mechanical shock. These tubes helped to demonstrate that the
          geometry of the gyroid is based on a network of curves that approximate
                 helices. (This correspondence isn't quite exact, however, as I explain in §7.)}   

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_{Two friends inside a gyroid sculpture
at the San Francisco Exploratorium}

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§2. My videos about soap films and TPMS

1972 Part 1 (56:42)
1972 Part 2 (27:09)
1972 Part 3 (35:20)
1972 Part 4 (28:54)
1974 Part 1 (36:02)

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§3. Interface Focus summary article (2012)

Reflections Concerning Triply-Periodic Minimal Surfaces,
a summary of my minimal surface research, which appeared in
October 2012 in Interface Focus, a journal of The Royal Society

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§4. The P−G−D family of associate minimal surfaces

Schwarz's P and D surfaces and their associate surface G (the gyroid) are the topologically
simplest examples of embedded TPMS that have cubic lattice symmetry. They are related
by the continuous bending transformation described in 1853 by Ossian Bonnet. The mean
curvature (which is zero at every point), the Gaussian curvature, and the orientation of the
tangent plane do not change anywhere throughout the bending. P and D are called adjoint
  surfaces: straight lines in one surface are replaced by plane geodesics in the other, and vice
versa. G contains neither straight lines nor plane geodesics. Among the countable infinity
of TPMS related to P and D by Bonnet bending, G is the only example that is embedded,
i.e., free of self-intersections. P, D, and G are each of genus three, which is the minimum
     possible genus for a triply-periodic minimal surface. (In the June/July 2009 issue — Volume
56, Number 6 — of the Notices of the American Mathematical Society, there appeared a  
   beautiful article, by Friedrich E.P. Hirzebruch and Matthias Kreck, on the subject of genus.
It's entitled "On the Concept of Genus in Topology and Complex Analysis".)                     
                                                          

_{P                                               G                                              D
  {6,4|4}P                                      {6,4|4}G                                     {6,4|4}D
   sc lattice                                     bcc lattice                                   fcc lattice}

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   Each of the surfaces P, G, and D is shown here in a tiling pattern called {6,4|4} by Coxeter.
The prototile for each of these tilings is a specific variant of hex₉₀, a generic regular skew
6-gon
with 90º face angles. There are
4 faces
incident at each vertex and holes with
4-fold
symmetry.

The three variants of hex₉₀ for P, G, and D are called Phex₉₀, Ghex₉₀, and Dhex₉₀, respectively.

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In a 1970 NASA Technical Note Infinite Periodic Minimal Surfaces Without Self-Intersections (p.38 ff),
I described how skeletal graphs can be used to represent TPMS. More recently David Hoffman and Jim
Hoffman (no relation) have demonstrated in their Scientific Graphics Project that for the TPMS P, G, D,
and also for a fourth surface (I-WP) of genus 4, there is a striking connection between the skeletal graph
of the surface and a modified version of its level surface approximation. Perhaps similar matches will be
found for other examples of TPMS, including surfaces of genus greater than 4.                                          

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P
'P' stands for primitif, the name assigned long ago by German crystallographers to crystal structures  
that have the symmetry of a packing of congruent cubes. The P surface exhibits this symmetry. Each
  of the two congruent interpenetrating labyrinths into which space is partitioned by the P surface may  
be regarded as an inflated version of the skeletal graph with tubular edges that enclose the edges of a
packing of congruent cubes. The symbol for the space group of P (No. 229) is Im3′m. (Alan Mackay,
the British physicist and crystallographer, has wittily dubbed P 'the plumber's nightmare'.)                 

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G
'G' stands for gyroid, a name I chose to suggest the twisted character of its labyrinths, which — unlike
  the labyrinths of P and D — are opposite mirror images (enantiomorphs). The skeletal graphs of G are
dual Laves graphs. The symbol for the space group of G (No. 230) is Ia3′d.                                           

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D
   'D' stands for diamond. The congruent pair of skeletal graphs of its labyrinths are dual diamond graphs ,
whose edges correspond to the bonds between adjacent carbon atoms in diamond. The symbol for the
space group of D (No. 224) is Pn3′m.                                                                                                       

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  I believe now that 'reciprocal' might have been a more suitable name than 'dual' for the relation between
    the two triply-periodic graphs — e.g., simple cubic, diamond, Laves, and others — of an intertwined pair.  
(The two graphs in many of these pairs are non-congruent.) 'Reciprocal' would have reduced the risk of
confusion with the accepted meaning of 'dual' in the expression 'dual graph' which by convention refers
either to plane graphs or to the graphs of edges of the triply-periodic Coxeter-Petrie polyhedra. In what
follows I will continue to use the word 'dual' to refer to the vaguely defined but symmetrical relation      
between pairs of triply-periodic graphs like those described here.                                                                

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P and D were discovered and analyzed by H. A. Schwarz in 1865. He derived an
explicit Enneper-Weierstrass parameterization for the surfaces, which morph into
each other via the Bonnet bending transformation. (Another well known example
of Bonnet bending is the helicoid-catenoid transformation.)                                   

P and D contain both embedded straight lines and plane geodesics. A straight line
in either surface morphs into a plane geodesic (a mirror-symmetric plane line of  
  curvature) in the other surface. Because every straight line embedded in a minimal
  surface is an axis of 2-fold rotational symmetry, a half-turn about the line switches
the two sides of the surface and also switches the two inter-penetrating labyrinths
into which space is partitioned. A TPMS is called balanced if its labyrinths are     
congruent. P and D are both balanced surfaces.                                                        

  The Weierstrass integrals shown below define the rectangular coordinates of P and
   D and of countably many associate surfaces. θ is called the angle of associativity or
   Bonnet angle. For θ = 0, the equations describe D, and for θ = π ⁄ 2, they describe P.
D and P are called adjoint (or conjugate) surfaces.                                                    

_{Enneper-Weierstrass equations for the embedded minimal
surfaces D, G, and P, which are related by Bonnet bending}

In the spring of 1966, I had never heard of the Schwarz surfaces P and D. In fact I knew
next to nothing about any minimal surfaces! But then I met Peter Pearce (cf. §31), who  
showed me two plastic models of what he called saddle polyhedra. The faces of these     
objects were skew polygons spanned by minimal surfaces. I decided to find out to what  
extent I could use Peter's concept of saddle polyhedron to simplify a rule for defining a   
duality relation I had been struggling to impose on certain pairs of triply-periodic graphs.

Using a toy vacuum-forming machine, I made plastic replicas of several soap films that  
span skew polygons. By a stroke of luck, two of these polygons, which happened to be   
regular skew hexagons, turned out to be modules of Schwarz's P and D surfaces. I was   
startled when I realized that these two simple and elegant surfaces must surely be widely
known (even though I had never heard of them). From a literature search in the UCLA    
library I discovered that J.C.C. Nitsche was a renowned authority on minimal surfaces,   
so I telephoned him to ask him to identify these two intriguing surfaces. He replied that   
judging from my descriptions, they were almost certainly two famous minimal surfaces   
analyzed about one hundred years earlier by H.A. Schwarz. He said that the coordinates  
of these two surfaces are defined by three complex integrals derived in 1863 by Karl       
Weierstrass, who — I learned later — was one of Schwarz's teachers.                                 

During the next few weeks, after a rather superficial study of what Schwarz wrote about  
his two periodic surfaces, which I dubbed D and P, I became convinced that there must  
   also exist a minimal surface with the symmetry and topology of G — the gyroid, but for   
almost two years I had no idea how to construct it. I knew that if it did exist, the skeletal  
  graphs ('backbones') of its two labyrinths must be an enantiomorphic pair of Laves graphs.

  In September, 1968, during a telephone conversation with Blaine Lawson about the gyroid
  problem, it at last became clear to me that the gyroid is simply a hybrid (cf. Eq. 1 in §5) of
  the two Schwarz surfaces D and P that happens to be embedded. (In §51, I summarize this
  conversation with Blaine.) The minimal surfaces D, G, and P are all described by Enneper-
Weierstrass equations, and the coordinates of any point in a lattice fundamental domain of
G are a linear combination of the coordinates of the corresponding points in D and P. The
Bonnet angle θ_G for the gyroid is given by the equation                                                       
θ_G = ctn^-1(K′ ⁄ K)
≅ 38.014773989108068108º,
where
K = K(1/4) ≅ 1.6857503548125960429,
K′ = K(3/4) ≅ 2.1565156474996432354.

   K is a complete elliptic integral of the first kind with parameter m=1/4; K′ is its complement.

  The images of Dhex₉₀ and Phex₉₀ shown directly below in §5 demonstrate the transformation
    under Bonnet bending of straight lines in each surface into plane geodesics in the other surface.
  In contrast to D and P, G contains neither straight lines nor plane geodesics. Its labyrinths are
        enantiomorphic (oppositely congruent). G can be regarded as an example of a balanced surface.  

Further details about how I discovered G are described in §3, §5, §7, and §51.

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§5. How I derived the value of θ_G, the Bonnet angle for the gyroid

_{An example of a lattice fundamental domain of the gyroid,
composed of eight differently oriented congruent hexagonal faces Ghex90}

_{The boundary curves for the Dhex90, Phex90, and Ghex90 faces of D (blue), P (red), and G (violet)
The trajectory of every point on each surface under Bonnet bending is an ellipse centered at O.
The trajectory for the six vertices of each face are shown as a black ellipse.}

_{Stereo view of the outline of the surface patch Dhex90 (blue) and its adjoint image Phex90 (red)
The [red] curve P1 P2 is the adjoint image under Bonnet bending of the [blue] curve D1 D2.      
λP and λD are the edge lengths of the circumscribing blue and red cubes, respectively.               
The ratio λP ⁄ λD is equal to K(3/4)/K(1/4) ≅ 1.2792615711710064662.}                                        

_{In 1866 Herman Amandus Schwarz proved in his pioneering analysis of D and P that the areas
     of Dhex90 and Phex90 are equal if the ratio λP ⁄ λD of the edge lengths of the circumscribing cubes
is equal to K(3/4)/K(1/4). This result of Schwarz — combined with the fact that one of the       
three rectangular coordinates of each vertex of every hexagonal face of a manifold I call M6    
  (see below) is equal to zero — provides just enough information to compute the value of θG,    
  the Bonnet angle for the gyroid, as I demonstrate in the proof immediately below.}                  

A lattice fundamental domain of D or P can be tiled by eight differently oriented
replicas of one hexagonal surface patch. Outlines of two such replicas are shown
above in blue (Dhex₉₀) and red (Phex₉₀). Dhex₉₀ is inscribed in a cube of edge   
length λ_D and Phex₉₀ is inscribed in a cube of edge length λ_P. In 1866, Hermann
   Schwarz proved that in order for Phex₉₀ and Dhex₉₀ to have the same area (which
   is required for Bonnet bending of either surface into the other), it is necessary that
   λ_P ⁄ λ_D = K(3/4)/K(1/4) ≅ 1.2792615711710064662. For both D and P, the normal
      vectors at the centers of the eight hexagonal patches in a lattice fundamental domain
     are all directed toward different corners of the circumscribing cube. For the patches
  Dhex₉₀ and Phex₉₀ shown above, the normal vectors point in the (1,1,1) direction
           (x is positive in the direction toward you, y is positive to the right, and z is positive up).

If we fix point O at the center of Dhex₉₀, then as Dhex₉₀ morphs into Phex₉₀ by
   Bonnet bending, the point D₁ on Dhex₉₀ moves along an elliptical trajectory r₁(θ)
with center at O. The equation for this ellipse is                                                     

                                       r₁(θ) = d₁ cos θ + p₁ sin θ                                      (1)

    The orthogonal vectors d₁ and p₁ are directed outward from the center of the ellipse
along its semi-minor and semi-major axes, respectively.                                         

If θ = π/2, the images under bending of d₁ and d₂ are p₁ and p₂, respectively.       

Let us fix the scale by setting λ_D = 2. Then                                                                         d₁  = (− 1,1,1)                  
|d₁| = √3                  
                        p₁  = (λ_P ⁄ λ_D) (0,− 1,1)                 
                                              |p₁| = √2 (λ_P ⁄ λ_D).                                             (2)

As stated above, H.A.Schwarz proved (cf. his Collected Papers, vol. I, p. 88) that

                                    λ_P ⁄ λ_D = K′(1/4) ⁄ K(1/4) (≅ 1.2792615711710064662).     (3)         

If we substitute for d₁ and p₁ in Eq. 1 from Eqs. 2 and 3, we obtain                     

                                    r₁(θ) = (−1,1,1) cos θ + (K′ ⁄ K) (0,−1,1) sin θ             (4)

            If we define θ = θ_G for the gyroid, then Eq. 4 becomes                                                      

                                   r₁(θ_G) = (−1,1,1) cos θ_G + (K′ ⁄ K) (0,−1,1) sin θ_G        (5)

The key to solving Equation 5 for θ_G is found in the geometrical properties of the
manifold M₆, an infinite regular skew polyhedron. whose faces are regular skew  
hexagons. Each of these hexagons can be inscribed in a truncated octahedron.      
   Below is an illustration of one hexagonal face of M₆, with its central normal vector
   (not shown) oriented in the (1,1,1) direction, followed by an illustration of a lattice
fundamental region composed of eight differently oriented replicas of this face.    

   _{The edges of a skew hexagonal face of M6 with central normal vector in the (1,1,1) direction
     This face of M6 is inscribed in a truncated octahedron, a space-filling polyhedron.
x is positive toward the observer, y is positive to the right, and z is positive up.
The coordinates of the six vertices of M6, in CCW order starting from G1, are  
              (−1,0,2), (0,−2,1), (2,−1,0), (1,0,−2), (0,2,−1), and (−2,1,0), respectively.                       
These six vertices must coincide with the six vertices of an identically         
oriented and appropriately scaled hexagonal face Ghex90 of the gyroid.}     

_{A lattice fundamental domain of M6, composed
of eight differently oriented skew hexagonal faces}

_{Each face is inscribed in one cell of an assembly of eight truncated octahedra.
             The blue arrow at the center of each face indicates the direction of the surface normal.
              Note that the two hexagons in each of the four identically colored pairs are oppositely
  oriented. Click here for an image that shows only the edges of the hexagons.}  

_{The boundary curves for the Dhex90, Phex90, and Ghex90 faces of D (blue), P (red), and G (violet)
The trajectory of every point on the surface under Bonnet bending is an
ellipse. The trajectories for the six vertices of each face are shown here.}

Now suppose that each edge of M₆ is replaced by a quarter-pitch of a helical arc,
as depicted in the image above. The table below lists the direction of the axis of  
the helix associated with each arc and also the sense (CW vs. CCW) of each arc.  

        arc   direction     sense   

G₁G₂      y             CW
    G₂G₃      x              CCW
G₃G₄      z             CW
    G₄G₅      y              CCW
G₅G₆      x             CW
    G₆G₁     z               CCW

Let us denote by Ghex₉₀ the minimal surface that spans this modified hexagon
     with helical edges G₁G₂, G₂G₃, ..., G₆G₁, because — just as for the hexagonal    
   faces Dhex₉₀ and Phex₉₀ of D and P— its face angles are all equal to 90º. Since
  the directions of the tangents to the pairs of edge curves that intersect at D₁, P₁,
and G₁ are identical, the surface orientation is the same at these points.            

            Vector equation Eq. 5 (above) is equivalent to the following three scalar equations, one
for each of the three components of the vector r₁(θ_G):                                         

                            x₁(θ_G) = −cos θ_G + (0) (K′ ⁄ K) sin θ_G                                 (5')   

                                      y₁(θ_G) = cos θ_G − (K′ ⁄ K) sin θ_G                                         (5'')           

                            z₁(θ_G) = cos θ_G + (K′ ⁄ K) sin θ_G                                         (5''')

But                   r₁(θ_G) = g₁                                                                                 
                                   = (−1,0,2).                                                                   (6)

Hence                y₁(θ_G) = 0,                                                                            (7)

If now we substitute for y₁(θ_G) from Eq. 7 in Eq. 5'', we obtain                       

                                    cos θ_G = (K′ ⁄ K) sin θ_G .                                                     (8)        

Therefore                  θ_G = ctn^-1(K′ ⁄ K).                                                          (9)

         Because of the 6-fold rotatory reflection symmetry of Dhex₉₀, Phex₉₀, and Ghex₉₀,
    Eq. 9 could just as easily have been derived by considering any of the other five
       vertices of Ghex₉₀ instead of the vertex G₁.                                                              

---

_{Models of the infinite regular skew polyhedra M4 (left) and M6 (right), precursors of the gyroid}

M₄ resulted from the execution of an intermediate step of an empirically developed procedure    
          aimed at deriving duals of triply-periodic graphs. The hexagons of M₆ are the conventional duals        
  of the quadrilaterals of M₄: every vertex of M₆ lies at the center of a face of M₄, and every vertex
of M₄ lies at the center of a face of M₆. The viewpoints are both in the (1,1,1) direction.              

---

The fortuitous failure of my dual graph recipe

       On February 14, 1968, I constructed a model (cf. photo at above left) of the infinite regular
         skew polyhedron I call M₄. It appeared at the penultimate stage of a recipe (that I was still   
        testing) for what I called the dual of a triply-periodic graph. The graph I was testing on that
       day is a triply-periodic symmetric graph of degree six that I called BCC₆, because it can be
              transformed into BCC₈, the ordinary bcc graph of degree eight, by adding two additional edges
at each vertex. I call such a symmetric graph deficient, because it remains symmetric   
         even when the number of edges incident at each vertex is increased. I had expected M₄ to be
                 finite, because this part of my recipe had never yet failed to yield one or more finite polyhedra.    

   I had been trying to understand whether it is possible to define precisely those classes of
   triply-periodic graphs for which my recipe would produce a unique pair of sets of saddle
polyhedra (cf. §27). I required that one of these two sets be composed of interstitial     
   polyhedra that fill the cavities of the graph. The other set was supposed to contain what I
called nodal polyhedra (cf. §28) enclosing the graph's vertices. I did not rule out the     
possibility that some or all of the faces of these saddle polyhedra might be flat.             

   My immediate reaction to M₄ was one of surprise to see that it was not finite! But I also
   realized how naive I had been to imagine that anything as jerry-rigged as this dual graph
        recipe would always perform the way I had expected. I was actually enormously relieved to
  encounter this failure! For a year and a half, I had been testing lots of graphs in a search
for a 'counter-example', and I had finally found one. At last I could stop searching.       

All of these thoughts were swept aside as soon as I examined M₄ carefully, because I   
      discovered that the skeletal graphs of its two enantiomorphic labyrinths were dual Laves  
        graphs. With mounting excitement I realized that I had at last found an object with the        
    same global combinatorial structure and symmetry as the elusive 'Laves periodic minimal
surface' I had hunted without success in the summer of 1966.                                          

           I still had no idea how to transform M₄ into a single minimal surface, but I judged by eye that
            M₆, its dual — a tiling with four skew hexagons incident at each vertex — would be somewhat
           less bumpy than M₄. In any tiling of a minimal surface by replicas of a straight-edged polygon
            spanned by a minimal surface (like the tiling of Schwarz's D surface by regular skew hexagons
               with 90º face angles, or the tiling of Schwarz's P surface by regular skew hexagons with 60º face
           angles), the two tiles of every adjacent pair are related by a half-turn about their common edge
                         (Schwarz's reflection principle). If adjacent tiles were instead related by rotation through an angle (let's
               call it θ_dihedral) different from π, there would be a kind of bump between them — a difference in
     the orientation of the tangent planes on the two sides of their common edge. A calculation
              confirmed that the M₆ bump is smaller than the M₄ bump. In M₄, θ_dihedral = cos⁻¹(−1/2) = 120º,
                     implying a bump equal to 180º − 120º = 60º, but in M₆, θ_dihedral = cos⁻¹(−5/7) ≅ 135.585º, implying
              a smaller bump of approximately 180º − 135.585º = 44.415º. (The face angles in M₄ and M₆ are
equal to cos⁻¹(1/3) ≅ 70.529º and cos⁻¹(−1/6) ≅ 99.594º, respectively.)                         

                I decided that a reduction in bumpiness of ~15.6º (60º − 44.415º) was large enough to justify
making a vacuum-forming tool for the skew hexagons of M₆.                                      

                  Three days later, I finished assembling the model of M₆ shown at the upper right. M₆ definitely     
looked somewhat smoother than M₄. More significantly, by this time I had come to     
    realize that M₄ and M₆ are described by Coxeter's dual regular maps {4,6|4} and {6,4|4}.
        This suggested a strong connection to Schwarz's D and P surfaces and to the infinite regular
  skew polyhedra of Coxeter and Petrie, but I wasn't yet sure what to make of all this.       

           After long exposure to examples of triply-periodic graphs on cubic lattices, the only examples
         of a symmetric graph on a cubic lattice I had succeeded in identifying were the simple cubic,
   diamond, and Laves graphs. The simple cubic graph is the skeletal graph of Schwarz's P
       surface, and the diamond graph is the skeletal graph of Schwarz's D surface. I couldn't help
     wondering whether the Laves graph is the skeletal graph of a third minimal surface. But I
            couldn't imagine using the concept of skeletal graph to prove the existence of such a surface!  

          Encountering this breakdown in my dual graph recipe had induced me to change direction. I
          decided to stop trying to construct a rigorous definition of the dual of a triply-periodic graph.
     M₄ and M₆ had forced me at last to recognize the futility of trying to transform a bunch of
    empirical relations between graphs into a comprehensive scheme with genuine predictive
      value. I concluded that the notion of duality for triply-periodic graphs would never lead to
     consistent results unless I could somehow tighten the definition of the classes of graphs to
             which it is applied. I summarized my struggle with dual graphs in my 1970 NASA Technical   
Note, Infinite Periodic Minimal Surfaces Without Self-Intersections.                             

         Summarizing: the reason I suddenly lost interest in trying to formalize the concept of duality
             for triply-periodic graphs is that I had noticed that the skeletal graphs of the two intertwined
                              labyrinths of M₄ are enantiomorpic Laves graphs. M₄ and M₆ both have the same topology and global
    symmetry (space group Ia3′d, No. 230) as the gyroid, which I had tried unsuccessfully to
construct in the summer of 1966. The geometry of M₆ strongly suggested to me the     
possibility of somehow constructing the gyroid out of hexagonal faces whose vertices
          coincide with those of M₆. Each face of M₆, like the hexagonal faces of D and P, is oriented
    in one of the eight (±1,±1,±1) directions. The y-coordinates y₁(θ_G) and y₄(θ_G) of vertices
     G₁ and G₄ (see images above), which are related by inversion in the center of a hexagonal
      face, are both equal to zero. For each of the other seven orientations of the hexagonal face,
    it is likewise true that one of the three rectangular components of the vectors that define  
the positions of a pair of vertices of M₆ related by inversion is equal to zero.                  

        My model of M₆ demonstrated that the straight edges of M₆ define infinite regular helical
polygons, which are centered on lines parallel to the rectangular coordinate axes. This
         suggested to me that if I were to replace the straight edges of M₆ by curved helical arcs, the
'bump' along the edges might shrink almost to zero. This was a wild and woolly guess,
      with absolutely no theoretical justification. I knew that there are perfect helices embedded
      in the helicoid, but I also knew that no solution was known for a minimal surface bounded
    by six helical arcs. (This whole idea actually turned out to be something of a red herring,
  and it threw me off the trail of the gyroid! Although it is now known that the difference
between cylindrical helices and the spiralling geodesics in the gyroid surface that are   
      centered on lines parallel to the three coordinate axes of the surface is quite small, it is not
zero. The image in §9 shows the difference.)                                                                    

              I later dubbed the object that is tiled by skew hexagons with strictly helical edges the 'pseudo-  
gyroid'. It is not a triply-perodic minimal surface. Even if all of its separate faces were
      minimal surfaces, they would not define a surface that is smoothly continuous across the  
boundary between adjacent faces. (Incidentally, it is still not known how to derive an  
  analytic expression for a simply-connected minimal surface spanning a 'Schwarz chain'
composed of consecutive helical arcs.)                                                                             

---

Below are images of the Voronoi polyhedron of a vertex of M₆.

_{Photos of 1968 models of the space-filling Voronoi polyhedron of a vertex
    of the graph {6,4|4}, which is composed of the edges of M6 (cf. photo above)
    The Voronoi polyhedron of a vertex of the graph {4,6|4}, the dual of {6,4|4},
is the truncated octahedron, which has full cubic symmetry.}                    

---

§6. New models of M₄ and M₆ constructed in 2011

_{30 skew quadrangles of M4 (stereo)
view: [111] direction}

_{30 skew quadrangles of M4 (stereo)
view: [110] direction}

_{30 skew quadrangles of M4 (stereo)
view: [110] direction}

_{30 skew quadrangles of M4
view: [111] direction, backlit by summer sky}

_{30 skew quadrangles of M4
view: [100] direction, backlit by summer sky}

_{The skew hexagons of M6
Viewpoint is in the (1,0,0) direction.}

_{Orthogonal projection onto [1,0,0] plane
of the edges of the skew hexagons of M6
The four small squares are the outlines of helical tunnels.}

---

§7. The relation between M₆ and the gyroid

In March 1968 I played my hunch about helical edges by sending a purchase order to a   
machine shop for a custom-designed brass tool for vacuum-forming plastic modules of   
hexagons with edges in the shape of helical arcs of alternating handedness. Below is a     
photo of the model of the pseudo-gyroid that resulted. It did look exactly like a minimal  
surface, but I had no idea how to derive equations for it. I sent a smaller version of this    
             model to Bob Osserman, a distinguished authority on minimal surface theory. Bob then               
suggested to Blaine Lawson, his talented PhD student, that he look into this problem (cf.
  §5 and §51). I introduced myself by telephone to Blaine, who explained that he was in the
throes of writing the final part of his PhD dissertation and couldn't predict when he         
would be able to start thinking about the gyroid problem. But I was relieved that my        
puzzle was at last in expert hands.                                                                                        

_{The pseudo-gyroid (1968)}

By early summer one of the progress reports I was required to write about my research   
had apparently disturbed somebody in NASA Headquarters. I was informed that some   
officials there were concerned about my 'playing with soap bubbles'. I also learned that  
Headquarters was thinking about having me transferred to a project more closely related
to NASA's mission (the mechanical support structure of the Hubble Space Telescope     
  was mentioned as an example of such a project). This news induced in me a state of mild
panic, and I abruptly switched my attention to a problem concerning collapsing graphs,
thinking (not very rationally) that I might be able to stave off threatening catastrophe by
  demonstrating how such graphs could be applied to the design of spaceframes that could
be stored compactly in a collapsed state and later deployed in an expanded state.           

---

§8. A long summer distraction

During the summer of 1968, as I waited to hear from either Bob or Blaine, I analyzed  
the collapse kinematics of several triply-periodic graphs, while trying to put the gyroid
problem aside, still convinced that it was best left to experts. But as luck would have it,
the analysis of these collapsing graphs required that I consider in detail the orientation  
of the surface normals on P, G, and D, and I was confronted daily by the most obvious  
imaginable clues to the fundamental relation (the Bonnet bending transformation) that   
connects these three surfaces. And yet I didn't recognize the clues! ('... eyes have they,
but they see not.')                                                                                                                 

---

§9. The cylindrical helix and the flattened cylindrical helix

_{Orthogonal projections of a spiralling geodesic curve on the
gyroid (black) and a cylindrical helix for comparison (red)
One-quarter of a single pitch is shown for each curve.}

_{A complete pitch (one period) of the geodesic is centered on a
line parallel to one of the coordinate axes. Both curves above
are shown in orthogonal projection on a coordinate plane. The
spiralling geodesic defines the shape of an edge of the regular
skew hexagons that are the faces of the {6,4|4}G tiling of G.}

---

§10. The machined parts of the brass vacuum-forming tool

_{The components of the brass mould, shown before brazing and final
machining, that I designed for vacuum-forming the plastic hexagons
of the 1968 model of the pseudo-gyroid.}

_{Two years earlier, I had discovered an arrangement of two sets of the
eight solid tetrominoes in enantiomorphic trigonal packings of a half-
cube, shown below. These arrangements of the tetrominoes pack the
cube. The shape of this partition of the cube suggested the design of
the tool parts shown in this photo.}

_{Packing of each of two oppositely congruent half-
cube triskelia by the set of eight solid tetrominoes}

            _{(In 1966 I naively imagined that the packing of the eight tetrominoes shown in
              this sketch might be unique. But in 1013, no sooner did I ask my friend George
Bell to investigate than he replied (same day!) that there are actually 36
solutions. On 2/18/2013, he emailed me the solution shown below.)}       

      _{If I hadn't discovered this puzzle set of tetrominoes and shared them with
    my friend and colleague Bill Van Atta, it is highly unlikely that I would
have met his father, Lester C. Van Atta, who found the set fascinating
and told Bill he wanted to meet me. We met for several hours, and the
the invitation for me to join NASA/ERC was the result!}                       

_{One of George Bell's 36 solutions for the packing
of the triskelion by eight solid tetrominoes}

My addiction to recreational mathematics, which worsened considerably
once I started playing with solid tetrominoes in 1965, was not helped by
exposure to the writings of Martin Gardner and Solomon Golomb. I am
extremely grateful to both of them.

Martin Gardner
10 October 1914 — 5 May 2010

Solomon Golomb
30 May 1932 — 1 May 2016

---

§11. Symmetrical lattice fundamental domains of the gyroid

The hexagonal faces of the gyroid Coxeter-Petrie map {6,4|4} can be joined
in a connected assembly in a variety of ways, each of which defines a lattice
fundamental domain (lfd) of the surface. Below are several examples of lfds
that are arranged to illustrate one or more of the symmetries of the surface.  

_{Front view of the lattice fundamental domain lfd1(G) of the
gyroid, composed of eight congruent skew hexagonal faces
stereo image}

_{The lattice is bcc (body-centered cubic).
There is a c2 axis (half-turn symmetry axis)
along the (1,0,0) line of sight through the point of
intersection of the two hexagonal faces in front.}

_{high resolution image}

_{Side view of lfd1(G)
stereo image}

_{There is a counter-clockwise 4-fold screw axis
along the (1,0,0) axis of this approximately
cylindrical open tunnel of lfd1(G).
Tunnels on opposite sides of the
surface are of opposite handedness.}

_{high resolution image}

---

_{Here the eight faces of lfd1(G) have been rearranged
to form lfd2(G), which was designed to emphasize
that through every vertex of a hexagonal face there
is an axis of 4-fold rotation-reflection symmetry.
It is a centered vertical line in this view.
Front view}

_{high resolution image}

_{Side view of lfd2(G)
high resolution image}

_{Two connected replicas of lfd2(G)
Front view}

_{high resolution image}

_{Top view of lfd2(G)
high resolution image}

_{Bottom view of lfd2(G)
high resolution image}

_{Symmetrical assembly of fifteen replicas of lfd2(G):
one centered in a cube,
one centered at each corner of that cube,
and six centered in the nearest outlying cubes.
Front view}

_{high resolution image}

_{The 24 vertices on the boundary of lfd2(G)
Pairs of vertices of a given color that are connected by a line segment
of the same color are related by a translation symmetry of the surface.
Front view}

_{high resolution image}

_{The 24 vertices on the boundary of lfd2(G)
Pairs of vertices of a given color that are connected by a line segment
of the same color are related by a translation symmetry of the surface.
Top view}

_{high resolution image}

_{Orthogonal projections of the graph of links among
the 24 colored vertices on the boundary of lfd2(G)
The projection at the left is onto the y-z plane;
the projection at the right is onto the x-y plane.}

_{high resolution image}

_{The graph of links among the 24 colored vertices
on the boundary of lfd2(G)
Front view}

_{high resolution image}

_{Orthogonal projection onto the y-z plane of the
graph of links connecting the 24 colored vertices
on the boundary of lfd2(G)
The black vertex at the center of the image lies in the
interior of lfd2(G) and is not a vertex of the graph.
Front view}

_{high resolution image}

_{Orthogonal projection onto the y-z plane of the
graph of links connecting the 24 colored vertices
on the boundary of lfd2(G)
Top view}

_{high resolution image}

---

_{Here eight faces of the gyroid have been arranged
to form lfd3(G), a third example of a symmetrical
lattice fundamental domain, vividly illustrating
the 3-fold rotational symmetry of the gyroid.
View along (100) cube axis}

_{high resolution image}

_{A second, less cluttered, view of lfd3(G)
View along (100) cube axis}

_{high resolution image}

_{A view of lfd3(G) along a c2 axis
along (110) axis}

_{high resolution image}

_{A view of lfd3(G) from the opposite end
of the c2 axis}

_{high resolution image}

_{A view of lfd3(G) along an axis of
3-fold rotation symmetry
in the (111) direction}

_{high resolution image}

_{Orthogonal projection of lfd3(G) onto a [111] plane
View along (111) direction}

_{high resolution image}

---

§12. Polyhedral surrogates of TPMS

It is interesting to explore the relation between TPMS — especially those
of low genus, like D, P, and G — and simpler structures that we will call
polyhedral surrogates with plane faces (ps). Each of these surrogates is a
plane-faced triply-periodic polyhedron that is homeomorphic to a TPMS.
The most symmetrical examples have the same symmetry as the TPMS.  
Both the TPMS and its surrogate can be represented by the same pair of  
inter-penetrating skeletal graphs. Each edge in such a graph joins a pair  
of vertices that lie at centers of symmetry of the TPMS and its surrogate.

The regular skew polyhedron {4,6|4} of Coxeter and Petrie is an example
of a ps of a TPMS. Since it corresponds to Schwarz's P surface, we call it
{4,6|4}_P. This is what it looks like:

_{{4,6|4}P
A regular ps of Schwarz's P surface}

A second example of a regular ps of P is {6,4|4}_P:

_{{6,4|4}P
Another regular ps of Schwarz's P surface}

An example of a regular ps of Schwarz's D surface
is the regular skew polyhedron {6,6|3}_D:

_{{6,6|3}D
A regular ps of Schwarz's D surface}

Another ps of Schwarz's D surface is the quasi-regular skew polyhedron
(6.4)² (below). Quasi-regular polyhedra are both edge-transitive and vertex-
transitive, but they are not face-transitive.

_{(6.4)²D
A quasi-regular ps of Schwarz's D surface
The red and green skeletal graphs are both replicas of the diamond graph —
the graph of edges connecting the sites of adjacent carbon atoms in diamond.}

For the gyroid G, there exist no examples of either regular or quasi-regular ps,     
but there is an infinite uniform skew polyhedron that is a ps of G. It is a kind of   
snub polyhedron, but unlike the twelve examples of finite snub polyhedra, which
  are chiral and exist in two enantiomorphic forms, it — like the gyroid — has only
a single form. Its two labyrinths are enantiomorphic. It is vertex-transitive, but it
is neither edge-transitive nor face-transitive. It is called (6.3².4.3)_G. Two images
are shown below.

_{(6.3².4.3)G
A uniform ps of the gyroid G
The faces of (6.3².4.3)G are regular plane polygons,
but its symmetry group is transitive only on vertices.
A single cubic unit cell is shown here.}

_{high resolution image}

_{(6.3².4.3)G,
including its skeletal graphs}

(6.3².4.3)_G was discovered by Norman Johnson in 1969, when he derived
the combinatorial type of every possible example of a 'uniform tessellation
of the {4,6} family'. He proved that the enumeration is complete. Norman
Johnson is well known for his 1966 enumeration of the 92 Johnson solids,
later proved exhaustive by Victor Zalgaller. He is currently writing a book
about uniform polytopes in R³ and R⁴.                                                           

          The (6.3².4.3)_G polyhedron was independently rediscovered several years ago by
        John Horton Conway, who named it mu-snub cube. ('mu' means multiple here.)

---

   The Voronoi polyhedron of a vertex of the union of the two dual skeletal
       graphs of Schwarz's D — or of (6.4)², its ps — is the truncated octahedron.
The boundary of each chamber in (6.4)² is a truncated octahedron from
which four of the eight hexagons have been removed.                            

_{(6.4)²
In both labyrinths there are
two differently oriented chambers.}

        The four faces incident at each vertex of (6.4)² are arranged cyclically: 6^.4^.6^.4.
          Hence the name (6.4)². There are two differently oriented varieties of chambers
              in each labyrinth. Chambers that share an open hexagonal face belong to the same
                labyrinth and are oppositely oriented. Chambers that are separated by a square face
belong to different labyrinths and are identically oriented.                         

The triply-periodic Coxeter-Petrie regular skew polyhedron {6,6|3}, which — like (6.4)2 — has the same topology and symmetry as Schwarz's diamond surface D. Here it is shown with just one of its two skeletal graphs. §13. Bibliography For an introduction to the mathematics of minimal surfaces, I enthusiastically recommend The Parsimonious Universe: Shape and Form in the Natural World, by Stefan Hildebrandt and Anthony Tromba. In a recent (August 2015) installment of Azimuth, John Baez's lively blog, there is a ton of information about the relation between the gyroid and photonic crystals. This installment is entitled "The Physics of Butterfly Wings". John also discusses the relation between the gyroid and graphene, which — like the Laves graph — is 3-connected. Included inter alia in his wide-ranging account of these matters are such exotic concepts as Dirac cones, Weyl points, Chern numbers, and Berry flux! (Note to Japanese readers: There's a link at the very bottom of the installment to a Japanese translation of the first section of the blog.) Below is a partial list of on-line sources — not limited to periodic minimal surfaces: mathematics, images, videos, applications in physics, chemistry, biology, engineering, etc. (Not all of these links are listed in alphabetic order of the last name of the author — or authors.) Marin Soljacic, Ling Lu et al Weyl points, magnetic monopoles, and photonic crystals Experimental discovery of Weyl points at MIT, using machine-fabricated gyroid (cf. Optical properties of gyroid structured materials: from photonic crystals to metamaterials) Ken Brakke: Triply Periodic Minimal Surfaces Comprehensive collection of illustrations of surfaces derived with Ken's Surface Evolver software Ken Brakke: 2005 translation (by Brakke) of J. Plateau's monumental 1873 treatise: Experimental and Theoretical Statics of Liquids Subject to Molecular Forces Only This is Plateau's account of his pioneering investigations of capillarity phenomena. It includes a description (pp. 103-107) of H.A. Schwarz's then almost brand-new reflection principle, which defines the role of rotation and reflection symmetries in the analytic continuation of finite pieces of many examples of periodic minimal surfaces. The problem of proving that there exists a minimal surface spanning a given boundary is called Plateau's problem, because it was Plateau who called special attention to the central importance of such a proof. James A. Dolan, Bodo D. Wilts, Silvia Vignolini, Jeremy J. Baumberg, Ullrich Steiner, and Timothy D. Wilkinson: Optical properties of gyroid structured materials: from photonic crystals to metamaterials Tomonari Dotera, Masakiyo Kimoto, and Junichi Matsuzawa Hard Spheres on the Gyroid Surface EPINET: Triply Periodic Minimal Surfaces An encyclopedia of triply-periodic minimal surfaces that includes a description of related periodic graphs Steven Finch: Soap film Experiments A collection of both analytic and numerical computations of the surface area of examples of minimal surfaces — some known to Schwarz and others newly introduced here Steven Finch: Gergonne-Schwarz Surface TPMS, elliptic integrals, and elliptic functions Steven Finch: Mathematical constants and functions Although it is not relevant here, this collection is too fascinating to be left out! Shoichi Fujimori and Matthias Weber: A Construction Method for Triply Periodic Minimal Surfaces Paul J. F. Gandy, Sonny Bardhan, Alan L. Mackay, and Jacek Klinowski: Nodal surface approximations to the P, G, D, and I-WP triply periodic minmal surfaces, Darren Garbus: Isoperimetric Properties of Some Genus 3 Triply Periodic Minimal Surfaces Embedded in Euclidean Space, M.S. Thesis, May 2010 Chaim Goodman-Strauss and John Sullivan: Cubic Polyhedra Wojciech Gòzdz and Robert Holyst: High Genus Periodic Gyroid Surfaces of Non-Positive Gaussian Curvature David Hoffman and Jim Hoffman: Geometry: Minimal Surfaces David Hoffman and Jim Hoffman: The Lidinoid Surface Stephen Hyde, Christophe Oguey, and Stuart Ramsden: Triply connected graph embeddings Stephen T. Hyde, Michael O'Keeffe, and Davide M. Proserpio: A Short History of an Elusive yet Ubiquitous Structure in Chemistry, Materials and Mathematics Hermann Karcher and Konrad Polthier: Touching Soap Films, An Introduction to Minimal Surfaces Hermann Karcher: The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions, Manuscripta Math. 64, 291-357 (1989) Hermann Karcher and Konrad Polthier: Exhibition of Historical Minimal Surfaces (video) Hermann Karcher and Konrad Polthier: list of references to materials that are not online. Jacek Klinowski: Periodic Minimal Surfaces Gallery Elke Koch and Werner Fischer 3-periodic minimal surfaces at the '[Details]' hyperlink in their Mathematical Crystallography Elke Koch and Werner Fischer: 3-periodic surfaces without self-intersections Xah Lee: Gallery of Famous Surfaces Eric Lord and Alan Mackay: Slide Show of Triply Periodic Surfaces Eric Lord and Alan Mackay: Periodic minimal surfaces of cubic symmetry William H. Meeks III: Introduction to Minimal Surfaces Isabel Hubbard, Egon Schulte, and Asia Ivic Weiss: Petrie-Coxeter Maps Revisited Alan H. Schoen: Infinite Periodic Minimal Surfaces Without Self-Intersections, NASA TN D-5541 (May 1970) Alan H. Schoen Reflections Concerning Triply-Periodic Minimal Surfaces A summary of my minimal surface research, published in October 2012 in Interface Focus, a journal of The Royal Society Alan H. Schoen Embedded Triply-Periodic Minimal Surfaces Mathematical Sciences Research Institute Publication Springer-Verlag New York, Inc., 1991. pp. 147-158 Abstract: Some aspects of the modelling of embedded triply-periodic minimal surfaces (ETPMS) by both soap films and plastic models are discussed. Eight new examples of possible ETPMS are introduced. Interference-diffraction patterns obtained by reflecting laser light from soap films with non-zero Gaussian curvature are decribed. Gerd Schröder-Turk: The bicontinuous fish tank Gerd Schröder-Turk: Bonnet transformation between the D, G and P minimal surfaces (video) Gerd Schröder-Turk, Stuart Ramsden, Andrew Christy, and Stephen Hyde: Medial Surfaces of Hyperbolic Structures Gerd Schröder-Turk, Andrew Fogden, and Stephen Hyde: Bicontinuous geometries and molecular self-assembly: comparison of local curvature and global packing variations in genus three cubic, tetragonal, and rhombohedral surfaces B. Schuetrumpf, M. A. Klatt, K. Iida, G. E. Schröder-Turk, J. A. Maruhn, K. Mecke, P.-G. Reinhard: Appearance of the Single Gyroid Network Phase in Nuclear Pasta Matter Martin Steffens and Christian Teitzel: Grape Minimal Surface Library Toshikazu Sunada: Crystals That Nature Might Miss Creating Matthias Weber: Gallery of Minimal Surfaces Matthias Weber: Bloomington's Virtual Minimal Surface Museum Margaret and Christine Wertheim: The Institute for Figuring (IFF) An organization dedicated to the exploration of the relation between mathematical forms and natural objects, with particular emphasis on shapes defined by hyperbolic geometry Adam Weyhaupt: New Families of Embedded Triply Periodic Minmal Surfaces of Genus Three in Euclidean Space Adam Weyhaupt: Meet the Gyroid Adam Weyhaupt: Deformations of the gyroid and Lidinoid minimal surfaces 3D-XplorMath-J Applets Wikipedia: Calculus of Variations Wikipedia: Gyroid

(I plan to post additional links here from time to time, but this
list will never be more than a tiny sample of what's out there.)

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§14. Regular warped polyhedra based on the gyroid

_{Abstract 658-30, submitted in summer 1968
to the American Mathematical Society for the
Madison meeting in August}

_{A summary of my enumeration of the
quasi-regular tessellations of the {6,4} family
(abstract published by the American Mathematical Society in 1969)}

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_{(6.4)²
front view
high resolution image (64-mesh quad-grid)
high resolution image (1024-mesh quad-grid)}

_{(6.4)²
top view
high resolution image (64-mesh quad-grid)
high resolution image (1024-mesh quad-grid)}

  (6.4)² — alternatively written (6.4.6.4)_G — is a triply-periodic
quasiregular warped polyhedron composed of regular convex
hexagons and regular skew quadrilaterals that are spanned by
    minimal surfaces. (The minimal surfaces are approximated here
    by hyperbolic paraboloids.) (6.4.6.4)_G has the same topological
    structure and symmetry as the gyroid. We adopt the convention
    that a triply-periodic polyhedron is called warped if at least one
      of its face species is a skew polygon. The 4 in (6.4)² is written in
   bold face italics to indicate that its quadrilateral faces are skew.

The face angle of the skew quadrilaterals in (6.4)² is cos^-1(2/7)
≅ 73.3985°. The tilt angle (the angle between an edge and the
orthogonal projection of the edge onto the equatorial plane) is
cos^-1√(5/7) ≅ 32.3115°.

A quasiregular warped polyhedron that is a sort of inverse to
(6.4)² is (6.4)², in which the regular hexagons are skew, with
90° face angles, and the regular quadrilaterals are squares. In
the continuous family (6.4)², the only examples in which one
   of the two face species is plane and the other is skew are (6.4)²
and (6.4)². In the future I will post images of some examples.

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§15. Regular tilings of P, D, and G

P                                               D                                              G
   {6,4|4}_P                                    {6,4|4}_D                                   {6,4|4}_G
    sc lattice                                   fcc lattice                                 bcc lattice

These three TPMS are tiled here in a pattern
called {6,4|4}. The prototile for each surface
    is a particular variant of hex₉₀, a regular skew
6-gon
   with 90º face angles. In this pattern, there are
4 faces
incident at each vertex and 'holes' with
4-fold
symmetry.

Alternatively, these surfaces can be tiled in the
patterns {4,6|4} and {6,6|3}. The prototiles for
{4,6|4} are variants of quad₆₀, a regular skew  
4-gon with 60º face angles. In this tiling, there
are 6 faces incident at each vertex and holes    
with 4-fold symmetry.

The prototiles for {6,6|3} are variants of hex₆₀,
a regular skew 6-gon with 60º face angles. In  
this tiling, there are 6 faces incident at each     
vertex and holes with symmetry of order 3.     

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It is conventional to call skew polygons
regular if they are both equilateral
and equiangular, irrespective of
whether their edges are
straight or curved.

---

The names {6,4|4}, {4,6|4}, and {6,6|3} are
Coxeter's modified Schläfli symbols for the
three infinite regular skew polyhedra. Each
of these three polyhedra is homeomorphic   
to — and has the same symmetry as — one
of the two Schwarz surfaces. {6,4|4} and    
{4,6|4} are homeomorphic to P, while          
{6,6|3} is homeomorphic to D.                      

---

P, D, and G are related by Bonnet bending,
without stretching or tearing. A prototile of
    one surface can be continuously transformed  
by bending into the corresponding prototile
of either of the other two surfaces:               

hex₉₀(D) ↔ hex₉₀(G) ↔ hex₉₀(P).
quad₆₀(D) ↔ quad₆₀(G) ↔ quad₆₀(P).
hex₆₀(D) ↔ hex₆₀(G) ↔ hex₆₀(P).

In fact an entire lattice fundamental region, not
just a prototile, can be transformed in this way.
  Below are stills from a 1969 movie showing the
bending of hex₉₀(D) ↔ hex₉₀(G) ↔ hex₉₀(P).

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§16. The bending of D → G → P

_{Sequence of Bonnet bending stages from D to G to P (stereo)
One lattice fundamental domain (plus one additional hexagonal face)}

    0º          D
~38.015º   G
  90º          P

_{The surfaces at all other angles between 0º and 90º are self-intersecting.}

The images above are eleven frames selected from a movie I designed in 1969
using a FORTRAN program written by the computer scientist Charles Strauss.
The cinematographer was Bob Davis of the MIT Lincoln Lab. He used a Bell   
and Howell 35mm movie camera that he had modified so that it could accept   
        input data from ERC's PDP-11 computer. Ken Paciulan and Jay Epstein carried out
  the data input tasks. The enthusiastic help of all of these wonderful guys is very
gratefully acknowledged.                                                                                       

The portion of the surface shown here is a lattice fundamental domain plus one
hexagonal face — the topmost one in the image for 0º Bonnet angle. The three
dashed lines are a set of lattice basis vectors. But the surface is not periodic (its
symmetry group is not discrete) unless the Bonnet angle θ satisfies the equation
θ_p,q = ctn^-1[(p/q)(K'/K)],
where p and q are any two coprime positive integers.                                            

This obviously necessary restriction hadn't occurred to me before Blaine Lawson
mentioned it — more than once! — in September 1968. Thanks, Blaine!              

Note that in the Bonnet transformation of the complete surface S₁ into S₂, not all
pairs of faces in S₁ that share a common edge preserve this connection also in S₂.

---

To animate the time-lapse bending sequence above,
click here, and press on the
Page Down/Page Up keys.

The portion of the 1969 'Part 4'
movie that illustrates this bending
starts at 3^min18^sec after the beginning.

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§17. Poincaré's hyperbolic disk model of uniform tilings

The combinatorial structure of the three tiling patterns
{6,4|4}, {4,6|4}, and {6,6|3} is illustrated by
Poincaré's hyperbolic disk model of
uniform tilings in the hyperbolic plane.

(image by Eric W. Weisstein,
Wolfram MathWorld)

{4,6|4}                          {6,4|4}                          {6,6|3}
(images from Wikipedia)

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§18. The three Coxeter-Petrie infinite regular skew polyhedra

Coxeter and his friend Petrie long ago discovered the three triply-
periodic regular skew polyhedra called {p,q|r}. Their faces are
regular plane polygons.

p is the number of edges of each face,
q is the number of faces at each vertex,
and
r is the number of edges of each hole.

{4,6|4}

{6,4|4}

{6,6|3}

---

{4,6|4}, including the skeletal graph of labyrinth A

{4,6|4}, including the skeletal graphs of both labyrinth A and labyrinth B

{6,4|4}, including the skeletal graph of labyrinth A

{6,4|4}, including the skeletal graph of labyrinth B

{6,6|3}, including the skeletal graph of labyrinth A

---

§19. Views of G tiled by hex₉₀

Among the countable infinity of surfaces that are
associate surfaces (cf. Fig. E1.2m) of P and D,
the gyroid G is the only one that is embedded.

Unlike P and D, G contains neither straight lines
nor plane geodesics. It has the same symmetry as
the union of its two enantiomorphic skeletal graphs
(Laves graphs). The lattice is bcc.

Like P and D, G can be tiled by
      (i) regular skew hexagons hex₉₀ ({6,4|4} tiling) or by
                (ii) regular skew quadrilaterals quad₆₀ ({4,6|4} tiling) or by
(iii) regular skew hexagons hex₆₀ ({6,6|3} tiling).

---

Below are views of G tiled by hex₉₀
in the Coxeter-Petrie {6,4|4} map.

Each hex₉₀ has 90º face angles.

Every hex₉₀ face is related to each of six faces with
which it shares an edge by a half-turn about an axis
of type (110) perpendicular to G at the midpoint of
the shared edge.

(1)

(2)                      (3)                       (4)                      (5)

                  (1)     [100] orthogonal projection
(2)     (100) viewpoint
                  (3)     [100] orthogonal projection
                  (4)     [111] orthogonal projection
(5)   ~(110) viewpoint

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§20. Views of G tiled by hex₆₀

Below are views of G tiled by the regular
skew hexagon hex₆₀, the prototile of
the Coxeter-Petrie {6,6|3} map.

hex₆₀ is so named because it has 60º face angles.
It is related to six other hex₆₀ faces, with each of
which it shares an edge, by a half-turn about an
axis of type (110) perpendicular to the surface at
the midpoint of the shared edge.

Stereo image of a cubic unit cell of G
tiled by eight replicas of hex₆₀
This unit cell is comprised of
two lattice fundamental regions.
(100) viewpoint
High-resolution version

Stereo image of the cubic unit cell of G
illustrated just above
(-1-11) viewpoint High-resolution version

Stereo image of the
hexagonal face hex₆₀ of G
(111) viewpoint
High-resolution version

Stereo image of the
hexagonal face hex₆₀ of G
and an associated cuboctahedron

The midpoints of the six edges of
hex₆₀ coincide with vertices
of the cuboctahedron.
(111) viewpoint

High-resolution version

Stereo image of the
hexagonal face hex₆₀ of G
(110) viewpoint High-resolution version

Stereo image of the
hexagonal face hex₆₀ of G
(415) viewpoint High-resolution version

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§21. Views of G tiled by quad₆₀

§22. Gyring Gyroid — a sculpture by Goodman-Strauss and Sargent Gyroid sculpture by Chaim Goodman-Strauss and Eugene Sargent Click here for the authors' account of how they implemented their design. Here's their razzle-dazzle video showing bits and pieces of its construction. §23. Shapeways 3D-printed models of TPMS If you're unfamiliar with TPMS, one place to begin looking is the set of Shapeways models made by Ken Brakke Alan Mackay Bathsheba Grossman and by others. Shapeways displays a large collection of gyroid and gyroid-related models here. Bathsheba Grossman's model of a kind of inversion of the gyroid bounded by an ellipsoid Below are a few of the TPMS models produced by Shapeways for Ken Brakke and for Alan Mackay. There's much more information about TPMS at one of Ken Brakke's webpages, including many illustrations he made with his Surface Evolver. P                                                  C(P) (models by Ken Brakke) D                                                  C(D) (models by Ken Brakke) I-WP                                          F-RD (models by Ken Brakke) Batwing (model by Ken Brakke)       Gyroid                                  Double Gyroid (models by Alan Mackay)       Sven Lidin's Lidinoid               Sven Lidin's Lidinoid 222 (models by Alan Mackay) NodalGyroid222a                             Fluoritesolid7 (models by Alan Mackay)        SBA-1x50                                  Batwing5x2 (models by Alan Mackay) §24. The four founding fathers of triply-periodic minimal surfaces       Georg Friedrich Bernhard Riemann      Karl Hermann Amandus Schwarz                                         (1826-1866)                                            (1843-1921)                                                     Alfred Enneper             Karl Theodor Wilhelm Weierstrass                                      (1830-1885)                                    (1815-1897)                                      (In §E7 there are photos of a few contemporary experts in this field.) §25. Definition of minimal surface Mathematicians have studied minimal surfaces since 1762, when Lagrange derived the 'Euler-Lagrange equation', which is satisfied by the surface of least area spanning a given closed curve. Aside from the plane, which defines a trivial solution of this equation, the first surfaces found as solutions of the Euler-Lagrange equation were the helicoid and catenoid, both of which were discovered by Meusnier in 1776. Meusnier also proved that the mean curvature of every solution surface is equal to zero. Since some closed curves are spanned by more than one surface with zero mean curvature everywhere, a minimal surface is conventionally defined as a surface with vanishing mean curvature at every point, rather than as a surface of least area. Of course every minimal surface is locally area-minimizing, i.e., the surface patch inside every sufficiently small closed curve enclosing any point of the surface has less area than any other surface bounded by that closed curve. Pioneering investigations of triply-periodic minimal surfaces (TPMS) were performed by Schwarz, Riemann, Weierstrass, Enneper, and Neovius in the middle of the 19th century. By the early 1960s, however, TPMS had almost faded from view in the mathematical literature. Since about 1970, there has been a revival of interest in TPMS as mathematical research on minimal surfaces of every kind has expanded. Now it is no longer just mathematicians who study TPMS. Materials scientists are also interested in them, because they have concluded that some of the few known examples of low genus — especially those on a cubic lattice — are useful as templates for the shapes of a variety of so-called self-assembled structures that are studied by physical, chemical, and biological scientists. Below are links to a tiny sample of the relevant mathematical and materials science studies, but the sample is neither comprehensive nor up-to-date. I have attempted here to summarize my own study of TPMS, which began quite unexpectedly in the spring of 1966. I have included an account — warts and all — of some of the events that led to my involvement in this study, during which I frequently wandered down bypaths that were well off the main route. (I have long believed that such bypaths sometimes offer a more rewarding view than the main route. John Horton Conway has explained that he finds it fruitful to juggle several ostensibly unrelated problems at the same time, because one problem may turn out to be the key to the solution of another. My discovery of a precursor of the gyroid minimal surface in 1968 was for me a validation of Conway's truism, as explained below.) §26 My stint at NASA/ERC (1967-1970) I am enormously indebted to the physicist Lester C. Van Atta, who created for me an unusual position as senior scientist, under his nominal supervision, at the NASA Electronics Research Center (ERC) in Cambridge, Masachusetts. Van Atta, who was both Associate Director and also Director of the Division of Electromagnetic Research, alowed me to indulge my new-found passion for TPMS, even though I lacked the credentials most employers would have considered a minimum requirement for such an undertaking. Because of his scientific reputation, he had sufficient clout to shield me from attacks both by local skeptics (of whom there were more than a few) and also by officials in NASA headquarters who wondered what on earth soap films might have to do with NASA's mission. Lester C. Van Atta 1905-1994 Associate Director, NASA Electronics Research Center But it was all too good to last! ERC was abruptly shuttered in July 1970 in what many of us concluded was probably an act of political malice directed by President Nixon against Senator Ted Kennedy of Massachusetts. I cannot avoid being somewhat sceptical of the purportedly objective history of the closing of ERC by the author of this contemporaneous account, in which no specific role is ascribed to Nixon. Until the announcement on December 29, 1969 by the Administrator of NASA that ERC would close on June 30, 1970, I felt quite free to decide what to investigate, with few strings attached. In retrospect, I believe that I would almost certainly have been unable to concentrate productively on my research at ERC if I had been aware of the turbulent political winds that were blowing about our heads. §27. Peter Pearce's concept of saddle polyhedra Before arriving at ERC in the fall of 1967, I decided that I would concentrate there on two areas of research:         (a)  symmetric triply-periodic graphs and their nodal and interstitial polyhedra (see explanation below), and         (b)  a search for new examples of TPMS (even though I had not yet discovered any such examples). I had been strongly interested in connections between triply-periodic graphs and convex polyhedra since the mid-1950s, but before 1966 I knew nothing about minimal surfaces, aside from a nodding acquaintance with the helicoid and the catenoid. In April of that year, Konrad Wachsmann, chairman of the architecture department at the University of Southern California, suggested that I visit the North Hollywood architect/designer Peter Pearce, who had a grant from the Graham Foundation for a one-year study of polyhedra, crystal structure, and related topics. Although Peter did not claim to be an expert on the mathematics of minimal surfaces, he had developed a novel application of minimal surfaces to the design of periodic structures that led me to make a radical change in the direction of my research. Below I summarize how this happened. Peter Jon Pearce Architectural designer At Peter's studio I saw several elegantly crafted handmade models of crystal networks, including two that especially caught my eye, because they each contained an example of a novel interstitial object Peter had invented and named saddle polyhedron. These saddle polyhedra had straight edges, but each face was curved in the shape of a minimal surface. (All of the stereoscopic image pairs below are arranged for cross-eyed viewing.)                                        A portion of the diamond graph                        The interstitial polyhedron                                                                                          of the diamond graph                                   A portion of the bcc graph                           The interstitial polyhedron                                                                                            of the bcc graph §28. Dual graphs and interstitial vs. nodal polyhedra I was thunderstruck by Peter's two saddle polyhedra, because I understood at a glance that they were the critical ingredient missing from a scheme I had tried to develop for illustrating the relation between the combinatorial and symmetry properties of crystal networks (triply-periodic graphs) and polyhedral packings. The intended purpose of my heuristic scheme was to represent every atomic site in a crystal strucure by a polyhedron with         (i)   the same number f of faces as the number Z of edges incident at the corresponding node of the graph, and         (ii)  the same symmetry as that node. For several lattices, the Voronoi polyhedron serves nicely for this purpose. For example, (i) the number f of faces of the cube, which is the Voronoi polyhedron for a vertex of the simple cubic (sc) lattice, is six, which is also the number Z of edges incident at each node (vertex) of a conventional ball-and-stick model of the lattice, and (ii) the cube also has the same symmetry as the node with respect to the surrounding lattice. A piece of the simple cubic (sc) graph, for which the interstitial polyhedron is the cube, which is the Voronoi polyhedron for a vertex of the graph. The cube is also the nodal polyhedron for a vertex of the sc graph A similar correspondence holds for the face-centered cubic (fcc) lattice if each vertex is enclosed by the rhombic dodecahedron, which is the Voronoi polyhedron for a vertex of this lattice. The nodal polyhedron of the fcc graph is the rhombic dodecahedron, the Voronoi polyhedron of a vertex of the graph. The fluorite graph is the dual of the fcc graph. Its nodal polyhedra are the regular tetrahedron and the regular octahedron. The fcc graph and the fluorite graph It is convenient to define this pair of graphs as duals. For the body-centered cubic (bcc) lattice, however, the combinatorial part of this correspondence breaks down. Although there are only eight nearest neighbors of each vertex in this lattice, the Voronoi polyhedron of a vertex is the truncated octahedron, which has fourteen faces. The reason for this numerical disparity is hardly profound. It's just that the second-nearest-neighbor sites in the bcc lattice happen to be situated in directions and at distances that cause truncation of the six vertices of the regular octahedron, which — as a first approximation to the Voronoi polyhedron — takes only nearest neighbor sites into account. I had observed a similar mismatch for the diamond crystal structure: even though there are only four nearest neighbors for each site, the Voronoi polyhedron has sixteen faces. The 14-faced Voronoi cell for a vertex of the bcc lattice For a pdf image, look here. I was unable to contain my excitement when I saw Peter's two interstitial saddle polyhedra, because I immediately recognized that they would make it possible to remove the numerical disparities I had observed for both the bcc lattice and the diamond crystal structure. I described to Peter a space-filling eight-faced saddle polyhedron, composed of regular skew hexagons with 90º corners, that would enclose each vertex of the bcc graph. A few days later I dubbed it the expanded regular octahedron, or ERO (see stereo image below). I proposed calling vertex-enclosing polyhedra nodal polyhedra, irrespective of whether they turn out to be saddle polyhedra or convex polyhedra. For about ten years, I had been calling the triply-periodic graph whose edges correspond to the edges of a packing of expanded regular octahedra the WP graph, because it mimics the pattern of string tied around a wrapped cubic box (see image below). Curiously, the ERO was about to introduce me to TPMS! The expanded regular octahedron ERO, which is the nodal polyhedron of the bcc graph and the interstitial polyhedron of the WP graph The eight faces of ERO match the number of edges incident at each vertex of the bcc graph, and both the saddle polyhedron and a vertex of the graph have the same symmetry. The bcc graph (Z=8) The edges of the tetragonal tetrahedron TT are shown in blue. The WP graph (Z=4) The edges of the expanded regular octahedron ERO are shown in blue. The bcc graph (green vertices) and its dual, the WP graph (orange vertices) §29. Soap film interlude (2012) When I started playing with soap films and minimal surfaces in May 1966, I was totally ignorant of the extensive literature on these subjects. I didn't know that there are boundary frames that can be spanned by more than one shape of minimal surface and that so-called unstable minimal surfaces, which do not minimize area, can span those boundary frames. On one of his web pages, called 'Catenoid Soap Films', Ken Brakke illustrates what may be the first known example of this phenomenon, the existence of both stable and unstable versions of the catenoid spanning the space between two parallel circular boundary frames. I recall poring over James Clerk Maxwell's classical article in the legendary Eleventh Edition of the Encyclopedia Britannica, inherited from my father, in which the author analyses this behavior of the catenoid. As I learned more about minimal surfaces, I eventually realized that in Peter Pearce's prescription of minimal surfaces for the faces of saddle polyhedra, it should probably be stated explicitly that the minimal surface is area minimizing — and therefore stable. (In fairness to Peter, I suspect that he was already aware of these distinctions! I don't recall ever having discussed these questions with him.) I did wonder a little about the variety of shapes of saddle polyhedra that would result if there existed more than one minimal surface spanning a given circuit of edges in a triply-periodic graph. In June 1966, I undertook some soap film experiments in order to explore these questions. I found that the boundary curve C0 (shown at left below), a simple closed curve in the shape of one of the several Hamilton cycles on the cuboctahedron, is spanned by at least two disk-type soap film surfaces of different shape. One of these two surfaces, S1, is an area-minimizing surface ('least-area surface') and is called stable. The other, S0, is not a least-area surface. It is called unstable, because it can be formed as a soap film on C0 only if one or more wires or threads are added to C0 along appropriate curves embedded in S0 — i.e., curves that partition S0 into an assembly of smaller surface patches, each of whose boundary curves is spanned by a unique stable least-area surface. S1 can drape the boundary frame C0 in either of two positions. Let's call it S1a if it's in one of these positions and S1b if it's in the other position. S1a and S1b are related by a halfturn about the axis A1A2. (See Ken Brakke's computed images of S1a and S1b below.) If a wire frame in the shape of C0 is withdrawn from a solution of soap and water, it will be spanned by a soap film in the shape of one or the other (but not both) of these surfaces. S0, which drapes the wire frame C0 in only one position, forms as a soap film if threads or wires are incorporated in C0 along one or both of the lines A1A2 or B1B2. C0, a curve that can be spanned by at least two differently shaped soap films, one stable and the other unstable Cuboctahedron The three [orthogonal] c2 axes of the boundary curve C0. The two axes A1A2 and B1B2 each intersect C0. The vertical axis V1V2 does not. The soap film S1a is an area-minimizing ('least-area') minimal surface. It is one of two differently oriented — but congruent — stable surfaces that span the boundary curve C0. S1a                                                   S0                                                    S1b (Stable)                              (Unstable, unless string                               (Stable) or wire is added along either A1A2 or B1B2 or both A1A2 and B1B2) Incorporating internal threads or wires along (a) either A1A2 or B1B2 or (b) both A1A2 and B1B2 in the wire frame C0, which has twelve edges, partitions C0 into an assembly of congruent skew polygon boundary frames, each with either seven edges [case (a)] or five edges [case (b)]. The question of how many minimal surfaces span a given boundary curve has been found to be extremely knotty, but it is known that there are two properties of a simple closed boundary curve C either of which guarantees that it is spanned by only one minimal surface of disk type:       (i) having a convex simple projection — whether central or parallel — onto a plane (Rado's 1932 theorem);       (ii) having total curvature less than 4π (Nitsche's 1967 theorem). Since the aforementioned 5-gons and 7-gons have total curvatures of only 21⁄6 π and 31⁄3 π, respectively, it follows from Nitsche's theorem that each of them is spanned by only one minimal surface of disk type. This implies that incorporating a wire or thread along either or both of the axes A1A2 and B1B2 will convert the bare frame C0 into a frame that is spanned by the surface S0. Note that the soap film in each of the photos below is the same piece of S0.          wire added at A1A2                      wire added at B1B2                 wires added at A1A2 and B1B2 In October 1967, three months after I joined NASA/ERC, I was a self-invited guest at the home of the late Hans Nitsche in Minneapolis. Although Hans showed considerable interest in my wire-frame model of C0 and in my vacuum-formed plastic models of S0 and S1, he never even mentioned the ground-breaking paper in which he introduced and proved his 4π theorem, which was about to be published! Since I always found Hans to be both kind and modest, I later concluded that perhaps he thought I was so ignorant of the mathematics of minimal surfaces that he would only confuse and embarrass me if he discussed such a subtle problem. I extend my warm thanks to Ken Brakke for pointing out a serious elementary blunder, in an earlier version of this discussion, of the number of soap films spanning the frame C0. §30. Triply-periodic graphs Ever since 1954, when I began an informal study of polyhedral packings (triggered by my Ph. D. research, at the University of Illinois/Urbana-Champaign, on atomic diffusion in crystalline solids), I had ruminated from time to time over the relation between polyhedra and crystal structures. I became familiar with a variety of commonly known crystal structures, and I sawed wooden models of the Voronoi polyhedra that enclose the vertices of some of these structures. In 1956 I designed and ran a FORTRAN program that confirmed my hunch that for self-diffusion in fcc crystals, the isotope effect and the Bardeen-Hering correlation factor are precisely equal. (The program modeled diffusion by an infinite random walk of a vacancy in a sequence of cubically symmetrical crystal volumes of increasing size.) This exact identity of the isotope effect and the correlation factor became the basis of the first experimental method — using radioactive tracers to sample the behavior of the diffusant atoms — of distinguishing between the interstitial and vacancy mechanisms of atomic self-diffusion in crystals. I modeled interstitial diffusion pathways (strictly random walk) by the edges of one triply-periodic graph and pathways for diffusion by the vacancy mechanism in the same crystal (correlated random walk) by the edges of a second triply-periodic graph intertwined with the first graph. I defined these two graphs as duals, and I attempted to discover whether it is possible to define which symmetry and combinatorial properties are required of a triply-periodic graph in order for it to have a unique dual, by analogy with the dual of a planar graph or the dual of a convex polyhedron. Using essentially ad hoc methods to identify dual pairs of triply-periodic graphs, I found that while the dual of the diamond graph is also a diamond graph, the dual of the fcc graph is the fluorite graph and the dual of the bcc graph is the WP graph. These relations are illustrated by the images shown above. But it soon became apparent to me that for many pairs of graphs, if one ignores the atoms in the crystals represented by the graphs there is no justification for labeling one graph substitutional and the other interstitial. §31. My first encounter with TPMS In April 1966, two days after meeting Peter Pearce, I made some examples of saddle polyhedra for myself, using the toy vacuum-forming machine I had bought for my children. My first model was the bcc nodal polyhedron, the expanded regular octahedron ERO illustrated above. But afterwards out of curiosity I joined two of its skew hexagonal faces by rotation instead of reflection. To my great astonishment, I found that if I continued to add faces in this fashion, the infinite smooth labyrinthine structure shown below began to emerge. (This vinyl model, as well as those shown in the next three images, are new ones I made the following year, after I had purchased a larger vacuum-forming machine.) A piece of Schwarz's D surface Next I replaced the 90º skew hexagon by one with 60º corners, and a second such labyrinthine surface appeared! A portion of Schwarz's P surface tiled by 60º skew hexagons A transparent model of P I had unwittingly stumbled onto the two classical examples of adjoint (or conjugate) TPMS, which were discovered and analyzed in 1866 by H.A. Schwarz (and also — independently — by Riemann and Weierstrass). It took a telephone call to the minimal surface authority Hans Nitsche in Minnesota for me to identify these surfaces. I decided to name them D (for diamond) and P (for primitif), after the crystal structures with matching topology and symmetry. I recognized that the chambers in the two complementary labyrinths of P define the sites of the cesium and chlorine ions, respectively, in the ionic crystal Cs-Cl. Only after consulting a handbook of crystal structures did I learn that the atoms of sodium and thallium in the binary solid solution Na-Tl occupy sites that correspond to the symmetrical 'chambers' in the respective labyrinths of D. I began to study in earnest both differential geometry and the complex analysis used in investigations of minimal surfaces. With the benefit of hindsight, I later recognized that if — at some time during the year after I stumbled onto D and P in April 1966 — I had taken the time to read Schwarz's Collected Works more carefully, I might possibly have noticed the following theorem on p. 174: TRANSLATION: I didn't read that passage until September, 1968, when I understood at long last that the coordinates of every point on G are simply a linear combination of the coordinates of corresponding points of D and P, i.e. that G is associate to D and P. A few days later Blaine Lawson pointed out to me that if D and P are scaled so that the 90º hexagons D_hex90 and P_hex90 that tile the map {6,4|4} in these surfaces are inscribed in a cube of the same size, that linear combination becomes the arithmetic mean of the coordinates of corresponding points of D and P! It is perhaps surprising that Schwarz doesn't seem to have taken the trouble to sum the coordinates of at least a few pairs of corresponding points of D and P. It seems very likely that if he had done so, he would have discovered the gyroid. §32. Dual graphs and skeletal graphs The concept of a dual relation for pairs of triply-periodic graphs had suddenly acquired new significance for me. I began to think of such graphs as potential skeletal graphs of the two labyrinths of an embedded TPMS. The geometry of such paired graphs would dictate the geometry of the TPMS. A literature search in the UCLA library indicated that besides D and P, only three other examples of embedded TPMS — H, CLP, and Neovius's surface — had been known since 1883. But I found it hard to imagine that there were not others! §33. Early hints of the existence of the gyroid For the D and P surfaces, as well as for H, CLP, and Neovius's surface, both labyrinths of the surface are directly congruent, which implies that their skeletal graphs are also directly congruent. I wondered whether any other dual pairs of triply-periodic graphs I had identified — including the oppositely congruent Laves graphs — might also be skeletal graphs of the two labyrinths of an embedded TPMS. The Laves graphs were a troublesome case, because the absence of reflection symmetries made it impossible for me to imagine how such a surface could be generated. Sometimes the makeshift rule I had refined by exploiting the relation between triply-periodic graphs and saddle polyhedra yielded a dual pair of triply-periodic graphs that were neither directly nor oppositely congruent — for example, the fcc–fluorite pair and the bcc–WP pair. Did this mean that there exist examples of TPMS in which the two labyrinths are not congruent? I did not yet know. Discovering examples of such surfaces would have to wait until I was free to investigate TPMS as something more than an evenings-and-weekends hobby. (Further details of this story are described below.) I recognized that Peter Pearce had made an inspired choice when he chose minimal surfaces for the faces of his diamond and bcc interstitial saddle polyhedra, and I was becoming confident that the correspondence between polyhedra and the nodes of crystal structures was about to become much simpler. At the same time, however, I had a nagging feeling about certain loose ends that needed tidying up. By May 1966 I had devised an ad hoc recipe for constructing both interstitial and nodal polyhedra that I hoped would be effective for every possible example of a triply-periodic graph. Although the recipe worked without a hitch for every graph I tested, I felt distinctly uneasy, because I suspected that there must exist cases for which it would be ineffective. Although I modified the recipe several times during the next several months, I was never able to give it a solid theoretical foundation. Beginning in June 1966, as a spare-time hobby I set out to discover a 'counterexample' — a graph for which the recipe fails to produce either interstitial or nodal polyhedra. I continued to test a variety of graphs, gradually accumulating a diversified collection of vacuum-formed interstitial and nodal polyhedra. I had the additional goal of finding a way to construct a hypothetical TPMS I originally named L (for Laves). Here, however, I will refer to it as G (for gyroid), even though I didn't invent that name until almost two years later. It was obvious that G couldn't have any reflection symmetries, since the union of its two enantiomorphic skeletal graphs has no such symmetries. I also recognized that there could be no straight lines embedded in G, since by a theorem of H. A. Schwarz, a straight line embedded in a minimal surface is an axis of 2-fold rotational symmetry. This implies that a half-turn rotation of G about such an axis would interchange the two labyrinths of G — and therefore also interchange the two skeletal graphs of G. But that is impossible, since the two skeletal graphs are enantiomorphic. I couldn't imagine how to define the boundary curves of an elementary surface patch whose edges are neither straight line segments nor curved geodesics (mirror-symmetric plane lines of curvature). For each of the five examples of TPMS known before 1968, there exists a skew polygon surface patch with straight edges. §34. Symmetric graphs I nevertheless had a strong conviction that G must exist. The principal reason for my thinking so was that the skeletal graph of each labyrinth of G shares with the skeletal graphs of Schwarz's P and D surfaces what I believed to be an exceptionally rare property: it is a symmetric graph. A second reason I believed in the existence of G was derived from purely visual evidence: when I compared the toy model of G that I had constructed out of stubby paper cylinders with the toy models I had made for P and D, I found that the G model was no less convincing than the other two. In all three models, when the cylinders are made as fat as they can possibly be (see the images just below), the total volume of the gaps between the two 'cylinder-ized' labyrinths is surprisingly small. I was startled to observe how snugly the cylinders of the intertwined labyrinths nestle against each other. When I compared the contours of my toy models of P and D with the smoother contours of my vacuum-formed models of P and D, it seemed entirely plausible that the junctions between cylinders in the toy model of G could be flared and filleted so that the envelopes of the two labyrinths would coalesce into one single surface — a TPMS — just as they do for P and D. I understood, of course, that vague intuitive arguments like these are only rarely fruitful, but on the other hand I was not prepared to dismiss the arguments as worthless. §35. Toy models of P, G, and G P D G 'Toy models' of P, D, and G The edges of the dual skeletal graphs are represented as right circular cylinders. In each of the images at the right, the cylinder radius is the maximum possible consistent with the requirement that the cylinders intersect only at isolated points of tangency. §36. BCC6 and M4 — hints of the gyroid's existence? On February 14, 1968, seven months after moving from Los Angeles to Cambridge, I reached two goals simultaneously. (a) I discovered a graph of degree six called 'BCC6' (see stereo image below) that provided the long-sought 'counterexample' to my empirical recipe for deriving the interstitial and nodal polyhedra of a triply-periodic graph. (b) Although this graph failed spectacularly to yield a finite interstitial polyhedron, it pointed toward something much more interesting — an infinite triply-periodic saddle polyhedron that I call M4. The symmetry and combinatorial structure of M4 strongly strongly suggested to me that the hypothetical G minimal surface might exist after all. I immediately lost almost all interest in saddle polyhedra and began to concentrate instead on confirming the existence and embeddedness of the G surface and on searching for other new examples of TPMS. A portion of the deficient symmetric graph BCC6 of degree six I define a deficient symmetric graph on a given set of vertices as a symmetric graph of degree less than the maximum possible for that set. (With two additional edges incident at each vertex, BCC6 would be transformed into BCC8, the bcc graph.) viewpoint: close to [100] direction 30 quadrangles of M4 (stereo) view: [111] direction 30 quadrangles of M4 (stereo) view: [110] direction 30 quadrangles of M4 (stereo) view: [110] direction 30 quadrangles of M4 view: [111] direction, backlit by summer sky 30 quadrangles of M4 view: [100] direction, backlit by summer sky §37. A premature announcement of the gyroid Abstract 658-30 submitted in summer 1968 to the American Mathematical Society This was an awkwardly premature announcement of the existence of the gyroid, which I then called L. I had merely conjectured, not proved, that L is a minimal surface. (After I recognized that it is associate to P and D, I renamed it gyroid.) §38. A botanical link to Schwarz's P surface My motivation for studying TPMS was not the result of a perceived connection between such surfaces and known structures in physics, chemistry, or biology. However, I did make regular use of encyclopedias of crystal structures to imagine the shapes of possible examples of TPMS. During a literature search at the UCLA library in the early summer of 1966, I discovered a 1965 article by Gunning and Jagoe [Gunning 1965a] that included electron micrographs of the prolamellar structure of etiolated green plants. These images led the authors to describe the prolamellar body as a collection of smoothly interconnected tubules on a simple cubic lattice. I interpreted this description as suggesting a rough similarity to Schwarz's P surface. (In 1971 Michael Berry [Berry, 1971] stated that Gunning and Jagoe later revised their analysis in favor of a network of tubules along the edges of the diamond graph instead of the simple cubic graph.) §39. More about NASA/ERC Remarkably, Lester Van Atta, who had recruited me to work at NASA/ERC and was my immediate supervisor there, never interfered with my choices of what to work on. Since it was he who had invented the name 'Office of Geometrical Applications' for my 'administrative unit', I concluded that he did expect me to try to produce something of practical value for NASA. But he was never less than enthusiastic about my concentration on the study of periodic minimal surfaces. My career at NASA was disappointingly short-lived, however. On December 30, 1969, the director of NASA visited Cambridge to announce to a gathering of all employees that ERC would be permanently closed in exactly six months. We were of course startled — as well as disheartened — by this unexpected news. The six year-old ERC was by far the youngest of the eighteen NASA centers. It was the only federal research center with electronics research for its mission, a legacy inherited from the Kennedy presidency (though it was President Lyndon Johnson who presided over its development). ERC was famously top-heavy (or perhaps I should say bottom-heavy) with a bloated support infrastructure of low- and mid-level administrators, clerks, etc., many of whom were from the Boston area, hired in the early days before President Kennedy was assassinated. Soon after it opened, the hiring of scientists and engineers slowed down abruptly, and it appeared that the original plan to develop a well-rounded scientific and technical staff had been abandoned. It was our impression that Lyndon Johnson preferred to support NASA activities elsewhere, especially in Texas. However, according to the Wikipedia entry for ERC: "Although it was the only Center NASA ever closed, ERC actually grew while NASA eliminated major programs and cut staff. Between 1967 and 1970, NASA cut permanent civil service workers at all Centers with one exception, the ERC, whose personnel grew annually." Whatever the case, I arrived at ERC in July 1967 in a state of blissful ignorance. Only after I began work did I begin to learn from my colleagues about discrepancies between ERC's officially stated 'mission' and what seemed to be its actual potential for significant accomplishment. §40. Geometers and other mathematical mentors My own position there was relatively comfortable, however, with one exception: the absence of in-house colleagues who shared my scientific and mathematical interests. I would have benefited from having someone close by for chit-chat about — and even collaboration in — those areas of research in physics and mathematics in which I had a special interest. There were spectacular compensations for this deficiency, however. For one, I was acquainted with a few extremely bright young mathematicians in the greater Boston area who showed a friendly interest in my work, and I benefited greatly from my few conversations with them. If only I had shown more initiative, I could have benefited even more from knowing them than I actually did. They included Thomas Banchoff (differential geometry), Norman Johnson (convex polytopes), Nelson Max (computer graphics), and Charles Strauss (computer graphics). In 1966, a year before I joined NASA/ERC, Norman Johnson introduced me to the analysis by Coxeter and Moser of the infinite regular maps {6,4|4}, {4,6|4}, and {6,6|3} (cf. the book by these authors that is cited below, following Fig. E1.1k). These three regular maps describe the combinatorial structure of the flat-faced Coxeter-Petrie infinite regular skew polyhedra. But they also describe the combinatorial structure of H. A. Schwarz's P and D surfaces, the canonical 19th century examples of triply-periodic minimal surfaces, as well as that of their only embedded associate surface, the gyroid G (which I nearly discovered in February 1968, when by chance I found a doppelganger that is spookily similar). (Parenthetical note: In 1969 Donald Coxeter was my guest in Cambridge, Mass., where he presented a lecture at MIT. To my surprise, he told me that he had never heard of the Schwarz surfaces!) H.S.M. "Donald" Coxeter Thomas Banchoff at Berkeley in 1973 photo by George Bergman Charles Strauss (seated) and Thomas Banchoff at Brown University in 1979 Norman Johnson (1930-2017) Nelson Max MIT was directly across the street from ERC, and Harvard was only a 20-minute walk away. ERC staff members had unrestricted borrowing privileges at the MIT library — an enormous convenience. A few prominent members of the MIT math faculty indulged me now and then when I had a pesky mathematical question, but for the most part, I was hesitant about bothering them, partly because I held them in such awe but also because I knew that for them many of my questions would turn out to be extremely elementary, if not downright trivial. A few months after I arrived at ERC, I was visited by Harald ('Hal') Robinson, a sculptor, designer, master machinist, and model-maker who lived in a nearby suburb. We hit it off immediately. After examining my plastic minimal surface models, Hal easily convinced me that he could make more accurate and more durable vacuum-forming tools than I could. Dr. Van Atta was acquainted with Hal's father, an engineer who was president of High Voltage Engineering Corp., the manufacturer of Van de Graaff generators. I persuaded Dr. Van Atta to hire Hal in a flexible part-time arrangement so that he could fabricate vacuum-forming tools for me. Hal wan't interested in a fulltime job, since he had other clients, and in any event I expected to have only enough projects to keep him busy intermittently. From then until the end of my stay at ERC about thirty months later, Hal was my invaluable collaborator. Dr. Van Atta also arranged to hire — one or two at a time — part-time work-study students from area universities (Boston University, Northeastern University, Harvard, and MIT) to help with FORTRAN programming and the assembly of new minimal surface models. These young superstars were Kenneth Paciulan, Richard Kondrat, Randall Lundberg, Jay Epstein, and Dennis ____(?). I am grateful to them all. Dr. Van Atta also hired James Wixson, an experienced applied mathematician and computer programmer. Jim helped me with a variety of chores. One of his several accomplishments was the invention and programming of a computer algorithm for generating every possible skew quadrangle that serves as a module for a compound periodic minimal surface on a cubic lattice — an assembly of finite surface patch modules whose four straight edges include at least one edge along a [111] direction, coincident with an axis of 3-fold rotational symmetry. Decades later, these solutions have become of some interest as models for structures investigated by physicists and chemists who are soft matter specialists. I recall now with some embarrassment that during my first week at ERC, I visited the Harvard mathematics department and stopped by the offices of one after another member of the faculty to ask naive questions about the rather prosaic problem of how to go about enumerating those examples of triply-periodic graphs that are symmetric. Professors Zariski and Ahlfors were both polite, but it was clear that my questions held little interest for them, and the interviews were mercifully short. Andrew M. Gleason (1921-2008) Andy Gleason was another matter, however. He cordially invited me into his office, where we spent the next ninety minutes or so discussing my problem. First he asked me why I was interested in this question. When I explained my still rather half-baked ideas about the connections to triply-periodic minimal surfaces, he showed considerable interest. Although he didn't provide me with definitive solutions for any of my problems, he did ask me a number of stimulating and provocative questions. I never met him again. It was only a few years ago that I learned of the great range of his highly original accomplishments in both 'pure' and 'applied' mathematics. He was a very kind person, and I shall never forget him. If I had known then that (a) Andy Gleason and I both graduated from high school in Westchester County, N.Y. (he in Yonkers and I in Mount Vernon), (b) he graduated from Yale in 1942, the year I entered Yale, and (c) we were both in Naval Intelligence during WWII (he helping to crack the Japanese code and I passively studying the Japanese language), I would undoubtedly have attempted some small talk about these coincidences, but that would hardly have advanced our discussion of mathematics! As a federal civil service employee, I had unfettered access to the WATS government long-distance telephone line. I made good use of it now and then, including having several fruitful conversations about the stability of minimal surfaces — beginning in 1968 — with Fred Almgren at Princeton. I first met him face-to-face in September 1969, when we arranged to have side-by-side seats on a flight to the USSR. For a week Fred was my roommate at the Hotel Iberia in Tbilisi, Georgia, while we were attending a conference on minimal surfaces. Afterward Fred went on to St. Petersburg for an extended sabbatical visit. Frederick J. Almgren, Jr. In April 1968, shortly after I discovered experimentally a remarkably close approximation to what I subsequently called the gyroid (but before I had any proof that such a minimal surface exists), I telephoned Robert Osserman at Stanford to ask for his help with a proof. I sent him a plastic model of the surface, and soon aferwards he asked his PhD student Blaine Lawson to investigate the problem. What followed is described below, just after Fig. E2.68c.9. From then on, I occasionally used the WATS telephone line to discuss some of my conjectures about minimal surfaces with Blaine, whom I found to be extremely knowledgeable about every conceivable aspect of minimal surface theory. Robert Osserman H. Blaine Lawson, Jr. §41. My visit to Stefan Hildebrand at the Courant Institute I was free — within reason — to attend meetings of the American Mathematical Society, of which I was a member. In contributed 15-minute talks at one or two of those AMS meetings, I described my work and showed some of my minimal surface models. Once or twice someone in the audience would express interest in the mathematics, but more often it seemed that they were curious mostly about how I had constructed the models! After one of those AMS meetings in New York City, I visited the Courant Institute, where I had the enormous good luck to meet Stefan Hildebrandt, already one of the up-and-coming leaders in the mathematics of minimal surfaces. During the next few years, Stefan more than once saved me from making a serious blunder as I groped my way toward a fuller understanding of minimal surfaces. Stefan Hildebrandt at Berkeley (1979) photo by George M. Bergman ©George M. Bergman Source: Mathematisches Forschungsinstitut Oberwolfach gGmbH Stephen Hyde told me in 2011 that according to Stefan, the reason it fell to his lot to interview me during my 1968 visit to the Courant Institute was that he was at that time one of the youngest members of the research staff. It was the custom for junior members to be assigned the chore of hosting the cranks and crackpots who invited themselves to the Institute. Since I was self-invited, for all anyone could tell I — with my bizarre colored models of surfaces in tow — might turn out to be one of those unwelcome visitors. I was gratified to learn from Stephen that Stefan concluded — after listening to my spiel and examining my surfaces — that I was probably neither crank nor crackpot! He took some photos of me and my models on the roof of the Courant Institute. Here is one of me holding my plastic model of the gyroid. A.H.S. holding a model of the gyroid at the Courant Institute, 1968 Photo by Stefan Hildebrandt §42. The end of NASA/ERC At ERC I buried myself in my research with little thought about the future. Dr. Van Atta provided even more support for my work than I ever asked for. On December 30, 1969, when we were first notified about the impending shutdown, Richard Nixon had been president for almost a year. According to an account published in the New York Times, a prominent science journalist overheard some interesting remarks in the White House by the physicist Lee DuBridge, the former CalTech president who was Nixon's scientific advisor. DuBridge was alleged to have said that the president's decision to close ERC was prompted by his wish to damage the presidential aspirations of the senior senator from Massachusetts, Teddy Kennedy. (Kennedy was widely regarded at the time as Nixon's most formidable potential rival.) NASA had been funneling about $60 million annually into Massachusetts, and a significant fraction of those funds supported ERC, with substantial collateral benefits to the state economy. Beginning in January 1970, ERC director James Elms made frantic efforts to find another federal agency to occupy the new $40 million building into which we had moved a week or so before the announcement of the shutdown. By late spring 1970, it was decided that a handful of members of the technical staff— mostly engineers and a few applied mathematicians — would be retained to work for a newly minted federal agency that would be called the Transportation Systems Center, as part of the U. S. Department of Transportation. The rest of us were told, "Good luck!" (Thanks to the good offices of Peter Pearce, I had already been invited to teach at the about-to-be-formed California Institute of the Arts, in Valencia, California, so my distress over the demise of ERC was somewhat less acute than that of many of my colleagues.) §43. A commission from the Museum of Modern Art for a sculpture of the gyroid In my last six months at ERC, I tried to record as much as possible of what I had learned about TPMS in a NASA technical note entitled 'Infinite Periodic Minimal Surfaces Without Self-Intersections'. Meanwhile, I had been commissioned to design and construct an 11-ft.-diameter model of the gyroid minimal surface for the Museum of Modern Art in New York City, where an Art and Mathematics exhibition was scheduled to open in mid-1970. Here's how the commission came about: Arthur Drexler, Director of the Department of Architecture and Design at MOMA, having heard about the gyroid from one of my colleagues, visited Cambridge in the late summer of 1969 to inspect my collection of minimal surface models. He immediately chose the gyroid as the surface he would like to see me sculpt for the exhibition. Dr.Van Atta then telephoned NASA headquarters and almost overnight obtained a grant of $25,000 to support the project. My friend Keto Soosaar, an expert structural engineer at MIT, introduced me to his colleague Jeannie Freiburghouse, an experienced Fortran programmer, and I immediately hired her to use the Weierstrass integrals to compute the coordinates of 8000 points on a hexagonal patch of the gyroid that corresponds to a face of the Coxeter-Petrie regular map {6,4|4}. I arranged for ERC to award a contract to the Gurnard Engineering Corporation of Beverly, Massachusetts to manufacture two CNC-milled aluminum dies for vacuum-forming two kinds of thin zinc-alloy modules — one in the shape of the hexagon of {6,4|4} and the other in the shape of the quadrilateral of {4,6|4}, its dual. These two kinds of module were to be joined by epoxy, in a face-to-face, overlapping arrangement, resulting in a design in which one side of the gyroid surface is tiled by hexagons while the other side is tiled by quadrilaterals. Such a design avoids the need for unsightly connectors along module edges. (I recognized that it would be more efficient to use a single module shape based on the hexagon of the self-dual map {6,6|3}, which is twice as large as that of {6,4|4}, but I didn't pursue this idea, because I knew from experience that the larger negative draft angle of such a big curvaceous module would probably prevent successful vacuum-forming, by introducing ugly ridge-like 'wrinkles' in each module.) In the spring of 1970, NASA funding for the project was abruptly subjected to a special kind of 'mid-course correction': it was cancelled. On investigation, I was informed that at a retirement party — presumably well lubricated — for the senior ERC comptroller responsible for my MoMA account, someone had 'accidentally hit the wrong key on his computer', with the result that the money still left in the account was sent back to Washington (i.e., NASA headquarters). The new heir to the comptroller's office told me that there was no way to recover this money. Having had earlier experience with bureaucracies, I recognized that the project was finished. (But see Figs. E1.18a-d below.) Dr. Van Atta resigned from ERC in the autumn of 1969 to become research vice-president of the University of Massachusetts/Amherst. Months later my colleagues and I guessed that he might have received early warning signals about the impending demise of ERC. His successor, Lou Roberts, an able electrical engineer and administrator, generously arranged for the last remaining technical typist in our division to be assigned the single task of typing my technical note, but the deadline was so tight that much of what I wrote was litle more than a first draft, since I had no opportunity for either proper editing or for review by another person. (Personal computers had not yet been invented. If computer work stations that allowed for some kind of word-processing existed in those days, I never heard of them.) Immediately after I submitted my manuscript to NASA, I handed in a list of typos and other errors for final corrections. Although I was promised that they would be dealt with, they were not. The one hundred complimentary copies I was promised turned out to be three copies. We all had the feeling that we were now ancient history, and nobody much cared. (Perhaps that is the way it always is with institutions that are in their death throes.) §44. A mathematical conference in Tbilisi In August 1969, before I had any suspicion that my sojourn at NASA would soon end, I received an invitation — thanks to the kind intervention of the mathematicians Robert Osserman and Lipman Bers — to describe my research in a post-deadline presentation at a September conference in Tbilisi, Georgia, USSR on Optimal Control Theory, Partial Differential Equations, and Minimal Surfaces. The conference chairman was Revaz Gamkrelidze of the Steklov Institute. When I flew to the USSR consulate in Washington to apply for a visa, the apparatchik in charge at first turned me down, using the excuse that there wasn't enough time. Just at that moment, the distinguished UCLA plasma physicist Burton Fried happened to enter the office. Recognizing me (we had chatted at an APS January meeting a few years earlier), he instantly addressed me by my first name. The apparatchik, who somehow realized that Burt was an important personage, was clearly startled at this show of familiarity. He turned away from me and quickly processed Burt's visa (for an upcoming conference on plasma physics in Russia). Meanwhile, I had retired to a couch a few feet in front of the counter, determined not to give up my own quest for a visa. Once Burt's application was processed, he stopped by the couch for a brief chat and then departed. For the next several minutes, the apparatchik pretended to ignore me while he was shuffling papers at the counter. Finally he looked up and asked, "Why are you still here?" I replied that I expected him to change his mind about my visa application, since he had managed to grant Fried's request in spite of the fact that Fried's schedule was even tighter than mine. Perhaps he was impressed by my skill in what he may have perceived as Marxist dialectics. In any event, he appeared to have a sudden change of heart and grumbled, "Perhaps I can do something for you after all." (This was my first — but not my last — observation of obsequious behavior by a petty Soviet bureaucrat.) When I landed at Vnukovo International Airport in Moscow about ten days later, I was welcomed by Revaz Gamkrelidze, but I still had to get my bulky collection of plastic models of TPMS through customs. When a stolid Ukrainian customs agent showed signs of balking at the sheer number of boxes I had brought with me (I suppose he suspected they contained contraband), Gamkrelidze put on an impressive show of commanding authority. He announced in a magisterial voice (in Russian) that the boxes contained "mee'-nee-mal soor'-fa-cez". The agent echoed in a bewildered voice, "Mee'-nee-mal soor'-fa-cez?" Gamkrelidze replied with great emphasis that the conference would be impossible without them, and that was that. After the conference began, I recognized with dismay that it would have been more sensible to bring fewer models. A few of the very dignified Russian and Western European mathematicians at the conference appeared to be somewhat offended by the sheer quantity of models I had brought. In any event, by the end of the four-day conference, several of the larger models had magically disappeared from the locked auditorium storage room in which they were kept overnight after each day's session, thereby lightening my load on the trip home. The conference was nominally hosted by Lev Pontryagin, the giant of mathematics at the Steklov Institute, and Ilia N. Vekua, the amiable Georgian mathematician who was then Rector of Tbilisi State University. Gamkrelidze had been Pontryagin's doctoral student. Lev Pontryagin                                 Ilia N. Vekua Although I was introduced to Pontryagin, I felt far too intimidated to attempt conversation with him. With Vekua, who was a convivial sort of man, it was another matter. During a banquet at his home one evening, he told me that Southern California was one of his favorite places in the world. After I told him that I had lived in San Diego and L.A. for ten years, he spent the next hour showing me his color slides of the California landscape and telling me stories about his visits to California. Here is the program of the 1969 Tbilisi conference: page 1 page 2 page 3 page 4 page 5 page 6 page 7 (Because I was invited at the last minute, I was not listed in the program.) Once I have digitized my stereoscopic Kodachrome slides of Tbilisi and the surrounding countryside, I will post images here. After the conference, Revaz Gamkrelidze generously arranged a special visit by four or five of us to a small local research institute, where his brother Tamaz, who is a distinguished orientalist, showed us a breathtakingly beautiful treasure that had recently been unearthed in Georgia. It was a tiny sculpture of a chariot and horses, composed entirely of thin gold wires (perhaps less than 1 mm. in diameter). I do not remember exactly how old it was estimated to be, but I vaguely recall hearing that it was about 4000 years old. (If one of my readers has information about this object, please share it with me.) Gamkrelidze invited those of us who were planning to be in Moscow during the week after the conference to attend a party at his Moscow apartment. Since I had a Moscow appointment scheduled at just the right time with V.A. Koptsik, the Lomonosov University specialist in Shubnikov — Belov color symmetry theory, I was able to attend the party. Koptsik graciously arranged for the 78 year-old Nikolai Belov, who had long since retired, to make a special trip to the university so that I could meet him. Belov was one of the truly memorable people I met during my two weeks in the USSR. I regret that I did not have the opportunity to spend more time with him. In his booming voice and heavily accented English, while loudly thumping his chest, he told me that he was "not a Communist, but a Russian!" (I thought that he was being somewhat indiscreet, but perhaps he was confident that he was too distinguished for Brezhnev to bother him.) I shot a stunning pair of stereoscopic photographs of Belov that I am still trying to locate in my cluttered files, because I would like to post them here. Since I am not a particularly skillful photographer, it is all the more remarkable that his portrait looks almost as if it had been taken by Yousuf Karsh. I first met Prof. Koptsik in 1968 at a geometry conference at the Ledgemont Laboratory of Kennecott Copper Co. in Massachusetts. It was organized and hosted by a metallurgical physicist, the late Arthur Loeb, whose specialty was crystallography. Here is the program of the Ledgemont conference: page 1 page 2 page 3 page 4 Prof. Koptsik showed special interest in my model of the minimal surface C(H). He explained that the distribution of the six colors in the model reminded him of a certain color symmetry group. An image of this model is shown below in Figs. E3.3 - E3.5, He invited me to visit him in Moscow, if I ever visited the USSR. (One year later, I did visit him there.) §45. Ken Brakke and his Surface Evolver Kenneth Brakke In 1999, thirty years to the day after the opening day of the Tbilisi conference, I telephoned Ken Brakke, whom I had met in 1991 at a University of Minnesota conference on minimal surfaces, to ask if he would like to collaborate on an illustrated book about TPMS. He replied that he would be interested in doing the illustrations for such a book, but not in writing the text. The book never materialized, because I never got around to writing it. Instead, over the next few years I occasionally sent Ken adjoint surface data derived mainly from soap film experiments carried out between 1969 and 1974. Using his powerful Surface Evolver program to 'kill periods', Ken quickly produced and posted online images of each conjectured surface. In a small fraction of the cases, he found that the hypothetical embedded surface does not exist. But my proposal to write a book with Ken evaporated. That wasn't his fault! I just decided that examining Ken's beautiful computer graphics images and reading his commentary was much more enjoyable than writing a book would have been. In 2010, at the instigation of my friend the mathematician Jerzy Kocik, I started this website, which will probably continue to grow for a while. With Ken's permission, I have included here a few examples of his images of TPMS, but I encourage you to visit Ken's set of websites. They're vastly more orderly than the collection of oddments below, and his ilustrations are supplemented by all sorts of information about a variety of other topics in geometry. §46. Research conferences Stephen T. Hyde and Gerd E. Schröder-Turk have effectively summarized the state of our understanding in 2012 of the role of triply-periodic minimal surfaces in chemistry and biology in their introductory review article, Geometry of interfaces: topological complexity in biology and materials, which was published in the Royal Society's Interface Focus (2012) 2, 529-538. Here is the Table of Contents for that journal volume. In October 2012, my wife Reiko and I attended the Primosten, Croatia conference, organized by Hyde and Schröder-Turk, on which these conference papers are based. In my own presentation, entitled Reflections concerning triply-periodic minimal surfaces, I described how I came to be involved in the investigation of minimal surfaces, beginning in 1966. (There is considerable overlap between some parts of this account and the material on this website.)

§47. Examples of dual skeletal graphs

Skeletal graphs of the two inter-penetrating labyrinths of a TPMS and Voronoi polyhedra that enclose the vertices of the graph are useful for representing the symmetry and topology of some examples of TPMS, especially when the graph edges for each labyrinth are symmetrically equivalent. Of course these geometrical constructions do not yield analytic solutions for the surfaces. (All stereoscopic image pairs are arranged for 'cross-eyed' viewing.) 47.1 Dual diamond skeletal graphs Fig. E1.0a A dual pair of diamond skeletal graphs 47.2 Dual simple cubic skeletal graphs Fig. E1.0b A dual pair of primitive cubic skeletal graphs Curiously, the set of vertices of a dual pair of diamond (D) graphs and the set of vertices of a dual pair of primitive cubic (P) graphs are identical. The lattice for this set is b.c.c (body-centered cubic). The Voronoi polyhedron for a vertex in this set is the truncated octahedron. I know of no other example of a pair of directly congruent dual graphs with cubic lattice symmetry. 47.3. Dual diamond graphs (skeletal graphs of the D surface) A triply-periodic minimal surface (TPMS) that is embedded, i.e., free of self-intersections, partitions space into a pair of disjoint labyrinths. If the labyrinths are congruent — either directly or oppositely — the surface is called balanced. The D (diamond) surface of H.A. Schwarz, which is a balanced TPMS, is shown in Fig. E1.1a. Fig. E1.1a Four translation fundamental domains of Schwarz's D surface (genus 3) For some purposes it is convenient to represent a TPMS of low genus by a surrogate with plane faces — a triply-periodic polyhedron with the same symmetry and topology as the surface. The two labyrinths of the TPMS and of its surrogate may be represented by a pair of triply-periodic skeletal graphs that have the same symmetry as the TPMS and its surrogate. Every edge in these graphs joins a pair of vertices that lie at centers of symmetry of the TPMS. A simple example of a surrogate of the D surface is the triply-periodic quasi-regular skew polyhedron (6.4)2 (cf. Fig. 1.1b), which is derived from the Coxeter-Petrie regular skew polyhedron {6,4|4} (cf. Fig. E1.35c). Quasi-regular polyhedra are edge-transitive, but not face-transitive. Fig. E1.1b (6.4)2, a surrogate of Schwarz's D surface (cf. Fig. E1.1a). The red and green skeletal graphs are both replicas of the diamond graph. Its edges join the sites of adjacent carbon atoms in diamond. The Voronoi polyhedron of a vertex of the union of the two dual skeletal graphs is the truncated octahedron. The vertices of the dual skeletal graphs in Fig. E1.1b lie at the centers of the chambers of the respective labyrinths. Each chamber in (6.4)2 is a truncated octahedron from which a tetrahedrally arranged subset of four hexagons has been removed. Hence the boundary of (6.4)2 is composed of four regular hexagons and six squares (cf. Fig. E1.1c). Fig. E1.1c The two differently oriented chambers in (6.4)2 (cf. Fig. E1.1b). The four faces incident at each vertex of (6.4)2 are arranged in cyclic order 6.4.6.4 — hence the name (6.4)2. In each of the two labyrinths, there are two differently oriented varieties of chambers. They are related by a quarter-turn about any one of the three Cartesian axes. In each labyrinth, adjacent chambers related by a translation of type [111] are of opposite variety. Fig. E1.1d Fig. E1.1e Fig. E1.1f Fig. E1.1g Like (6.4)2, the Coxeter-Petrie triply-periodic regular skew polyhedron {6,6|3} (cf. Figs. E1.2a,c) has the same topology and symmetry as Schwarz's diamond surface D (cf. Fig. E1.2b). In this example, in contrast to the case of (6.4)2, the unit cell has only one orientation (cf. Fig. E1.2c), since it is a translation fundamental region. Fig. E1.2a A lattice fundamental region of {6,6|3} The lattice is fcc. Fig. E1.2b A lattice fundamental region of Schwarz's diamond surface D Fig. E1.2c Thirteen lattice fundamental regions of {6,6|3} Dual pairs of diamond graphs are shown below in both 'medium thick' and 'thick' versions. In the thick version (cf. Figs. E1.3e-h), the diameter d of the cylindrical tubes is the largest possible, consistent with the requirement that the dual graphs not overlap. Overlap occurs when the ratio d/e ≥ 21/2/2 (~.707), where e is the edge length of an ideally thin skeletal graph. (For enlarged views, select the hyperlinks just below the images.) Medium thick diamond graphs view along ~(100) direction Fig. E1.3a                        Fig. E1.3b                        Fig. E1.3c       graph 1                              graph 2                       graphs 1 and 2 Fig. E1.3d Orthogonal projection of graphs 1 and 2 on [111] plane Thick diamond graphs view along ~(111) direction Fig. E1.3e                        Fig. E1.3f                        Fig. E1.3g       graph 1                              graph 2                       graphs 1 and 2 Fig. E1.3h Orthogonal projection of graphs 1 and 2 on [111] plane Fig. E1.3i (6.4)2 — a triply-periodic quasi-regular polyhedron that has the same topology and symmetry as Schwarz's diamond surface D Fig. E1.3j (6.4)2 with embedded [skinny] dual graphs Fig. E1.3k (6.4)2 with embedded [fat] dual graphs 47.4. Dual simple cubic graphs (skeletal graphs of the P surface) Dual pairs of simple cubic graphs are shown below in both 'medium thick' and 'thick' versions. In the thick version (cf. Figs. E1.4e-h), the diameter d of the cylindrical tubes is the largest possible, consistent with the requirement that the dual graphs not overlap. Overlap occurs when the ratio d/e ≥ 1/2, where e is the edge length of an ideally thin skeletal graph.

Medium thick simple cubic graphs oblique view Fig. E1.4a                        Fig. E1.4b                        Fig. E1.4c       graph 1                              graph 2                       graphs 1 and 2 Fig. E1.4d Orthogonal projection of graphs 1 and 2 on [100] plane Thick simple cubic graphs oblique view Fig. E1.4e                        Fig. E1.4f                        Fig. E1.4g       graph 1                              graph 2                       graphs 1 and 2 Fig. E1.4h Orthogonal projection of graphs 1 and 2 on [100] plane Fig. E1.4i The Coxeter-Petrie triply-periodic regular skew polyhedron {6,4|4}, which has the same topology and symmetry as Schwarz's primitive surface P Fig. E1.4j {6,4|4} with embedded [skinny] dual graphs Fig. E1.4k {6,4|4} with embedded [fat] dual graphs

{6,4|4} is a is an infinite regular polyhedron. There is only one variety of chamber — a truncated octahedron whose six square faces have been removed. Hence it is bounded by eight regular hexagons. At the center of every chamber there is a vertex of one of the dual skeletal graphs illustrated in Figs. E1.4a-h.

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47.5. Dual Laves graphs (skeletal graphs of the G surface)

The two intertwined skeletal graphs of the gyroid, in a Shapeways 3D printed version designed by virtox. Click here for a video of 'Bones', an animated view of these graphs, also made at Shapeways by virtox. In the past fifteen or twenty years, I've noticed that many writers have adopted terminology that is obviously useful for distinguishing (a) the actual gyroid surface from (b) one or the other of the two swollen skeletal Laves graph surfaces that are illustrated above in the images of the intertwined 3D printed surfaces. According to this convention, the 'hermaphroditic' minimal surface I named the gyroid in 1968 is instead referred to as the double gyroid, while each of the swollen Laves graphs, both of which are chiral, is called either gyroid or single gyroid. Although this convention may at first sight appear confusing, it has the advantage of being both orderly and concise. A good example of this usage can be found in John Baez's blog, Azimuth for August 2015. Dual pairs of Laves graphs are shown below in 'thin', 'medium thick', and 'thick' versions. In the thick version (cf. Figs. E1.5i-l), the diameter d of the cylindrical tubes is the largest possible, consistent with the requirement that the dual pair of tubular graphs not overlap. Overlap occurs when the ratio d/e ≥ 31/2/2 (~.866), where e is the edge length of an ideally thin skeletal graph. The fact that the d/e ratio is significantly larger for the pair of thick Laves graphs than it is for the thick simple cubic and thick diamond graph pairs suggests (but does not prove) that the thick Laves graphs occupy a larger fraction of space. In order to make the comparison precise, it would be necessary to take into account the detailed geometry in the neighborhood of the intersections of the cyclindrical tubes. I have not done this. In order to display the pairs of intertwined graphs as clearly as possible, views are shown for each of the three principal 'crystallographic' directions: [100], [111], and [110].

Thin Laves graphs [100] view Fig. E1.5a                        Fig. E1.5b                        Fig. E1.5c       graph 1                              graph 2                       graphs 1 and 2 Fig. E1.5d Orthogonal projection of graphs 1 and 2 [100] view Medium thick Laves graphs [100] view Fig. E1.5e                        Fig. E1.5f                        Fig. E1.5g       graph 1                              graph 2                       graphs 1 and 2 Fig. E1.5h Orthogonal projection of graphs 1 and 2 [100] view Thick Laves graphs [100] view Fig. E1.5i                        Fig. E1.5j                        Fig. E1.5k       graph 1                            graph 2                       graphs 1 and 2 Fig. E1.5l Orthogonal projection of graphs 1 and 2 [100] view Thin Laves graphs [111] view Fig. E1.5m                Fig. E1.5n                Fig. E1.5o       graph 1                      graph 2               graphs 1 and 2 Fig. E1.5p Orthogonal projection of graphs 1 and 2 view: [111] Medium thick Laves graphs [111] view Fig. E1.5q                Fig. E1.5r                Fig. E1.5s       graph 1                      graph 2               graphs 1 and 2 Fig. E1.5t Orthogonal projection of graphs 1 and 2 [111] view Thick Laves skeletal graphs [111] view Fig. E1.5u                Fig. E1.5v                Fig. E1.5w       graph 1                      graph 2               graphs 1 and 2 Fig. E1.5x Orthogonal projection of graphs 1 and 2 view: [111] Thin Laves skeletal graphs [110] view Fig. E1.6a                     Fig. E1.6b                     Fig. E1.6c       graph 1                           graph 2                      graphs 1 and 2 Fig. E1.6d Orthogonal projection of graphs 1 and 2 view: [110] Medium thick Laves skeletal graphs [110] view Fig. E1.6e                Fig. E1.6f                Fig. E1.6g       graph 1                      graph 2               graphs 1 and 2 Fig. E1.6h Orthogonal projection of graphs 1 and 2 view: [110] Thick Laves skeletal graphs [110] view Fig. E1.6i                    Fig. E1.6j                    Fig. E1.6k       graph 1                           graph 2                   graphs 1 and 2 Fig. E1.6l Orthogonal projection of graphs 1 and 2 view: [110] Fig. E1.7 Straw model of the pair of dual Laves skeletal graphs (1960) view: [100] Additional stereo images of the Laves graph Fig. E1.8 The 'clockwise' Laves graph, skeletal graph of one labyrinth of the G surface Fig. E1.9 The 'counter-clockwise' Laves graph, skeletal graph of the other labyrinth of the G surface Fig. E1.10 The enantiomorphic skeletal graphs of the two disjoint labyrinths of the G surface Fig. E1.11 Orthogonal projection on [100] plane of the enantiomorphic Laves graphs Fig. E1.12 Another view of the 'counter-clockwise' Laves graph 47.6. Other pairs of dual skeletal graphs Fig. E1.13a Fig. E1.13b The dual skeletal graphs of a hypothetical but nonexistent embedded TPMS called TO-TD (TO stands for truncated octahedron, the interstitial cage of the blue graph. TD stands for tetragonal disphenoid, the interstitial cage of the orange graph.) blue graph: degree 4 orange graph: degree 14 The blue graph is symmetric. The orange graph is regular but not symmetric. The edges of the blue graph are all symmetrically equivalent, but the edges of the orange graph are clearly not all symmetrically eqivalent. They are not even all of the same length. If TO-TD existed, it would be a non-balanced TPMS— i.e., its two labyrinths would be non-congruent. Hence there could be no straight lines embedded in the surface, since such lines are c2 axes and would have the effect of interchanging the two labyrinths. Instead, the surface would be tiled by replicas of a patch S bounded by curved geodesics that — because of the reflection symmetries of the union of the blue and orange graphs — are mirror-symmetric plane lines of curvature. In 1974, I tested for the existence of TO-TD by using a laser-goniometer method I had devised in 1968 at NASA/ERC. This extremely tedious method requires the construction of a set of several straight-edged boundary frames of various proportions. The laser is used to measure the orientation of the normal to the surface of a [long-lasting] polyoxyethylene soap film S' bounded by each of these frames at many points that are as close as possible to the edges of the frame. Each S' is a candidate for the surface adjoint to S. The adjoint curves computed for the edges of S demonstrated that it is impossible to 'kill the periods' and therefore that TO-TD does not exist. In 2001, Ken Brakke used his Surface Evolver program to confirm this conclusion with enormously greater speed and accuracy than is possible with the soap film-laser technique. Fig. E1.13c The dual pair of skeletal graphs for another hypothetical but nonexistent embedded TPMS The green vertices define the sites of the Cu atoms, and the blue vertices define the sites of the Mg atoms in the binary alloy Cu2Mg, which has the structure called Cubic Laves phase C15.

I call the Cu graph FCC6(II). The Mg graph is the diamond graph (cf. Fig. E1.3d). Both graphs are symmetric. The interstitial cavities in the Cu graph are of two kinds: small tetrahedral cages and large truncated tetrahedral cages. All the interstitial cavities in the Mg graph are identical: the expanded regular tetrahedron (ERT) (cf. Fig. E2.19c, E2.20). In 2001, Ken Brakke used his Surface Evolver program to demonstrate that it is impossible to kill periods for this hypothetical surface. Hence it is almost certainly safe to conclude that the surface does not exist. Note that in this example, in contrast to other pairs of dual graphs treated here, it is not true that for both graphs, every interstitial cavity of the graph contains a vertex of the dual graph. (The small tetrahedral cages of the Cu graph do not contain any vertex of the Mg graph.) §48. TPMS mathematical background The Gauss map Fig. E2.1 (stereo image) Curved triangular Flächenstück ABC of Schwarz's Diamond surface Fig. E2.2 (stereo image) Riemann sphere (unit sphere) The elementary minimal surface Flächenstück ABC shown in Fig. E2.1 is mapped onto the spherical triangle ABC on the Riemann sphere shown in Fig. E2.2 by the Gauss map. Each point on the minimal surface is mapped onto a point on the Riemann sphere that has the same normal vector. The red arrows at points A, B, and C indicate the directions of the surface normal vectors. There are twelve replicas of the Flächenstück ABC in the skew hexagonal face EDAE'D'A' of Schwarz's D surface (cf. Fig. E2.1), but there are only six corresponding spherical triangles in the large spherical triangle AED on the Riemann sphere (cf. Fig. E2.2). The two Flächenstücke ABC and A'B'C, for example, are both mapped onto the same spherical triangle. An entire lattice fundamental region covers the Riemann sphere twice. As a consequence the mapping defines a two-sheeted Riemann surface, with branch points at the eight 'cube corner' points like C. Fig. E2.3a Stereographic projection onto the complex plane of the elementary triangular Flächenstück ABC of Fig. E2.2 Fig. E2.3b

Triply periodic minimal surfaces are infinitely-multiply-connected, but it is nevertheless easy to characterize the topological complexity of every example of such a surface by computing the genus p of a single lattice fundamental domain. Except where it is specifically stated to the contrary, it will be assumed in all that follows that TPMS refers to an embedded surface, i.e., one that is free of transverse self-intersections. Since the smallest posssible value for the genus is three, Schwarz's P and D surfaces are members of a very small select group of topologically simplest examples of TPMS. Below is a recipe for computing the genus of a TPMS. It is based on one of Gauss 's most astounding discoveries, the Gauss-Bonnet theorem, which links the topology and the geometry of a surface. One can very crudely express the essence of the Gauss-Bonnet theorem in this context by saying that the larger the value of the integrated Gaussian curvature for one lattice fundamental region of the surface, the steeper the saddle-like surface contours, and — therefore — the larger the number of tubular 'handles' in the surface as it 'bends around' this way and that. On p. 233 of the 13th edition of 'Mathematical Recreations and Essays' by W.W. Rouse Ball and H.S.M. Coxeter, the authors use Euler's formula F − E + V = 2, which relates the number F of faces, the number E of edges, and the number V of vertices of a convex polyhedron to prove that adding a handle to an orientable surface reduces the Euler-Poincaré characteristic Χ = 2 − 2p by 2 and therefore increases the genus p by 1. The proof simply updates the values of F, E, and V after two different n-gons of a map on the surface are joined by a 'bent prism' (which is a convenient device for representing a handle). F is increased by n − 2, E is increased by n, and V remains unchanged. Since X = F − E + V, Χ is reduced by 2. Fig. E2.4 Recipe for calculating the genus of one lattice fundamental domain of a TPMS, applied to Schwarz's P surface Another way to calculate the genus is to substitute for dG from Eq. 5 in the equation p =1− dG. |dG| is equal to the number of times the Gauss map of the minimal surface ('Gauss image') covers the Riemann sphere. For Schwarz's P surface, dG= − 2. The sign of dG for surfaces of negative Gaussian curvature, like minimal surfaces, is negative because the sense of a geodesic edge-circuit on the surface is opposite to that of its Gauss image on the Riemann sphere. If you're not familiar with the Gauss map, look here. For discussion of the Euler-Poincaré characteristic Χ=2 − 2p, look here. For information about the Gauss-Bonnet theorem, look here.

§49. The D-G-P family of associate minimal surfaces Fig. E3.1a A page from Schwarz's Collected Works 49.1 H. A. Schwarz's diamond surface D

In 1966 I named this surface 'diamond' because both of its interwined labyrinths, which are congruent, have the shape of an inflated tubular version of the familiar diamond graph (cf. Figs. E1.3d to E1.3k). Below are three of H. A. Schwarz's illustrations of the D surface in in his Gesammelte Mathematische Abhandlungen, Springer Verlag, 1890.                        Fig. E3.1b                                            Fig. E3.1c Fig. E3.1d Fig. E3.1e Stereo view of the linear asymptotics embedded in the 'crossed triangles D-catenoid' wire-frame of Schwarz's diamond surface D (cf. Fig. E3.1d) The ratio 2h/λ of the triangle separation 2h to the triangle edge length λ is equal to √ 6 / 6 (~.408). On p. 105 of Part I of his Collected Works (published in 1890), Schwarz comments as follows: It appears that for arbitrary values of the separation of the bounding triangles, the equations of these surfaces [D and P] cannot be expressed as elliptic functions of the coordinates. Fig. E3.1f Orthogonal projection of the linear asymptotics embedded in the 'crossed triangles D-catenoid'. Fig. E3.1g Four translation fundamental domains of Schwarz's D surface Each face is one of the hex90 faces shown below in Fig. E3.1h.

The lattice for D is face-centered cubic (fcc), and the translation fundamental domain has genus 3. The edges of the skeletal graph of degree 4 for each of the two congruent labyrinths correspond to nearest-neighbor links in the diamond crystal structure. Fig. E3.1h The hexagonal face hex90 of D is defined by the Coxeter map {6,4|4}. Its face angles are 90º, and its area is half the area of the hexagonal face hex60 defined by the Coxeter map {6,6|3}, shown below in Fig. 3.1i. Fig. E3.1i The hexagonal face hex60 of D is defined by the Coxeter map {6,6|3}. Its face angles are 60º, and its area is twice the area of the hexagonal face hex90 defined by the Coxeter map {6,4|4}, shown above in Fig. 3.1h. Fig. E3.1j A rhombic dodecahedral translation fundamental domain of Schwarz's diamond triply periodic minimal surface D As a toy model for generating the 'pipejoint' module of D in Figs. E3.1j, k, l, imagine that you are inside a spherical soap bubble at the center of a rhombic dodecahedron. Deform the bubble by blowing toward its interior surface in the four tetrahedral directions [1,1,1], [-1,-1,1], [-1,1,-1], [1,-1,-1] simultaneously, forming four cylindrical tubules attached symmetrically to the inside faces of the rhombic dodecahedron around four of its eight trigonal corners. Fig. E3.1k A stereo image of the translation fundamental domain of D in Fig. 3.1j larger image Fig. E3.1l A translation fundamental domain of D on which approximations to closed geodesics (the red curves) are inscribed. These geodesics are not plane curves. larger image Fig. E3.1m A regular skew curvilinear hexagon of D, which is a face of the regular map with holes {6,6|3} larger image The inscribed regular skew hexagon with straight edges is a face of {6,4|4}, one of the three regular maps with holes described in Generators and Relations for Discrete Groups, H.S.M. Coxeter and W.O.J. Moser, Springer-Verlag, New York, 1965 and in Infinite Periodic Minimal Surfaces Without Self-Intersections, NASA TN D-5541, p. 49. 49.2 H. A. Schwarz's primitive surface P In 1966 I named this surface primitive because its two interwined labyrinths, which are congruent, each have the shape of an inflated tubular version of the familiar primitive (or simple cubic) graph (cf. Figs. E3.3d to E3.3k). D and P are adjoint surfaces: each surface can be mapped into the other by an isometry (the Bonnet transformation). Straight lines in one surface are mirror-symmetric plane lines of curvature (plane geodesics) in the other. Fig. E3.2a Stereoscopic image of the linear asymptotics (blue) and plane geodesic curves (green) in the 'square catenoid' of P (cf. Fig. 1.2d, e, f) Fig. E3.2b Stereoscopic image of the linear asymptotics embedded in the 'crossed triangles P-catenoid' wire-frame of Schwarz's primitive surface P (cf. Figs. 1.2c, d, e) Here the triangles are only half as far apart as the crossed triangles in Schwarz's D surface (cf. Figs. 1.1d, e, f). The ratio h/λ of the triangle separation h to the triangle edge length λ is equal to √ 6 / 12 (~.204). The P and D surfaces are related by a dilatation along any of the four [111] directions. An annular 'crossed-triangles catenoid' (CTC) minimal surface exists for every value of h/λ less than some allowed maximum value (h/λ)max, but there are embedded straight lines only in the CTCs of D and P. The proof depends in part on Schoenflies's theorem (cf. Fig. E2.1), which proves that there are only six skew quadrilaterals, spanned by minimal surfaces, that generate TPMS by half-turn rotations about their edges, i.e., by repeated applications of Schwarz's reflection principle. Fig. E3.2c Orthogonal projection of the linear asymptotics in the 'crossed triangles P-catenoid' of Fig. E3.2b Fig. E3.2d A cubically symmetrical translation fundamental domain of the primitive triply periodic minimal surface P discovered and analyzed by H. A. Schwarz in 1866 together with its adjoint surface D (cf. Fig. E3.1h). Fig. E3.2e A translation fundamental domain of P stereo image Fig. E3.2f Six translation fundamental domains of Schwarz's P surface

The lattice for P is simple cubic (sc), and the translation fundamental domain has genus 3. If each pair of opposite holes were joined by a hollow tube, the translation fundamental domain would be transformed into an object that is homeomorphic to a sphere with three 'handles'. P is the unstable stationary state of an inflated jungle-gym-like soap film. Any finite portion of such a soap film can be made stable if threads are stretched along a sufficient number of the embedded straight lines ('linear asymptotics'). As a sort of metaphor for the 'pipejoint' module of P in Figs. E3.2d, e, imagine that you are inside a spherical soap bubble at the center of a cube. Now deform the bubble by blowing against its interior surface in the six directions x, −x,   y, −y,  z, −z simultaneously, forming six cylindrical tubules attached symmetrically to the inside of the cube faces at their centers. P partitions R3 into two congruent inter-penetrating labyrinths. The skeletal graph (NASA TN D-5541, pp. 38-39) of each labyrinth is the graph of degree 6 whose edges are those of a packing of congruent cubes. In Figs. E1.4a-h below are images of tubular simple cubic graphs shown in both 'medium thick' and 'thick' versions. In the thick version, the diameter d of the cylindrical tubes is the largest possible, consistent with the requirement that the dual pair of tubular graphs not intersect. Intersection occurs when the ratio d/e ≥ 1/2, where e is the edge length of the [thin] skeletal graph. 49.3 The gyroid surface G Fig. E3.3a A lattice fundamental domain of the gyroid G (100) viewpoint The Coxeter-Petrie {6,4|4} map defines the arrangement of the hex90 hexagonal faces. G has the same symmetry as that of the union of its two enantiomorphic skeletal Laves graphs. The lattice is bcc. G is the only embedded surface among the countable infinity of surfaces that are associates (cf. Fig. E1.2m) of Schwarz's P and D surfaces. G contains neither straight lines nor plane geodesics. Every hex90 face is related to each of six faces with which it shares an edge by a half-turn about an axis of type (110) perpendicular to G at the midpoint of the shared edge. Fig. E3.3b Stereo view of the lattice fundamental domain of G shown in Fig. E3.3a higher resolution image (100) tunnels in G Fig. E3.3c A ninth hex90 face has been added here at the top of the piece of G shown in Figs. E3.3a,b. This orthogonal projection of G onto the (100) plane shows that the projected outline S of the spiralling geodesic that bounds each (100)-type tunnel in G is approximately circular. For a higher-resolution version of this image, look here. For a stereoscopic perspective view, look here. (111) tunnels in G Fig. E3.3d This orthogonal projection onto the [111] plane of the piece of G in Fig. E3.3a demonstrates that the (111) tunnels are fatter than the (100) tunnels and not so nearly circular. For a high-resolution view, look here. For a stereoscopic perspective view, look here. Fig. E3.3e The hexagonal tile hex90 of G (front view) The regular skew hexagon hex90 is a face of the Coxeter-Petrie {6,4|4} map. Its face angles are 90º, and its area is one-half the area of the hexagonal face hex60 defined by the Coxeter-Petrie {6,6|3} map. (The hexagon hex60 is shown in Figs. 3.3m, n.) Fig. E3.3f The hexagonal face hex90 of G (back view) The front and back surfaces are not the same! Fig. E3.3g The hexagonal face hex90 of G (side view) Fig. E3.3h This image suggests that hex90 of G can be inscribed in a truncated octahedron, but that is impossible. Although the vertices of hex90 coincide with six vertices of the truncated octahedron, its edges are not plane curves. Each edge approximates the shape of a quarter-pitch of a helix. One half of each edge lies inside the truncated octahedron, and the other half lies outside. Alternate edges are curves of opposite handedness. G contains one replica of hex90 in three of every four truncated octahedra in a packing of truncated octahedra. Fig. E3.3i The quadrangular tile quad60 of G, a regular skew polygon (front view) Fig. E3.3j The quadrangular tile quad60 of G (back view) Fig. E3.3k The quadrangular tile quad60 of G (side view) The tile quad60 is a face of the Coxeter-Petrie {4,6|4} map, which is the dual of {6,4|4}. Its face angles are 60º, and its area is equal to two-thirds that of hex90. Every quad60 face is related to each of four faces with which it shares an edge by a half-turn about an axis of type (110) perpendicular to G at the midpoint of the shared edge. Each of these midpoints is also the midpoint of an edge of a dual hex90 face (cf. text below Fig. E3.3a). (Fig. E3.3l illustrates a well-known property of dual regular tilings of the plane: the midpoints of edges of dual polygons coincide. Not surprisingly, this property holds for dual regular polyhedra as well.) Fig. E3.3l A pair of dual regular graphs in the plane Points like P lie at the coincident midpoints of a pair of triangle and hexagon edges that intersect. Fig. E3.3m A semi-regular skew 12-gon composed of six replicas of quad60 of G Its face angle sequence is ..., 60º, 120º, 60º, 120º,... Unlike the three regular skew polygons quad60, hex90, and hex60, this 12-gon does not tile G. For a high-resolution view, look here. Fig. E3.3n The hexagonal face hex60 of G is defined by the self-dual Coxeter map {6,6|3}. Its face angles are 60º, and its area is twice the area of the hexagonal face hex90 shown above in Figs. 3.3e,f,g. Fig. E3.3o The hexagonal face hex60 of G (side view) Every hex60 tile is related to each of six faces with which it shares an edge by a half-turn about an axis of type (110) perpendicular to G at the midpoint of the shared edge. Fig. E3.3p Assembly of approximately octahedral shape tiled by the hexagonal faces hex90 of G view: (100) direction Fig. E3.3q Another view of the model of G shown in Fig. E3.3p view: (111) direction §50. More images of the three Coxeter-Petrie infinite regular skew polyhedra After I met Norman Johnson in June, 1966, I realized that each of these three polyhedra is homeomorphic to — and has the same symmetry as — either the Schwarz surface D or the Schwarz surface P. {6,6|3} is homeomorphic to D, while {6,4|4} and {4,6|4} are homeomorphic to P. Regular skew polyhedron {6,6|3} (homeomorphic to D) Fig. E3.2g Translation fundamental domain The lattice of the oriented surface is fcc Fig. E3.2h Assembly of thirteen translation fundamental domains Regular skew polyhedron {6,4|4} (homeomorphic to P) Fig. E3.2i Translation fundamental domain The lattice of the oriented surface is sc Fig. E3.2j Assembly of twenty-seven translation fundamental domains Regular skew polyhedron {4,6|4} (homeomorphic to P) Fig. E3.2k translation fundamental domain The lattice of the oriented surface is sc Fig. E3.2l Assembly of twenty-seven translation fundamental domains For a discussion of the Coxeter-Petrie regular skew polyhedra, which includes animated graphics, see this Wikipedia article. §51. My early interest in TPMS (1966-1970) In the spring of 1966, I accidentally 'discovered' the Schwarz surfaces P and D and then observed their close connection to the Coxeter-Petrie regular skew polyhedra. In order to represent the symmetry and combinatorial structure of both the surfaces and their flat-faced relatives, the Coxeter-Petrie polyhedra, I employed the metaphorical device of dual skeletal graphs, which I'll call g1 and g2. These are triply-periodic graphs regarded as lying centered in the interiors of the two intertwined labyrinths of these structures. In the discussion that follows, it is assumed (although not stated!) that g1 and g2 are either directly or oppositely congruent. If this restriction is dropped, the assumption (stated below) that the two labyrinths each contain exactly half of space in their interiors must also be dropped. Now imagine that every edge of g1 is replaced by an infinitely thin hollow tube with walls composed of some soap-film-like material, and that the space inside the entire connected network of these tubes defines a single hollow — but shrunken — labyrinth t1. (Assume that t1 has no self-intersections. Remember: this is a metaphorical concept, not a rigorous mathematical construction.) Here is how I described the relation between g1 and g2, the two skeletal graphs of a TPMS in 1970, on p. 79 of Infinite Periodic Minimal Surfaces Without Self-Intersections: "Assume that the skeletal graph is given for one labyrinth of a given intersection-free TPMS. Let each edge of the skeletal graph be replaced by a thin open tube, and let these tubes be smoothly joined (without [self]-intersections) around each vertex so that the whole tubular graph forms a single infinitely multiply-connected surface, which contains the skeletal graph in its interior. Such a tubular graph is globally homeomorphic to the corresponding minimal surface. If the tubular graph is sufficiently "inflated", it becomes deformed into a dual tubular graph which contains in its interior the skeletal graph of the other labyrinth of the surface. The "outside" of the first tubular graph is the "inside" of the second tubular graph. The two tubular graphs of a given TPMS are required to have the same space group as the TPMS, and to correspond, respectively, to two tubular graphs which are globally homeomorphic to the TPMS." Now imagine inflating t1 so that at its summit,         (i)    in its interior it contains exactly half of all space,         (ii)   its surface has zero mean curvature everywhere,         (iii)  it has the same symmetry as the configuration of the two dual skeletal graphs g1 and g2, and         (iv)  it has no self-intersections. At the inflation summit, the surface t1 is an embedded TPMS. Until it reaches the summit, t1 exhibits the symmetry of g1. At the summit, it has the symmetry of the union of the [intertwined] g1 and g2. As the inflation proceeds beyond the summit, the surface exhibits the symmetry of g2. It eventually shrinks down to the thin tubular graph t2, which has the symmetry of g2. Curiously, the outside of t1 is transformed into the inside of t2. In early 1969, the distinguished topologist Dennis Sullivan Dennis Sullivan was sharing an office at MIT (just across the street from NASA/ERC) with the mathematics professor Dirk Struik, Dirk J. Struik 1894-2000 In 1967, I had the rare privilege of becoming acquainted with Dirk Struik. We both enjoyed hiking along the nature trails in Concord near my home. When Prof. Struik was more than 100 years old, I attended his lecture on      the history of mathematics at an AMS meeting in Cincinnati. He was in top form. (He lived to be 106 years old.)                                                             who had shown him a draft copy of my Infinite Periodic Minimal Surfaces Without Self-Intersections technical note. Dennis invited me to his office to explain that the transformation of the tubular graphs g1 and g2 is an example of the classical Alexander-Pontryagin duality (which I had never heard of before). For the Schwarz surface P, the skeletal graphs g1 and g2 are identical to the 6-valent simple cubic graph defined by the edges of an ordinary packing of cubes. For the Schwarz surface D, both g1 and g2 are copies of the 4-valent diamond graph, whose edges correspond to the nearest neighbor links in the diamond crystal structure. Both of these graphs are symmetric, i.e., there is a group of symmetries that is transitive on all of its edges and on all of its vertices. I knew of only one other example of a symmetric triply-periodic graph on a cubic lattice — the Laves graph. In the spring of 1966, I was seized by the notion that there must exist a TPMS with [enantiomorphic] Laves graphs for its skeletal graphs. I called it the Laves surface L. Unlike the skeletal graphs of P and D, however, the configuration of two dual Laves graphs has no reflection symmetries, and its axes of rotational symmetry lie in directions that I determined could not possibly correspond to lines embedded in the surface I was seeking. As a consequence, I had no idea how to generate a surface patch bounded by either straight line segments or plane geodesics a ("Schwarz chain"). By the time I moved to NASA in July 1967, I had made a reasonably thorough study of selected parts of Schwarz's writings on periodic minimal surfaces, and I understood the Bonnet associate surface transformation that defines the relation between P and D (and also the relation between the catenoid and the helicoid). The brightly colored plastic models of P and D I had constructed were almost literally screaming out to me that I should explore the territory between these two surfaces (where one surface is bent continuously into the shape of the other), but I did not hear their screams! In February 1968, I stumbled accidentally on a very close approximation of the gyroid. I'll call it the pseudo-gyroid. The models illustrated in Figs. E1.2k, E1.2l, and E1.17 show the final steps in the procedure that led to this pseudo-gyroid. The resemblance between this virtual doppelgänger and the true gyroid is so close that with the naked eye it is impossible to tell them apart. I still had no idea yet that the gyroid is just a surface associate to P and D that happens to be free of self-intersections. In those days there were not yet any known examples of embedded TPMS derived by examining intermediate stages of the 'morphing' transformation that bends one minimal surface into its adjoint surface via the associate surface transformation, and I didn't have the imagination to think of that possibility. In 1990, it occurred to Sven Lidin and Stefan Larsson to look for an embedded surface among the surfaces associate to Schwarz's [embedded] surface H and its self-intersecting adjoint surface, and they found exactly one, which is now known as the lidinoid: The lidinoid (which was originally dubbed 'the HG surface' by its Swedish discoverer, Sven Lidin) Schwarz's H surface To return to the gyroid story, in May 1966 — as mentioned above — I had already begun to suspect that a minimal surface with the symmetry and topology of the gyroid might exist. My suspicions were based on the fact that the Laves graph — like the skeletal graph of each labyrinth in Schwarz's P and D surfaces — is not merely regular (all vertices are of the same degree), but also symmetric (it is both vertex-transitive and edge-transitive). In H and CLP, the two other examples of Schwarz's TPMS, which — like P and D — are of genus 3, the skeletal graphs are merely regular and not symmetric. My intuition suggested that a symmetric graph is so homogeneous that it is very likely to be the skeletal graph of a labyrinth of some embedded TPMS. (I eventually discovered that although some of the few known examples of symmetric triply-periodic graphs are skeletal graphs of labyrinths of such surfaces, by no means all of them are.) During the spring and summer of 1968, I concentrated on the writing of a so-called preliminary report (an internal NASA document, not intended for general circulation), entitled "Expansion-Collapse Transformations on Infinite Periodic Graphs", NASA/Electronics Research Center Technical Note PM-98 (September 1969), draft versions of two patent applications, and computer graphics animations of collapsing graphs. The considerably less time-consuming one of the two patent drafts was eventually entitled, "Honeycomb Panels Formed of Minimal Surface Periodic Tubule Layers". I had discovered no useful ideas about how to prove that the pseudo-gyroid (cf. Fig. E1.17) was the basis for a bona fide minimal surface. Blaine Lawson told me in early August that he too had made no progress toward figuring out how to prove that a skew hexagonal face of the pseudo-gyroid, with its strictly helical edges, could somehow be analytically continued to generate an embedded periodic minimal surface. But ever since my first phone conversation with Blaine in late spring, I had found it extremely helpful to discuss with him a variety of questions concerning minimal surfaces other than the gyroid. I used him as a sounding board on some of my still tentative ideas about how to derive new examples of embedded TPMS by        (a) enumerating all the ways of constructing a Schwarz chain as a connected sequence of arcs on the n faces (one arc on each face) of what is now called a 'Coxeter cell' — a convex polyhedral space-filler related to each of its neighbors by reflection in a face, and, somewhat later,        (b) 'hybridizing' two TPMS (cf. Fig. E2.71). In early September 1968, I returned to Cambridge from an AMS summer meeting at Madison, Wisconsin, where I had used the pseudo-gyroid model shown in Fig. E1.17 to illustrate my 15-minute talk (cf. Fig. E2.10). I was still calling the surface the 'Laves surface' in those days. A souvenir postcard Lake Mendota, from the Wisconsin Union Boat House Madison, Wisconsin (1968) It was at this Madison summer meeting that I met several mathematicians of my father's generation who knew something about minimal surfaces. I particularly enjoyed meeting Wolfgang Wasow, who is shown below in a 1952 photo with Magnus Hestenes. (I met Hestenes in 1969 in Tbilisi, Georgia.) Wolfgang Wasow (left) and Magnus Hestenes (right) H. Blaine Lawson, Jr. About a week after my return to Cambridge, I used the government WATS line to phone Blaine to ask him whether he had made any progress toward a proof that the gyroid is a minimal surface. He replied that he hadn't, because finishing his dissertation had left him little time to think about any other matters. He said he was going to have to abandon work on the problem. I begged him not to give up, because I felt certain the solution was close at hand (even though I had no rational grounds for believing that to be the case!). To change the subject, I told Blaine about the graph collapse transformation I had discovered, and how it could be 'run backwards' to provide the basis for the design of expandable space-frames. I had investigated the transformation for graphs associated with the P, G, and D surfaces (cf. Figs. E2.68b.0, E2.68b.1, and E2.68b.2, for example). I described what I called 'just a coincidence' (or words to that effect): that the trajectory of every graph vertex is an ellipse not only in the associate surface transformation of Bonnet but also in the totally unrelated graph collapse transformation. I emphasized that there is no fundamental connection between these two transformations. I described how I had found that of the triply-periodic graphs that are associated with P and D, either as embedded graphs or skeletal graphs, those that have reflection symmetries are not candidates for expandable space-frames because of pairwise collisions of edges (called webs or struts by space-frame engineers) that occur early in the collapse. In contrast to this behavior, for all of the twisted graphs derived from the pseudo-gyroid, including the Laves graph, no such collisons of edges occur. The only collisions are the ones that would occur in actual physical spaceframes, in which struts collide somewhat before the 'complete collapse' stage because of their finite thickness. I had not previously even mentioned graph collapse to Blaine, and it's hardly surprising that he didn't seem to understand the details of what I said to him. It was obvious that I hadn't explained the elliptical trajectories coincidence very well, because Blaine's response was something like: "Are you saying that the gyroid is associate to Schwarz's P and D surfaces?" I hadn't said that at all, but it hardly mattered, because at that instant, everything suddenly fell into place. The fog had finally lifted! Thanks to Blaine's question, I finally understood that the gyroid is just a surface associate to P and D that happens to be embedded (free of self-intersections). It is the only such surface, as I was soon able to confirm by means of simple 'morphing' sketches similar to the computer drawings in Fig. E1.21. Because I had spent the summer analysing the details of graph collapse transformations on P, D, and G, I was aware that the surface orientation at the vertices of the hexagonal faces of the Coxeter-Petrie map {6,4|4} on the pseudo-gyroid is identical to the surface orientation at the corresponding vertices of P and D. That was a powerful hint pointing to the Bonnet transformation that had been 'staring me in the face' every day since March, when I assembled my first plastic model of the gyroid. I was hugely embarrassed, realizing how obvious it should have been to me that the gyroid is associate to P and D! After all, I was familiar with the properties of the Bonnet transformation. I had long since traced out the geometrical relation between the equatorial circle in the catenoid and the central axis of the helicoid, which I had found illustrated in Dirk Struik's marvellous Lectures on Classical Differential Geometry. I had also sketched the corresponding curves in P and D countless times. Those relations should have been the clue. I had also spent days studying not only H. A. Schwarz's Collected Works, but also Erwin Kreyszig's Differential Geometry, Luther Pfahler Eisenhart's A Treatise on the Differential Geometry of Curves and Surfaces, and Barrett O'Neill's Elementary Differential Geometry. Although I knew from experience that ideas that should be obvious are sometimes anything but obvious, I nevertheless felt stupid when I realized that I had posed the wrong question to Bob Osserman back in March. when I asked him whether there might be a way to derive the Weierstrass parametrization for a Schwarz chain composed of six helical arcs. I had mistakenly assumed that the edges of the hexagonal faces of the {6,4|4} map on the gyroid were perfect helices. Immediately after Jim Wixson joined NASA/ERC in January 1968, I asked him to write a FORTRAN program for calculating — from Schwarz's equations — the coordinates of a set of closely spaced points on the equatorial geodesic of the 'square catenoid' in P. A simple soap-film demonstration suggests that although this curve appears to be approximately circular, it cannot be a circle. Consider its shape in the limit of very small separation of the boundary squares of the 'square catenoid'. In that limit it can be described roughly as a square with slightly rounded corners. As the separation of the boundary squares is increased, the curve looks more and more like a circle, but I found it impossible to imagine that it becomes exactly circular when the separation becomes equal to its value in the P surface. When I plotted the points computed by Jim, I found that the equatorial geodesic departs from perfectly circular shape by slightly less than 0.5%. In October 1968, after understanding at last that the coordinates of every point in a lattice fundamental region of G are a simple linear combination of the coordinates of a pair of corresponding points in D and P, I plotted a graph of the orthogonal projection of the quasi-helical image SG in G of the equatorial geodesic SP in P. I found that this projection of SG (cf. Fig. E3.2b) also departs from perfectly circular shape by slightly less than 0.5%. Fig. E3.2a Stereoscopic view of the linear asymptotics (blue) and plane geodesic curves (green) in the 'square catenoid' of P (cf. Fig. 1.2d, e, f) Fig. E3.2b Orthogonal projection on the [100] plane of the quasi-helical geodesic SG in G (cf. Fig. E3.3c) (I didn't learn about Björling's Strip Theorem until several years after I left NASA/ERC. This theorem proves very simply that the equatorial geodesic SP in Schwarz's P surface cannot be circular.) I felt only slightly less stupid when I discovered that Blaine's response to my harangue about the elliptical trajectories of the vertices of collapsing graphs was not actually the result of his understanding that the gyroid was associate to P and D. He had been justifiably confused by my rambling description of those irrelevant elliptical trajectories. When I explained to him the evidence for the associate surface relationship, he agreed that it was a reasonable idea. I immediately proposed that we publish together an announcement about the gyroid. He courteously refused, explaining that his crucial question to me was prompted by a misunderstanding of what I was saying. But I insisted that if he had not asked me that question in precisely those words, it would have been impossible to say how long it might have taken me to understand what was going on. He then reluctantly agreed to collaborate on a paper about the gyroid. Two days later, Dr. Van Atta returned to ERC from an out-of-town trip. He had been following my struggles with the pseudo-gyroid for months. As soon as I told him my exciting news, including my plan to co-publish with Blaine Lawson, he scolded me in no uncertain terms! He insisted that I phone Blaine and explain that I had made a serious error, and that I must publish alone. (It was the only time Dr. Van Atta ever displayed anger or impatience toward me.) Blaine was courteous when I relayed my new message to him, but I realized that my vacillation must have offended him. Gradually I succeeded in feeling very slightly less stupid than I had at first, after reflecting on the fact that neither Schwarz, Riemanm, Weierstrass, nor any of their successors seem to have suspected the existence of an embedded surface associate to P and D, in spite of the fact that they were all experts on the Bonnet transformation. On the other hand, I realized that it had been pure dumb luck for me to stumble onto M4 and M6, the precursors of the gyroid. I was able to derive the angle of associativity (cf. Fig. E1.23) easily, because I had already made a detailed study of the geometrical calculations Schwarz carried out in his analysis of the P and D surfaces. In January 1968, because I was curious about the precise shape and arc length of the quasi-circular edges of a {6,4|4} hexagon of P (cf. Fig. E1.2c), I sketched the outline of a computer program for getting answers to these questions. My colleague Jim Wixson coded the program in FORTRAN and ran it on ERC's PDP-11 minicomputer. The output of Jim's program, combined with Schwarz's analysis, provided the required clues to the value of the angle of associativity of G (cf. Fig. E1.23). These results demonstrated that the departure from perfect circularity of the quasi-circular holes in a pipejoint unit cell of P is in the range of approximately ± 0.5% of the hole's mean radius, implying a comparably small departure from perfect helicity of their image curves in the gyroid. Not only did the quasi-helical curves in the pseudo-gyroid (cf. Fig. E1.17) turn out to be very close approximations to the corresponding curves in the true gyroid, but these curves are also extremely close to — but not quite the same as — the corresponding curves in the 'level-set' gyroid (cf. Fig. E1.31.) In late 1968, I decided that I must somehow force a nodal polyhedron for BCC6 into being, and by trial-and-error I produced the space-filling saddle polyhedron shown In Figs. E2.70a, b, and c. It is described in Infinite Periodic Minimal Surfaces Without Self-Intersections. Fig. E2.70a BCC6 Pinwheel polyhedron: the 6-faced nodal polyhedron of the deficient symmetric graph BCC6 of degree 6 (stereo pair) The vertices of the graph are the complete set of vertices of the bcc lattice. BCC6 is described on p. 82 of Infinite Periodic Minimal Surfaces Without Self-Intersections. Fig. E2.70b An oblique view of the BCC6 Pinwheel polyhedron (stereo pair) Fig. E2.70c The BCC6 graph (orange) and its dual graph (black) (stereo pair) The edges of the black graph are the edges of the BCC6 Pinwheel polyhedron. A property of the black graph that I regard as not strictly kosher is that it intersects the orange graph. (Perhaps one can find a 'nicer' example of an improvised nodal polyhedron for the BCC6 graph.) Fig. E2.10 Abstract 658-30 submitted to the American Mathematical Society in 1968 announcing the discovery of the gyroid I referred to the gyroid here as 'L' (for 'Laves'). A few weeks later I renamed it 'gyroid'. This announcement was slightly premature! I mailed in the abstract in Fig. E2.10 a month or so before the Madison summer meeting of the AMS, even though I had not yet succeeded in proving that the surface represented by the pseudo-gyroid (cf. Fig. E1.4c) is a single continuous minimal surface. I had naively assumed that any expert on minimal surfaces would be able to construct such a proof. A few months earlier, I had sent a plastic model of the pseudo-gyroid to Bob Osserman, who passed it along to his PhD student Blaine Lawson, Blaine promised to think about the problem in his spare time, even though he was already fully occupied with his dissertation research. It wasn't until about ten days after the Madison meeting that at last I understood — thanks to a fruitful phone conversation with Blaine — that the gyroid is an associate surface in the Schwarz P−D family. (The details are described below in §51.) By means of drawings based on hand calculations, I confirmed that there are no other intersection-free associate surfaces between P and D. A few months later, the differential geometer Tom Banchoff introduced me to Charlie Strauss, the mathematician/computer graphics expert who is his friend and collaborator. I hired Charlie to write a computer graphics program for producing stereoscopic perspective animations, and I used one such animation sequence (cf. Fig E1.21) to strengthen the evidence that every other associate surface is self-intersecting. Twenty-eight years later, this claim was at last proved rigorously by Karsten Grosse-Brauckmann and Meinhard Wohlgemuth, in their article, 'The gyroid is embedded and has constant mean curvature companions', Calc. Var. Partial Differential Equations 4 (1996), no. 6, 499-523. The precise version of the surface that I had in mind at the time of the Madison meeting had a fatal flaw: the boundary of each of its hexagonal faces is a chain of one-quarter pitches of circular helices, alternately right-handed and left-handed (cf. Fig. E2.7). Even now, in 2011, it is not known how to derive an analytic solution for a minimal surface bounded by a circuit ('Schwarz chain') composed of such arcs. Although an assembly of these hexagons looks like a single infinitely-connected minimal surface, it is not one. To explain why I had chosen these helical curves for the surface patch boundary, I summarize below — in approximately chronological order — the tangled sequence of events that culminated in the construction of the model of L. In February 1968 I found my first promising lead in the hunt for the gyroid — a pair of related surfaces I named M4 and M6. One might call them wrinkled versions of the gyroid. The original models of these surfaces are shown in Figs. E1.16a and E1.16b. A variety of stereoscopic images of more recent models of M4 and M6 are shown in Figs. E1.19 and E1.20. Fig. 1.16a                                                          Fig. 1.16b M4                                                                     M6 view: [111] direction As the result of a lucky guess about how to remove the 'wrinkles' in M6, I produced the pseudo-gyroid, which is shown in Fig. E1.17. The details of how I got from the pseudo-gyroid to the gyroid are described in §7 and §8. Fig.E1.17 The pseudo-gyroid By September 1968 I had concluded that the gyroid is the only embedded surface among the countably many surfaces associate to P and D, but my 'proof' was based on (a) physically bending assemblies of plastic replicas of surface modules and on (b) computer graphics animation of that bending (cf. Fig. E1.21b). In 1996, Karsten Grosse-Brauckmann and Meinhard Wohlgemuth published a rigorous proof that the gyroid is embedded (free of self-intersections) and contains neither straight lines nor reflection symmetries, in The gyroid is embedded and has constant mean curvature companions, Calc. Var. 4, 1996, 499-523. It seems likely that before 1968, no one had ever bothered to look at any of the countably many intermediate surfaces associate to Schwarz's P and D surfaces to determine whether any of them were embedded. I confess that before 1968 it had never occurred to me to look there (even though it should have!). The gyroid has received much attention from physicists, chemists, and biologists since the early nineties, because it has been found to be — at least approximately — a kind of geometrical template for a great variety of self-assembled bicontinuous structures, both natural and synthetic. I first announced its existence at an AMS meeting in August, 1968 (cf. abstract in Fig. E2.10), in the mistaken belief that the pseudo-gyroid is a bona fide minimal surface. (It is remarkably close to one!) The actual gyroid G is described on pp. 48-54 of Infinite Periodic Minimal Surfaces Without Self-Intersections, NASA TN D-5541 (May 1970), and also here in Fig. E1.23. Weierstrass parametrization of G Fig. E1.2m The rectangular coordinates of G, defined by Schwarz's solution (Weierstrass-Enneper parametrization) for the entire family of surfaces associate to P and D If θG in the term eiθG is replaced by zero, the coordinates are those of D. If θG is replaced by π/2, the coordinates are those of P — the adjoint of D. (The value given above for θG agrees up to eight significant figures with the value derived from a more recent analysis based on a Schwarz-Christoffel mapping, in Adam Weyhaupt's 2006 PhD thesis (pp. 115-116). Stereo views of translation fundamental domain of P, D, and G Fig. E1.2p The D surface Fig. E1.2q The G surface Fig. E1.2r The P surface For high-resolution pdf versions of these three images, look here (D), here (G), and here (P). These pdf images will probably load slowly, because they are large (7 to 10 Mb). For maximum image clarity, zoom in to make the width of the image almost equal to the screen width. The associate surfaces D and P are called adjoint. The angle of associativity (cf. Fig. E1.23) by which they are related via Bonnet's bending transformation is π ⁄ 2. Like all of the other intermediate surfaces associate to the helicoid and the catenoid, G contains neither straight lines nor plane lines of curvature. The round tunnels centered on [100], [010], and [001] axes, which are arranged in square checkerboard arrays, are bounded by approximately helical curves of opposite handedness in the two intertwined labyrinths of the surface. The outermost curved edges of the eight congruent hexagonal faces in Fig. E1.3a correspond to the edges of the regular map with holes {6,6|3}. If you examine these edges closely, you will see that they are not quite plane. Each of them lies half inside and half outside the enclosing cube. Let us call a geodesic curve on a triply-periodic minimal surface a trivial geodesic if it is either a straight line or a mirror-symmetric plane line of curvature, and a non-trivial geodesic otherwise. Since there are neither straight lines nor mirror-symmetric plane lines of curvature in the gyroid, all of its geodesic curves are non-trivial. If you apply rubber bands to physical models of the three surfaces P, G, and D (as I have done), you will easily discover a variety of periodic geodesic curves, both closed and unbounded. I will eventually show here images of several non-trivial geodesics, both closed and unbounded, on these three surfaces. For a discussion of geodesics on multiply-connected surfaces, see Steven Strogatz's 2010 NYTimes essay on geodesics, with links to Konrad Polthier's videos on this topic. The four images in Figs. E1.15d to E1.15g were made by Ken Brakke, period killer extraordinaire. The embedded surfaces I-WP (cf. also Fig. 2.5) and F-RD (cf. also Fig. 2.6) are the adjoints of two of the four self-intersecting Schoenflies surfaces (cf. Fig. E2.1). In 1975, I used the laser-soap film technique mentioned just below Figs. E1.15a and E1.15b to kill periods in order to obtain the approximate shapes of the curved edges of the octagonal surface patch of F-RD(r). Several years ago, Ken used his Surface Evolver program to obtain the vastly improved modeling of F-RD(r) shown below. Soon afterward I asked Ken to use Surface Evolver to obtain the hexagonal patch of I-WP(r). In order to emphasize how the tubular structure of each cubic surface cell on the right is related to its companion at its left by a one-eighth-turn (45º) about a vertical axis, Ken included '(r)' in its name. Fig. E1.15d                   Fig. E1.15e     F-RD                           F-RD(r) Fig. E1.15f                   Fig. E1.15g     I-WP                           I-WP(r) A spectacular large model of G is shown in the YouTube video, Gyroid playground climbing structure at the San Francisco Exploratorium and also in Figs. E1.18a-d (below). This structure was designed and built by a team that included Thomas Rockwell, Paul Stepahin, Eric Dimond, and John Kinstler. Kinstler describes here how the plastic modules of the Exploratorium gyroid were designed and fabricated. Here is a photo of a more recent laminated plywood gyroid sculpture at the Exploratorium. The photographer was the formidable polymath Jef Poskanzer, who has informed me that the young man inside the sculpture is his nephew Henry. Fig. E1.18a Exploratorium gyroid photo courtesy of Amy Snyder High-resolution photos of the Exploratorium gyroid (courtesy of Amy Snyder) are at: E1.18a E1.18b E1.18c E1.18d Figs. E1.19a-e show recently constructed models of M6, which was the immediate precursor of the gyroid (cf. abstract in Fig. E2.10). The vertices coincide with the vertices of the Coxeter-Petrie map {6,4|4} on the gyroid, but the edges of the hexagonal faces are line segments, not the quasi-helical arcs of the gyroid. Figs. E1.20a-e show M4, the immediate precursor of M6. Fig. E1.19a 65 hexagons of M6 view: [111] direction, silhouetted by bright summer sky backlighting Fig. E1.19b 65 hexagons of M6 (stereo) view: [100] direction Fig. E1.19c 65 hexagons of M6 view: [100] direction, silhouetted by bright summer sky backlighting Fig. E1.19d 65 hexagons of M6 view: [111] direction Fig. E1.19e 65 hexagons of M6 view: [100] direction Fig. E1.20a 30 quadrangles of M4 (stereo) view: [111] direction Fig. E1.20b 30 quadrangles of M4 (stereo) view: [110] direction Fig. E1.20c 30 quadrangles of M4 (stereo) view: [110] direction Fig. E1.20d The edges of the model in Fig. E1.20c, which define a portion of the 'deficient' graph BCC6 view: close to [100] direction Fig. E1.20e 30 quadrangles of M4 view: [111] direction, backlit by summer sky Fig. E1.20f 30 quadrangles of M4 view: [100] direction, backlit by summer sky Cubic unit cell of G Fig. E1.22 Cross-eyed stereogram of a gauzy cubic unit cell of G G is associate to P and D, i.e., its rectangular coordinates are a linear combination of those of P and D (cf. Fig. E1.23). This figure shows twice as much of G as is contained in one translation fundamental domain. The lattice for G is body-centered cubic (bcc). Among the countable infinity of surfaces that are associate to P and D, G is the only embedded surface. A translation fundamental domain has genus 3. Cubic unit cell of G Fig. E1.24 A view from a different corner (the lower right rear corner) of the unit cell of G shown in Fig. E1.22 The skeletal graphs of the two enantiomorphic labyrinths of G are enantiomorphic Laves graphs of girth ten (and degree 3). Since the lattice for G is bcc, a truncated octahedron is a reasonable choice for a translation fundamental domain. This cube unit cell has volume equal to the volume of two translation fundamental domains. The sequence of three stereo images immediately below illustrates the application of Ossian Bonnet's 1853 bending transformation — with no stretching or tearing — to the 'morphing' of Schwarz's D surface into Schwarz's P surface. The orientation of the tangent plane at each point of the surface remains fixed throughout the bending. The G surface shows up a little more than one-third of the way (cf. Fig. E1.23) through the transformation. A simply-connected translation fundamental domain of genus 3 is shown in each of these figures. Each such domain is composed of eight congruent regular skew hexagonal faces, which are defined by Coxeter's regular map with holes {6,4|4} (cf. Fig. E1.4). The lattices for the three [oriented] surfaces are face-centered cubic (fcc) for D, body-centered cubic (bcc) for G, and simple cubic (sc) for P. Stereo view of G Fig. E1.28 Cross-eyed stereogram of a larger piece of G A hexagonal face of G Fig. E1.29 Cross-eyed stereogram of a hexagonal face of G that corresponds to a face of the regular map with holes {6,4|4} {cf. Fig. E1.5). The six vertices of the face coincide with vertices of a truncated octahedron. The edges of the face appear to be plane curves, but they are not. One half of each edge lies inside this truncated octahedron, and one half outside. view: [100] direction Two hexagonal faces of G Fig. E1.30 Cross-eyed stereogram of two hexagonal faces of G David Hoffman and Jim Hoffman (not relatives!) have made this animation of the associate surface transformation P→G→D for a single translation fundamental domain. Gerd Schröder-Turk has also animated the P→G→D Bonnet transformation . In addition, he has created a spectacular 'FlyThrough' animation of a large chunk of G. His movie shows the enantiomorphic skeletal graphs of the two labyrinths of G. Don't be surprised if Gerd's movie file takes a while to download. It's not small (~91 Mb!). The "level-set" surface G*, which very closely resembles G, is defined by the equation cos x sin y + cos y sin z + cos z sin x = 0. David Hoffman and Jim Hoffman have illustrated the striking resemblance between G and G*. Below are images of G* made with Mathematica. The level surface G* (a) (b)                      (c)                      (d)                     (e)                    (f)                        (g)                      (h)                    Fig. E1.31 Various views of the level surface G* (a) stereo view (b) skew hexagonal face (c) assembly of eight of the skew hexagonal faces, which defines a translation fundamental domain bounded by a cube (d) the assembly shown in (c) viewed from the (1,1,1) direction (e) the assembly shown in (c) viewed from the (− 1,− 1,− 1) direction (f) view of the assembly in (c) from the (1,0,0) direction, showing the square array of quasi-helical tunnels. Adjacent tunnels spiral alternately CW and CCW. (g) view of the assembly in (c) from the (1,1,1) direction, showing the kagomé array of quasi-helical tunnels. Adjacent tunnels spiral alternately CW and CCW. (h) oblique view of the assembly shown in (g) The quasi-helical curves ('flattened helices') in G* that are centered on c4 axes coincide with right circular cylindrical helices of radius π/4 at just four points in each period, at exactly quarter-period intervals. At all other points helices fail to satisfy the level surface equation for G*. Halfway between each pair of consecutive points of coincidence, the radius of each flattened helix has a local minimum ≈ π/4 − 0.015941383804981744. Hence this reduced radius is approximately 2.02972 % smaller than π/4. L1V17, the chiral Voronoi polyhedron of a single Laves graph L1V17 is the Voronoi polyhedron of a Laves graph vertex. It is shown below and also in Fig. II-2h, p. 90 in NASA TN D-5541. L1V17 has 17 faces: 2 hexagons, 3 octagons, 6 quadrangles, and 6 pentagons Like the Laves graph itself, L1V17 is chiral. Here's a list of the coordinates of the 30 vertices of L1V17: Below are three perspective views of L1V17      The isometries of L1V17 include an axis of 3-fold rotational symmetry    through the centers of its opposite hexagonal faces and axes of 2-fold             rotational symmetry through the centers of each of its three octagonal faces.    Note that although L1V17 itself lacks reflection symmetry, each of its    faces has reflection symmetry. The quadrangles and pentagons have    d1 symmetry, the octagons have d2 symmetry, and the hexagons      have d3 symmetry. (L1V17 is also shown in Fig. II-2h on p. 90 of     Infinite Periodic Minimal Surfaces Without Self-Intersections.) The images below show assemblies of two,   three, four, and sixteen specimens of L1V17. Two contiguous L1V17 cells Every cell is related to each of its three nearest neighbor cells by a halfturn about either of the two orthogonal C2 axes that bisect the octagonal face they share. These axes also bisect an edge of the associated Laves graph that is   orthogonal to that face.                                                        Three contiguous L1V17 cells Four contiguous L1V17 cells     This assembly of four cells is a translational fundamental domain of an infinite packing of these polyhedra.         Sixteen contiguous L1V17 cells (Four translational fundamental domains) Evolute (aka 'net') of L1V17 Schlegel diagram of L1V17 A small portion of the triply-periodic surface composed of a space-filling assembly of L1V17 cells from each of which the three octagonal faces have been removed.           Orthogonally projected end view of the edges of L1V17    If the colors are ignored, this image has d3 symmetry. If this Voronoi polyhedron were opaque,                     the green edges would be visible only from the front, the red edges would be visible only from the rear,                    and the black edges would be visible from both front and rear. L1V20, chiral Voronoi polyhedron of a single partially collapsed Laves graph L1V20 is the Voronoi polyhedron of a vertex of a   partially collapsed Laves graph. It is shown below and also — at a slightly more advanced stage of        collapse — in Fig. II-2g, p. 90 in NASA TN D-5541. L1V20 has 20 faces: 2 hexagons, 6 decagons, and 12 triangles Like L1V17 (cf. above), L1V20 is chiral. Here's a list of the coordinates of the 36 vertices of L1V20: Here's a view of a collapsing Laves graph at      an earlier stage than the one that defines L1V20. Below are several perspective views of L1V20,   plus two views of the associated Laves graph at the stage of collapse that defines L1V20.              Above is a close-packed assembly of six translucent specimens of L1V20,      each enclosing a vertex (blue sphere) of this partially collapsed Laves graph.       The three green line segments incident at each vertex are edges of the graph.         The shorter red line segments (which are not edges of the graph) have been            inserted to demonstrate that some pairs of vertices that were separated before    collapse by a distance greater than the edge length are now separated by a distance that is smaller than the edge length.                                               This is an optically exact Schlegel diagram of L1V20       produced by a generic notebook I wrote in Mathematica. Here is a combinatorially correct but optically inexact      Schlegel diagram of L1V20 that I designed by eye before     I wrote the Mathematica notebook mentioned just above. L2V17, chiral Voronoi polyhedron of the vertices of the union of two dual Laves graphs         L2V17 is the chiral Voronoi polyhedron whose two enantiomorphic versions enclose the vertices of a pair of suitably intertwined        dual Laves graphs. It is shown below (and also in Fig. II-2a, p. 89     in NASA TN D-5541). An infinite assembly of replicas of either      enantiomorph, in which adjacent polyhedra share decagons only,         fills just half of space. In a packing (space-filling assembly), which          is populated by equal numbers of the two enantiomorphs, hexagons          and quadrangles are shared by oppositely congruent enantiomorphs. Three views of L2V17, the chiral Voronoi polyhedron of a Laves graph vertex L2V17 has 17 faces: 2 hexagons, 12 quadrangles, and 3 decagons. Schlegel diagram of one of the two enantiomorphs of L2V17 ABCE, one of the twelve quadrangle faces of L2V17 The isometries of L2V17 are identical to those of L1V17. They      include an axis of 3-fold rotational symmetry through the centers of its opposite hexagonal faces and an axis of 2-fold rotational     symmetry through the center of each of its three decagonal faces.    Furthermore, just like L1V17, L2V17 lacks reflection symmetry.     Although its decagons have d2 symmetry and its hexagons have       d6 symmetry, its quadrangles are asymmetrical. However, L2V17      has a curious combinatorial property that is distantly related to a       property of Solomon Golomb's rep-tiles: each quadrangle can be    tiled by three similar isosceles triangles. (Each of the two large    triangles is exactly 50% greater in area than the small triangle.) The Voronoi cell of a vertex of a collapsing graph On pp. 85-90 of Infinite Periodic Minimal Surfaces Without Self-Intersections,     I introduced the concept of the collapse of a triply-periodic graph, and I included    images of the Laves graph at five stages of collapse. Computer animations of the        collapse of the Laves graph and also of the collapse of three other symmetric triply-      periodic graphs are included in my 1972 video Part 4, starting at time code 7m04s. Also included is an animated sequence of views of the evolution of the Voronoi    polyhedron of a graph vertex. (Some other examples of the animation of Voronoi         polyhedron evolution were unfortunately lost, because of the confusion and disorder       that prevailed at NASA/ERC during its closing months in the spring of 1970, when these computer movies were made. However, a few stills from these movies are    shown on p. 90 of Infinite Periodic Minimal Surfaces Without Self-Intersections.) Rotation of the edges of the Laves graph during the collapse transformation The red edges and the vertices A0, B0, and C0 belong to the Laves graph before collapse. The blue edges and the vertices V, A1, B1, and C1 define the stage of collapse that produces the Voronoi polyhedron L1V20 shown above. On pp. 662-664 of Reflections Concerning Triply-Periodic Minimal Surfaces, I          summarize some of the properties of the collapse transformation (which is applicable only to triply-periodic graphs that are symmetric). In the image shown above,        the vertex V at the center of the cube is a vertex of the Laves graph. The three red      line segments VA0, VB0,, and VC0 represent the three edges of the graph incident      at V. From the vantage point of an observer fixed at V, each of the three red edges         rotates 90º in the course of the graph collapse. The continuous change in the relative positions of the graph vertices produces continuous changes in the shape of the Voronoi polyhedron, and intermittent changes in its combinatorial structure.           In the terminal [collapsed] state of the graph, each edge is mapped onto one of the   six edges (shown above in green) of a regular tetrahedron. For the configuration             of blue edges illustrated above, the edge rotation angle θ1 = cos-1[3/√10] (≅18.4349º).            This configuration appears shortly after the first change, but before the second change,       in the combinatorial structure of the Voronoi polyhedron. If V is at the origin, then A0 = (0,1,-1)/√2, B0 = (-1,0,1)/√2, C0 = (1,-1,0)/√2. A1 = (0,1,-2)/√5, B1 = (-2,0,1)/√5, C1 = (2,-1,0)/√5. In 1969, with the help of the computer graphics expert Charles Strauss and the camera engineer Bob Davis, I made 35mm movies of the collapse of several examples of triply-periodic graphs (cf. NASA TN D-5541, pp. 85-88). Several of these movies, including the one in my 1972 video "Shapes of Soap Films Part 4", which is now on YouTube, showed the continuous transformation of the Voronoi cell that encloses each vertex of a collapsing graph. However, the only one of these movies showing Voronoi cell transformations that survives today is the one currently on YouTube. (I am planning to remake these missing movies.) During the transformation of the Voronoi polyhedron that encloses each vertex of a collapsing graph (a symmetric graph of degree Z), if a particular vertex V of the graph is regarded as fixed, then the Z vertices that are endpoints of the edges incident at V all travel along circular arcs centered on V. In its collapsed state, the infinite graph degenerates into the edge complex of a single convex polyhedron (cf. Reflections Concerning Triply-Periodic Minimal Surfaces for a few explanatory details, including illustrations). I plan to add here a detailed analysis of graph collapse, including a derivation of the surprisingly simple algebraic relation between (a) the fictitious linear displacement of vertices that would occur if instead of rotating edges, one translated vertices, allowing edges to stretch without limit and (b) the actual rotation of edges that occurs with edges treated as unstretchable. The sequence of Voronoi cells displayed in the 1972 video cited above demonstrates that the rearrangement of vertices produced by graph collapse has an interesting consequence: at discrete intervals, both the combinatorial structure and the symmetry of the Voronoi cell undergo an abrupt change. This can be seen in the 1972 video in the scenario entitled 'Voronoi cells of a vertex of the collapsing graph (degree 6) of the edges of the Coxeter map {4,6|4} on the gyroid surface'. The Voronoi cell L2V17 suddenly appears with its enlarged complement of symmetries at approximately 10 min:33 sec after the beginning of the scenario, at the very center of a symmetrical sequence of combinatorial types of polyhedra. But it is immediately preceded and followed by a continuous sequence of polyhedra of the same combinatorial type but lower symmetry. As stated above, L2V17 is the chiral Voronoi cell of a vertex of one enantiomorph of the Laves graph in the presence of its enantiomorphic dual. But the 1972 video demonstrates that L2V17 is also the Voronoi cell of a vertex of the partially collapsed graph of degree 6 with the combinatorial structure and symmetry of the 'deficient' graph BCC6 — the graph composed of the edges and vertices of M4 (cf. [cross-eyed] stereo image just below). 30 skew quadrangles of M4 (stereo) view: [110] direction Fig. E1.32 shows a set of screen-captured images from the 1972 video, extracted from the continuous sequence of geometrical transforms of the initial Voronoi cell, a truncated octahedron. Polyhedra of three combinatorial types appear in this sequence. L2V17, at the center of the sequence, is the polyhedron labeled '10:32'. The arrangement of the combinatorial types in the sequence is symmetrical with respect to the center of the sequence, but the orientation of each polyhedron that precedes L2V17 is related to the orientation of the matching polyhedron following L2V17 by inversion in the center of the polyhedron, followed by rotation through a quarter-turn. The approximate value of the video time code for each polyhedron is listed directly below its image, and the combinatorial type of each polyhedron is illustrated by its Schlegel diagram, shown below its time code. The values of the time code indices demonstrate that the transformation is accelerating at a significant rate. The rearrangement of graph vertices produced by graph collapse is vaguely reminiscent of the atomic rearrangements that occur in polymorphic transformations of the crystallographic structure of many metals. NOTE: When time permits, I will post here a detailed analysis of the collapse transformation of triply-periodic graphs. I will include graphic images of selected examples of graphs and their associated Voronoi polyhedra, plus examples of nets of some of these polyhedra, together with vertex coordinate data. 10:14           10:21           10:27           10:29           10:32           10:35           10:37           10:40           10:42 Top row: Screen-captured video images of Voronoi cells of                the vertices of the partially collapsed M4 graph The Voronoi cell at 10:32 is L2V17!           Bottom row: Schlegel diagrams of the Voronoi cells in the top row.   Each of these Schlegel diagrams displays the combinatorial   structure of the polyhedron displayed above it, but the only optically exact projection is that of L2V17. Moreover, some of the diagrams have reflection symmetries not present in    the corresponding polyhedron.                                               Voronoi G, a polyhedral toy model of the gyroid derived from L2V17 Voronoi G is a triply-periodic polyhedron with plane faces that roughly approximates the gyroid surface (cf. Figs. E1.34c to E1.34g below). It is derived by removing the three decagonal faces from each polyhedron in an infinite packing composed of the two enantiomorphs of the convex polyhedron L2V17 (cf. Figs. E1.34a to E1.34b below).          Fig. E1.34a                                                                                            Fig. E1.34b Stereoscopic views of the two enantiomorphic shapes of the chiral Voronoi cell L2V17 of a vertex of one Laves graph in the presence of its dual graph (This polyhedron is depicted in Fig. II-2a on p.89 of Infinite Periodic Minimal Surfaces Without Self-Intersections.) Fig. E1.34c Voronoi G, viewed in the [100] direction (stereoscopic pair) (For a higher-resolution [pdf] version of the above image, look here.) Fig. E1.34d Orthogonal projection of Voronoi G onto [100] plane (cf. Fig. E1.34c) Fig. E1.34e Voronoi G, viewed in the [110] direction (stereoscopic pair) (For a higher-resolution version of the above image, look here.) Fig. E1.34f Orthogonal projection of Voronoi G onto [110] plane This image demonstrates that the gyroid is symmetrical by a half-turn about each of its six axes of type [110]. Fig. E1.34g Orthogonal projection of Voronoi G onto [111] plane (For a higher-resolution [pdf] version of the above image, look here.) Compare the shapes of the tunnel holes with the corresponding holes in G shown in Fig. E1.19. Fig. E1.34h One chamber of Voronoi G, which is L2V17 minus its three decagon faces. All but six of its thirty vertices lie on the sphere of radius r1 shown here. The six outliers — shown as red dots — lie on a slightly larger sphere (not shown) of radius r2. The ratio r2/r1 ≅ 1.029. Uniform polyhedra based on G Figs. E1.35a,b The uniform gyroid (6.32.4.3), an infinite uniform polyhedron The faces of (6.32.4.3) are all regular plane polygons, and its symmetry group is transitive on both faces and vertices. (For a pdf version of the image in Fig. E1.35a, look here.) The combinatorial structure and symmetry of (6.32.4.3) are defined by the Poincaré hyperbolic disk model of uniform tilings in the hyperbolic plane. Fig. E1.35c The Poincaré hyperbolic disk model of 6.32.4.3) (Wikipedia image) Let me define two objects to be homologous if they have the same symmetry and the same topology. Since April 1966, I had been mulling over the homologous relations between the two Schwarz minimal surfaces P and D and the Coxeter-Petrie regular skew polyhedra {6,4|4}, {4,6|4}, and {6,6|3}, whose faces are regular plane polygons, My first model of the gyroid Fig. E1.4i A Voronoi surrogate of P — the Coxeter-Petrie triply-periodic regular skew polyhedron {6,4|4}, which is homologous to P. And here is one D surrogate: Fig. E1.2c A Voronoi surrogate of D — the Coxeter-Petrie triply-periodic regular skew polyhedron {6,6|3}, which is homologous to D. After I create an image of {4,6|4}, I will post it here. (cf. Wikipedia's animated images of the three Coxeter-Petrie regular skew polyhedra.) Both {6,4|4} and its dual {4,6|4} are homologous to P, and the self-dual {6,6|3} is     homologous to D. In 1968 I derived an example of a triply-periodic polyhedron with   plane faces that is homologous to G: Voronoi G (cf. Figs. E1.34c-g). It was known     from the work of Coxeter and Petrie that there exists no regular triply-periodic skew     polyhedron with plane faces that is homologous to G. I wondered, however, whether      there exists a uniform triply-periodic polyhedron with plane faces that is homologous      to G. The abstract in Fig. E1.35c is a condensed summary of the results of my search for quasi-regular tilings of P, G, and D.                                                                                 Many of my notebooks, files, and physical models related to these tilings were destroyed    when an intruder ransacked offices at Southern Illinois University in 1982. I plan to             reconstruct this work and to post here images of more examples to supplement the single          example, (6.4)2, shown in §14.                                                                                                          After I showed my abstract (cf. Fig. E1.35c) to Norman Johnson in 1969, he supplemented       it by deriving the combinatorial structure (although not the geometry) of every quasi-        regular or uniform tiling in the {4,6} family. When I investigated the geometry of his         examples, I discovered that (6.32.4.3) is the only uniform tiling of the gyroid by plane             polygons in this family. The (6.32.4.3) tiling was later discovered independently by John              Horton Conway, who named it mu-snub cube. ('Mu' is an abbreviation for multiple here.)                     Norman Johnson is well known for his enumeration of the 92 Johnson solids, which Zalgaller                   later proved is exhaustive. He devoted many years of his life to writing a book on uniform         polytopes in R3 and R4, but I believe it was still unpublished at the time of his death     in 2017. For decades, I tried to persuade Norman to publish an account of his proof           that his enumeration of the uniform tessellations of the {4,6} family is complete, but to no avail. Without Norman's permission, I show that list below in Fig. E1.35b.       Fig. E1.35b Norman Johnson's 1969 enumeration of the uniform tessellations of the {4,6} family (unpublished) Fig. E1.35c A summary of my enumeration of the quasi-regular tessellations of the {6,4} family (abstract published by the American Mathematical Society in 1969) Fig. E1.35d (below) shows AMS Abstract 658-30 (1968), cited in the first line of the Abstract in Fig. E1.35c (above). Fig. E1.35d Abstract 658-30, submitted in summer 1968 to the American Mathematical Society for the Madison meeting in August I gave a 15-minute oral presentation there that was based on this work. Uniform polyhedron models of P Fig. E1.36 (6.43), an infinite uniform polyhedron model constructed by my Cal Arts student Bob Fuller in 1971 Fig. E1.37 (6.43), an infinite uniform polyhedron model constructed by Bob Fuller in 1971 Fig. E1.38a (6.43), an infinite uniform polyhedron model constructed by Bob Fuller in 1971 Fig. E1.38b Stereo photo of Bob Fuller's model of (6.43) Isaac Van Houten's bronze sculpture of G Fig. E1.39a Isaac Van Houten's 2006 bronze sculpture of the gyroid G Fig. E1.39b Template used by Isaac to cast his sculpture of G Bathsheba Grossman's 3D-printed sculpture of G Fig. E1.41 Two views of one of Bathsheba Grossman's printed models of G In 1934, the German mathematician Berthold Stessmann published an article in Mathematische Zeitschrift 38, 1934 (414-442) entitled "Periodische Minimalflächen". A book by Stessmann, also entitled " Periodische Minimalflächen ", was published by J. Springer in 1934. * * * * I have been unable to learn what became of Stessmann in the WWII period. (Perhaps there are some leads in here.) If you discover any information about him, please email me. * * * * On the first page of Stessmann's article there are drawings of the six skew quadrilaterals that Arthur Moritz Schoenflies proved in 1890 are the only skew quadrilaterals, spanned by minimal surfaces, that generate TPMS by half-turn rotations about their edges, i.e., by repeated applications of Schwarz's reflection principle. These six Schoenflies quadrilaterals are reproduced here in Fig. E2.1. The six Schoenflies quadrilaterals Fig. E2.1 The six Schoenflies quadrilaterals Of the six Schoenflies quadrilaterals, only I and III are patch boundaries for embedded surfaces — Schwarz's D and P, respectively (cf. Figs. E1.2a and E1.3a). By applying Schwarz's reflection principle, it is easy to demonstrate that the other four quadrilaterals — II, IV, V, and VI — define surfaces with transverse self-intersections. But what about the adjoints (cf. text below Fig. E1.23) of II, IV, V, and VI? It is a fundamental property of any two adjoint minimal surfaces S1 and S2 that if a boundary edge in S1 (say) is a straight line segment E1, then its image in S2 is a segment of a plane line of curvature C1 (a plane geodesic) that lies in a plane perpendicular to E1. Let us call this property 'Property A'. The adjoint of II is an embedded surface of genus 9 and is called Neovius's surface. It was first analyzed in 1883 by Edvard Rudolf Neovius, an 1869 doctoral student of Schwarz's, in his dissertation, Zweier Speciellen Periodischen Minimalflächen auf welchen unendlich viel gerade linien und unendlich viele ebene geodätischen linien liegen. I sometimes identify Neovius's surface by the alternative name C9(P) — or just C(P) — because it has a remarkable property that I believe was never mentioned in writing either by Neovius or Schwarz: it has exactly the same embedded straight lines as P and has genus 9. (The 'C' in C9(P) stands for 'complement'.) C9(P) is illustrated in Figs. E2.2a,b,c, and its self-intersecting adjoint surface C9(P)† is illustrated in Fig. E2.2d. Edvard Neovius (1851-1917), who became Professor of Mathematics at the University of Helsinki, was a cousin of Ernst Lindelöf and uncle of Rolf Nevanlinna. (For further information, see this short article about the history of the Finnish Mathematical Society.) The embedded Neovius surface C9(P) and its self-intersecting adjoint surface C9(P)† (a)                                                                              (b) Figs. E2.2a and b One lattice fundamental region of the embedded Neovius surface C9(P) (genus 9) The images are copied from Tafeln II and III of Neovius's 1883 dissertation. On June 27, 2012, one day after my wife Reiko and I visited the magnificent National Library of Finland (cf. images above), Mr. Jari Tolvanen, a reference librarian there, kindly informed me by email that Neovius's doctoral dissertation, a copy of which is in the library's collection, is also available online here. (c)                                                                              (d) Figs. E2.2c and d (c) An assembly of eight lattice fundamental regions of the embedded Neovius surface C9(P) and (d) an assembly of seven lattice fundamental regions of its self-intersecting adjoint surface C9(P)† These two images are copied from Tafeln IV and III of Neovius's 1883 dissertation. A complete copy of the dissertation is shown here. Fig. E2.3a My first model of C9(P) The 'see-through' tunnels are aligned in [110] directions. Fig. E2.3b A later model of C9(P) Fig. E2.4a Announcement in a February, 1969 abstract published by the American Mathematical Society describing C(D), a TPMS that is complementary to Schwarz's D surface "Abstract 658-30" mentioned in the third line and shown here as Fig. E2.10, refers to the gyroid, which I originally named "L" (for Laves) in 1968. The possibility that there exists an embedded counterpart of C(P), which it would be reasonable to call 'C(D)', occurred to me as I was crossing the street while returning to my NASA office from the MIT library. I was staring at the illustrations in the photocopy of Neovius's 1883 PhD dissertation made for me a few minutes earlier by the MIT science librarian. When I saw the drawing shown above in Fig. E2.2b, I was startled to see that the set of straight lines in the surface is identical to the set of straight lines in Schwarz's P surface. I imagine that Neovius and Schwarz (his teacher) also must have noticed this matching of lines! I suddenly decided that what is sauce for the goose may also be sauce for the gander. I reasoned that since both P and D can be regarded as [infinite] regular polyhedra, it is plausible that if one of them (P) has an embedded companion surface that contains the same set of straight lines, then the other (D) probably does too. I concluded that if I could confirm the existence of such a surface for D, I would name Neovius's surface 'C(P)' and the new surface 'C(D)' — where 'C( )' means 'complement of ( )'. As soon as I reached my office a few minutes later, I needed only to glance at my straw model of the straight lines in D to recognize instantly that the embedded surface C(D) exists. A few months later, I realized that higher order complements of both P and D probably exist too (cf. discussion of 'Notched adjoints' following Fig. E2.82). The MIT librarian told me that no one had borrowed Neovius's thesis during the fifty years since the library had acquired it. It was printed in large folio format, and as a result she had to cut all the pages into quarters (which took her a good while!) before she could make a photocopy for me. Perhaps that's a good indication of how interested mathematicians were in periodic minimal surfaces in those bygone days. (My account of how the MIT library copy of Neovius's thesis came to see the light of day for the first time in fifty years would hardly be complete if I failed to mention that I initially asked the librarian to obtain a copy for me via inter-library loan, because I naively assumed that it was much too obscure to be included in the MIT collection. I even mentioned that I was aware that she might have to send to Finland for it, and that I was prepared to wait. She did not deign to reply but instead simply marched off in the direction of the stacks, pausing only once to look back over her shoulder, wearing an expression of polite disdain as if to say, "What kind of institution do you think we are?" I can't imagine that she actually knew for certain that she would find the thesis on the shelves, but she did locate it in only a matter of a few seconds!) C(D) Fig. E2.4b C19(D) (genus 19) Right: One translation fundamental domain Left: One-fourth of a translation fundamental domain C19(D) is called the [first-order] complement of D, because it has exactly the same embedded straight lines as D. Ken Brakke's images of a sequence of higher-order complements of C19(D) of genus 35, 51, 67, ..., which belong to two families, A and B, are shown Fig. E2.4c C19(D) Two translation fundamental domains Fig. E2.4d C19(D) One-fourth of a translation fundamental domain, cut from two different parts of the surface Fig. E2.4e What's left in 2011 of the brittle 1970 model of C19(D) shown in Fig. E2.4b view: [111] direction Fig. E2.4f C(D) view: [-1-1-1] direction Fig. E2.4g C(D) view: [110] direction What about the adjoints of Schoenflies IV, V, and VI? When (in 1968) I first considered these three quadrilaterals, I was startled to realize that although both the triply-periodic surface derived from Schoenflies IV and the one derived from its adjoint are obviously self-intersecting (both of them have a 120º corner and therefore a branch-point), it seemed highly likely that the adjoints of both Schoenflies V and Schoenflies VI are patches for embedded surfaces! I developed persuasive experimental evidence in support of this conjecture by (a) blowing on soap films inside tetrahedral cells to model the adjoint patches and (b) performing Bonnet bending of physical replicas of Schoenflies V and Schoenflies VI made from vacuum-formed plastic sheet material. I chose the name F-RD for the embedded adjoint of Schoenflies V (which has genus 6) and the name I-WP for the embedded adjoint of Schoenflies VI (which has genus 4). My naming conventions are explained on p. 38 of Infinite Periodic Minimal Surfaces Without Self-Intersections. Andrew Fogden confirmed that F-RD is embedded, in his 1992 article entitled "A systematic method for parametrizing periodic minimal surfaces: the F-RD surface", Journal de Physique 2 (1992) 233-239. Fogden succeeded in deriving an implicit equation (fifth degree polynomial) for the F-RD Weierstrass polynomial, a task that was begun by Berthold Stessmann in 1934 but not completed (cf. Stessmann B., Periodische Minimalflächen, Mathematische Zeitschrift 33 (1934) 417-442). Because it is impossible to solve for the F-RD Weierstrass polynomial in explicit form, its surface coordinates can be computed only numerically. In my 1997 F-RD poster essay, I summarize some of Fogden's remarkable results for F-RD and I-WP. In a 1992 collaboration between the versatile mathematician Djurdje Cvijović and the Cambridge University chemist Jacek Klinowski, the authenticity of I-WP was rigorously established and several of its quite surprising mathematical properties were derived (Cvijović, D. and Klinowski, J.: The computation of the triply periodic l-WP minimal surface, Chemical Physics Letters 226 (1994) 93-99). The analysis of I-WP turned out to be far simpler than that of F-RD. In every known example of an embedded TPMS that contains no straight lines, the two labyrinths are found to be non-congruent. The versatile crystallographers Elke Koch and Werner Fischer have classified such surfaces as non-balanced and surfaces that contain straight lines — like P and D — as balanced. Because I-WP and F-RD contain no straight lines, they are non-balanced. These two surfaces are illustrated in Figs. E2.5 and E2.6. F-RD and I-WP were the first examples of non-balanced TPMS to be identified, but in recent years many additional examples of such surfaces have been discovered. I-WP Fig. E2.5a I-WP (genus 4) oblique view Fig. E2.5b I-WP Modeled by Ken Brakke, using his Surface Evolver software oblique view Fig. E2.5c I-WP view: [111] direction (approximately) Fig. E2.5d I-WP view: [100] direction Portions of two non-trivial geodesics (cf. discussion below Fig. E1.24) are shown. The one is at the left is closed, and the triply-periodic one at the bottom is unbounded. Fig. E2.5e I-WP view: [100] direction Fig. E2.5f I-WP view: [100] direction Fig. E2.5g I-WP oblique view F-RD Fig. E2.6a F-RD (genus 6) view: [111] direction (See my 1997 F-RD poster essay for details.) I am indebted to John Brennan and Robert Fuller, two of my truly outstanding students at California Institute of the Arts, for volunteering in 1971 to complete this large model of     F-RD. In 1983, when I returned to Southern Illinois University/Carbondale from an extended   trip, I was informed by two junior teachers that a marauder (whose identity I never learned) had recently trashed several of our offices and laboratories, including mine. They also said they had already 'cleaned up the mess' in order to prevent an official investigation. To my       horror, I found that several of my research notebooks and almost all of my models, including F-RD and several others made by John and Bob, were nowhere to be found in either my       office or my lab. (I was informed by the same two junior teachers that it was too late for me      to inspect any of the remains, because all the trash had long since been carted off to the dump and buried. I was also told that it had been decided not to report what had happened to the   university police, because the attendant publicity would reflect badly on the Design             Department. To my great regret, I acceded to the demands made by these two young men.)   Fig. E2.6b F-RD view: [111] direction Fig. E2.6c F-RD view: [110] direction Fig. E2.6d F-RD view: [100] direction Fig. E2.6e F-RD translation fundamental domain view: [111] direction Fig. E2.6f F-RD One-half of a translation fundamental domain Fig. E2.6g F-RD One translation fundamental domain (stereo view) Fig. E2.6h F-RD One-and-a-half translation fundamental domains MY MEANDERING ODYSSEY A more or less chronological account of some of my physics research before I became interested in minimal surfaces I realize that except in books on the history of science or mathematics, it is not customary to describe the development of mathematical or scientific results in strictly chronological fashion, and only an unusually dedicated reader will have the stamina required to reach the end of this story. So much of what I did depended on chance events that it may sometimes seem to the reader like a sort of random walk. It was more than a decade after my journey began that I first had an inkling of where it might lead. For those readers who make it all the way to the end, I can only say, "Mazel Tov!" 1953-1957 Fig. E2.10 The author in 1954 sectioning a single crystal of silver at the jeweler's lathe used for radioactive tracer studies of atomic diffusion in metals and alloys. This apparatus — and most of the other equipment we used —was designed and/or assembled by Prof. David Lazarus, assisted by Carl T. Tomizuka, my distinguished predecessor at the University of Illinois in Urbana/Champaign. At the University of Illinois in the 1950s, research in condensed matter physics was heavily weighted toward the study of point defects in metals, semiconductors, and alkali halides. For my PhD research in David Lazarus's group, I made radioactive tracer measurements of atomic diffusion coefficients as a function of temperature and alloy composition in single crystals of Ag-Cd and Ag-In, using experimental techniques developed by Dave, his post-docs, and the students (principally Carl Tomizuka) who preceded me. Although I enjoyed this work at first, my progress was slow, and as I looked in awe at the accomplishments of some of my classmates, I gradually began to question whether I was temperamentally suited for a career as an experimental physicist. Besides, it seemed to me that my thesis topic did not have much scientific significance, and I saw little prospect of making any fundamental advance in the field of diffusion. I was fascinated, however, by the mathematics of random walks on lattices — a famous example of Brownian motion. This fascination eventually led me to my first significant discovery in physics — that by measuring the isotope effect for self-diffusion in an elemental crystal, one could distinguish between 'substitutional' diffusion and 'interstitial' diffusion. This had not previously been possible. How this came about is described below. One day in the spring of 1957, I read a 1952 paper by John Bardeen and Conyers Hering entitled, "Diffusion in Alloys and the Kirkendall Effect". Appendix A of that paper is an analysis by Hering of the difference between the diffusion coefficient of a vacancy (vacant atomic site) and that of an atom. It had long been widely accepted that the mechanism for self-diffusion in noble metals, for example, is the exchange of an atom with an isolated vacancy. The concentration of vacancies was known to be relatively dilute, even at the elevated temperatures required for observing self-diffusion. As a consequence, after an atom has exchanged positions with a particular vacancy, it is somewhat more than randomly likely that the next jump of that atom will be an exchange with the same vacancy. When that happens, the two consecutive jumps of the atom will have either partially or wholly canceled each other, and the atom is described by Hering as undergoing a correlated random walk. An atom in an interstitial position (e.g. a lithium atom in a germanium crystal), on the other hand, is believed to hop from one interstitial site to another with no correlation between the directions of consecutive jumps. Its diffusion is characterized as a strictly random walk. Hering proved that if atoms diffuse by the vacancy mechanism ('substitutional' diffusion) and the vacancies are relatively dilute, then the diffusion coefficient for an atom is smaller than the diffusion coefficient for a vacancy by a fractional amount that depends on the coordination number (number of nearest neighbors of an atom) of the host crystal. The smaller the coordination number, the larger this fraction. In a crystal with cubic symmetry in which a vacancy jumps a distance a with frequency Γ in an uncorrelated random walk, the diffusion coefficient for the vacancy is given by Dvacancy = a2 Γ / 6.    (E2.11) Hering showed that for the correlated random walk of an atom in a homogeneous crystal in which the atom-vacancy jump vector has at least two-fold rotational symmetry, the diffusion coefficient for the atom is given by Datom = f  a2 Γ / 6,    (E2.12) where the correlation factor f  is given by f = (1 + < cos θ >Av ) / (1 − < cos θ >Av );    (E2.13) < cos θ>Av is the average value of the cosine of the angle between two consecutive jumps of an atom. Since the diffusion of an atom in an interstitial position does not involve an exchange with a vacancy, the directions of its consecutive jumps are uncorrelated. I had a hunch that in crystals of cubic symmetry, Bardeen-Hering correlation would reduce both the self-diffusion coefficient and the isotope effect for self-diffusion by the same fractional amount. This turned out to be the case. Let Dα and Dβ be the self-diffusion coefficients of isotopes α and β of mass mα and mβ and correlation factors fα and fβ, respectively. I defined the isotope effect = ((Dβ / Dα) − 1) / ((mα / mβ)1/2 − 1).    (E2.14) In the absence of correlation effects, the isotope effect would be equal to unity. I conjectured that correlation effects would cause it to be equal instead to fβ. If this conjecture were correct, one could distinguish between interstitial self-diffusion and self-diffusion by the vacancy mechanism in cubic crystals simply by measuring the isotope effect for self-diffusion. I first estimated the influence of Bardeen-Hering correlation on the isotope effect by using an approximate model of correlation published by Alan LeClaire and Alan Lidiard in Phil. Mag., 1, 518 (1956). This rough estimate appeared to confirm my hunch that in crystals with the required symmetry, the isotope effect is equal to the correlation factor. 1958-1964 In 1958, in order to refine my calculation of the influence of correlation on the isotope effect, I designed a Fortran program for extending it to a higher order of approximation. In this program, I modeled the infinite crystal by a sequence of four cubically symmetrical sub-crystals of successively larger volumes, each centered at the initial site of the diffusing atom. Fig. E2.14 The four cubically symmetrical sub-crystals in my Fortran program for the random walk of a vacancy The top row lists the number of atomic sites in each sub-crystal. The vacancy is located at the center of each sub-crystal. The four sub-crystals contain nearest neighbors, 2nd nearest neighbors, 3rd nearest neighbors, and 4th nearest neighbors of the vacancy, respectively. A vacancy, starting from a site adjacent to the diffusing atom, was allowed to execute an infinite random walk, during which it had a finite probability of escaping through the boundary of the sub-crystal. Program runs on an IBM 704 computer for each of the five successively larger subcrystals confirmed that correlation does indeed reduce both the isotope effect and the self-diffusion coefficient by exactly the same fractional amount (with an accuracy of at least eight significant figures), in agreement with the calculation I had made earlier using the Lidiard-LeClaire model. I showed my results to Alan Lidiard when he visited me in San Diego. Shortly afterward, he and K. Tharmalingam published an algebraic proof that my results were exact, in an article entitled 'Isotope Effect in Vacancy Diffusion' (Phil. Mag., 4, Issue 44, 1959, pp. 899-906). Junjiro Kanamori In 1960, I derived complicated combinatorial expressions for the the Bardeen-Hering correlation factor for self-diffusion by the vacancy mechanism on the sites of four 2-dimensional and four 3-dimensional crystal structures. A few weeks later, while I was visiting the University of Chicago, I showed these expressions to the theoretical physicist Junjiro Kanamori, who immediately described for me a widely used method of transforming such combinatorial expressions into multiple integrals. Thanks to Kanamori's help, Robert W. Lowen, Jr. and I were then able to evaluate correlation factors for seven of these eight structures. (I don't recall whether we ever completed our calculations for the eighth structure, the face-centered cubic lattice, for which Hering had obtained a value of 0.78.) We reduced the correlation factors for the other three 3-dimensional cases to triple elliptic integrals and published our results in the Bulletin of the American Physical Society, April 1960, 4, No. 5, p. 280 (cf. Figs. E2.15a,b). Fig. E2.15a Computed values of Bardeen-Hering correlation factors (APS abstract)   STRUCTURE    Z              − < cos θ >Av          f  linear chain    2   1          0  honeycomb layer    3   1/2     .333333  square layer    4   1 − 2/π     .466942  triangle layer    6   5/6 − √ 3/π     .566057  diamond    4   1/3     .500000  simple cubic    6   .209841     .653120  body-centered cubic    8   1 − Γ4(1/4)/8π3 − 8π/Γ4(1/4)     .727194 Fig. E2.15b Correlation factors for seven crystal structures After 1959, I tried — with limited success — to invent a systematic duality recipe for associating infinite periodic graphs in pairs to represent plausible geometrical pathways for diffusing atoms in both vacancy-exchange diffusion and interstitial diffusion. It never occurred to me to imagine a surface of some sort separating such pairs of graphs until 1964, when I learned about the Coxeter-Petrie [infinite] regular skew polyhedra, (Only in 1966 did I realize that these three infinite polyhedra are 'flattened and folded' incarnations of Schwarz's P and D surfaces. When I met Donald Coxeter for the first time at a 1966 geometry conference in Santa Barbara, he told me that he had never heard of the Schwarz surfaces. He looked thoroughly startled when I showed him my plastic models of P and D, which he examined closely for a couple of minutes or more before saying a word.) I catalogued a variety of examples of crystal structures that could be neatly partitioned into two disjoint substructures, and I computed the shapes of the Voronoi cells for many examples of unary, binary, and ternary crystal structures. From time to time I made wooden models of many of these polyhedra and used some of them as nodes for ball-and-stick network models of crystal structures. An especially useful resource for me in those days was 'Third Dimension in Chemistry', by Alexander F. ('Jumbo') Wells (cf. John Tanaka's oral history interview of Wells). It was from this book by Wells that I learned, in 1958, of the existence of the Laves graph. A graph is called symmetric if all of its vertices are symmetrically equivalent and all its edges are symmetrically equivalent. Another way of saying this is: A symmetric graph is one that is both edge-transitive and vertex-transitive. A regular graph, on the other hand, is one in which every vertex has the same degree. Hence every symmetric graph is regular, but not every regular graph is symmetric. I know of only three symmetric graphs on cubic lattices: the simple cubic graph (degree six), the diamond graph (degree four), and the Laves graph (degree three). I imagined in fantasy an elemental crystal whose atomic sites correspond to the vertices of a single Laves graph, with self-diffusion occurring by means of atom-vacancy exchanges. As an additional part of the fantasy, I imagined measuring the isotope effect for self-diffusion in this hypothetical crystal. It seems that the Bardeen-Hering correlation factor (cf. Eq. E1.3) has never been computed for the Laves graph, but it is likely to have a value of less than one-half, since the degree of the Laves graph is smaller than that of the diamond graph. (The data in Fig. E2.15b suggest the possibility that this correlation factor may be exactly 1/3. I may calculate it some day, just for fun!) Consequently the isotope effect, which theory says is equal to the correlation factor, would also be less than one-half. But I realized that the fantasy was far-fetched, because the interstices in this hypothetical crystal would be so large that self-diffusion would not necessarily occur by a simple vacancy exchange mechanism. On the other hand, suppose there exists a strongly ordered binary intermetallic compound in which the atoms of the two elements sit on the respective sites of two dual Laves graphs. Such a structure would be analogous to Zn-S (zinc sulfide), or In-Sb (indium antimonide), but with coordination number (degree) of each subgraph equal to three, not four. In a hard sphere model of such a structure, the interstitial cavities would be of modest size if the sphere radii for the two species were not grossly different. Radioactive tracer measurements for each species of the isotope effect for self-diffusion would provide evidence for or against the hypothesis that self-diffusion occurs by the vacancy mechanism. Below is a stereo pair of recent photos of an ancient model of the two inter-penetrating Laves graphs. A pair of dual Laves graphs Every node (wooden triangle) in each graph is joined by a wooden dowel not only to its three nearest neighbors in the graph, but also to its two nearest neighbors in the dual graph. UPDATE: Fig. E2.16 Toshikazu Sunada In a highly original article in the Notices of the American Mathematical Society in 2007, Crystals that Nature Might Miss Creating, the mathematican Toshikazu_Sunada — who was unaware of the history of the Laves graph — independently predicted its existence, making use of results of his research on random walk on crystal lattices. In his remarkable analysis, Sunada's 'K4 crystal' (i.e., the Laves graph) emerges as the unique mathematical twin of the diamond crystal. He proves that diamond and K4 are the only three-dimensional crystals with the property he calls strong isotropy, and also that the honeycomb (cf. graphene) is the only two-dimensional crystal with this property. (I confess that I understand only the easy parts of Sunada's article!) Sunada's 2008 letter to AMS Notices about the Laves graph my 2008 letter to AMS Notices about the Laves graph 1964 − April 1966 In July 1964, after spending a few stimulating months consulting for a new division of Beckman Instruments on the design of apparatus for measurements of the Mössbauer effect, I joined the Physics Research Laboratory of Space Technology Laboratories (STL) in Los Angeles. Within a few months, STL was acquired by TRW and changed its name to TRW Systems. I continued — from time to time — to ponder the question of how to develop a 'partitioning algorithm' for inter-penetrating pairs of triply-periodic graphs. April 1966 −April 1967 One day in April, 1966, in a hallway of the TRW Physics Research Laboratory, I noticed an engineer who was drawing polygons on a large plastic sphere. When I [politely] asked him what he was doing, he replied with some impatience that he was trying to model a fly's-eye lens by arranging a few hundred hexagons on a sphere but was having some difficulties. (In 'Ernst Haeckel (1843 − 1919) is still a problem', Eclectica (2009), Alan Mackay describes a similar error made — and subsequently corrected — by Ernst Haeckel.) Trying not to sound patronizing, I suggested to the engineer that he might consider including some pentagons, and I explained why the patterm he was searching for didn't exist. I told him the famous story about Euler and the bridges of Königsberg, and I explained that Euler had derived an equation that imposes combinatorial constraints on every possible combination of polygons that tile the sphere. Because it was obvious that he was somewhat less than pleased by my butting in, I decided not to pursue the matter further. However, I did casually mention the incident to my supervisor. A few days later, I was invited by the research vice-president of TRW Systems to spend one or two days every month as a kind of informal consultant to a group of company engineers who were designing a manned space station. (I knew very little about structural engineering, but in 1965 — four years before the first lunar landing — I had submitted an invention disclosure to TRW describing a modular building system designed for use on the moon. It employed hollow columnar space-frames, based on the geometry of space-filling tetragonal disphenoids. Each column, which was clad in aluminum and stored in a flat collapsed configuration, was designed to be self-deployed after delivery to the moon. Columns could be filled with lunar sand, so that a shelter constructed from an assembly of columns would provide effective shielding from dangerous radiation.) In order to catch up on gossip about the current state of the art in modular building systems, I paid a visit to the distinguished architect Konrad Wachsmann, chairman of the architecture department at nearby University of Southern California. Wachsmann in turn referred me to the North Hollywood architect/designer Peter Pearce, who was studying polyhedra, crystal structures, periodic three-dimensional networks, and the design of a modeling kit for both polyhedra and networks. Peter had received a grant from the Graham Foundation to study natural and man-made periodic structures. He showed me many ball-and-stick models of crystal structures he had constructed with the help of his assistant, Bob Brooks. Illustrations of these models appeared twelve years later in Peter's book, 'Structure in Nature is a Strategy for Design', MIT Press (1978). Peter told me that he had been inspired especially by R. Buckminster Fuller, Alexander F. Wells, D'Arcy Thompson, and Charles Eames, his former employer. Two of Peter's models each contained a specimen of what he called saddle polyhedra and made a profound impression on me. Peter had seen a museum exhibit designed by Charles and Ray Eames in collaboration with the mathematician Ray Redheffer, in which a motor-driven quadrangular wire frame emerged repeatedly from a beaker of soap solution with a physical approximation to a minimal surface spanning its boundary. Peter recognized that by spanning appropriate circuits of edges in triply-periodic graphs with plastic polygons that approximated minimal surfaces, he could fill the interstitial cavities in those graphs with saddle polyhedra. Peter's concept of saddle polyhedron struck me instantly as the critical ingredient required to complete the duality rule ('partitioning algorithm') I had been mulling over in my struggle to develop a systematic relation between substitutional and interstitial sites in crystal structures. Although I never expected to find a rule applicable to every possible triply-periodic graph, I did hope to find one that would work at least for every symmetric graph — a graph which is both edge-transitive and vertex transitive, i.e., a graph in which all vertices are symmetrically equivalent and all edges are symmetrically equivalent. As explained in pp. 76-85 of Infinite Periodic Minimal Surfaces Without Self-Intersections), however, I discovered that although it is not necessary for the graph to be symmetric, it is apparently necessary to add the stipulation that for symmetric graphs, each vertex of the graph g is joined by an edge to every one of the Z nearest neighbor vertices (such a graph is called a graph of maximum degree with respect to the vertices). each vertex lies at the centroid of the positions of the Z nearest neighbor vertices (such a graph is called locally centered). (But I discovered later that the two saddle polyhedra shown below in Figs. E2.50-E2.53 and in Figs. E2.55a,b,c demonstrate that these conditions are too restrictive.) One (cf. Fig. E2.20) of Peter's two saddle polyhedra filled an interstitial cavity of the diamond graph, a symmetric graph of degree four on the vertices of the diamond crystal structure, while the other (cf. Fig. E2.25) filled an interstitial cavity of the body-centered cubic (bcc) graph, a symmetric graph of degree eight on the vertices of the bcc lattice. Each of these saddle polyhedra is called the interstitial polyhedron of the graph g and has the following properties: (a) the interstitial polyhedron and the graph g have the same point group symmetry with respect to the center of the cavity; (b) the number of faces of the interstitial polyhedron is equal to the degree Z (number of edges incident at each vertex) of a second [dual] graph ginterstitial, in which there is a vertex v at each cavity center and Z edges — incident at v — that protrude through the faces of the interstitial polyhedron. Each of these Z edges is incident also at a vertex v of one of the Z adjacent interstitial polyhedra. Because the diamond graph happens to be self-dual, if every vertex of the graph is enclosed by a replica of the interstitial polyhedron, the assembly of such polyhedra — just like the assembly of interstitial polyhedra that occupy the interstitial cavities — define a packing of R3. In this role, these saddle polyhedra are called nodal polyhedra. The nodal polyhedron of the bcc graph is shown in Fig. E2.27. For some infinite symmetric graphs — depending on the proximity of vertices in coordination shells beyond the first — the number F of faces of the space-filling Voronoi polyhedron that encloses each vertex is greater than Z. The simplest example of a pair of symmetric graphs that illustrate the duality expressed by properties (a) and (b) is a pair of simple cubic (sc) graphs (cf. Fig. E2.58). My goal was to incorporate the concept of saddle polyhedron in a procedure that defines this duality in a systematic way. Among the many triply-periodic graphs that exhibit both properties (a) and (b) defined above are the seven symmetric graphs listed in the table below. The fcc graph and the FCC6(I) graph are the only examples among these seven for which the dual graphs are not also symmetric. For the fcc and sc graphs, both the nodal and interstitial polyhedra happen to be convex.   GRAPH    Z    F     RELEVANT FIGS.  Laves    3   17     E2.41, E2.42  WP    4   12     E2.36  diamond    4   16     E2.19c  sc    6    6     E2.58 − E2.60  FCC6(I)    6   12     E2.46 − E2.49  bcc    8   10     E2.24b, E2.36  fcc    12   12     E2.56d − E2.56f Fig. E2.17 Examples of symmetric graphs on cubic lattices Z = degree of the graph (coordination number). F = number of faces of the Voronoi polyhedron associated with each vertex.    Diamond graph   dual graph   diamond graph   nodal polyhedron   expanded regular tetrahedron ERT   interstitial polyhedron   expanded regular tetrahedron ERT Fig. E2.18 Fig. E2.19a                                 Fig.2.19b Schwarz's D surface (images courtesy of Ken Brakke) The diamond graph is the skeletal graph of both labyrinths. Fig. E2.19c A portion of the diamond graph (Z=4) The edges of the expanded regular tetrahedron ERT (cf. Fig. E2.20), interstitial polyhedron of the diamond graph, are shown in blue. Fig. E2.20 The expanded regular tetrahedron ERT, interstitial polyhedron of the diamond graph It is also the nodal polyhedron of the diamond graph. (The diamond graph is self-dual.) ERT is the saddle polyhedron Peter Pearce constructed in an interstitial cavity of the diamond graph (cf. Fig. E2.19c). Each face is a regular skew hexagon with face angle θ = cos-1(− 1/3) =~109.47°. Fig. E2.21 The regular tetrahedron and the edges (black lines) of ERT Fig. E2.22 The 16-face Voronoi cell for the vertices of the diamond graph For a sharper [pdf] version of this image, look here.   bcc graph   dual graph   WP graph   nodal polyhedron   expanded regular octahedron ERO   interstitial polyhedron   tetragonal tetrahedron TT Fig. E2.23 Fig.2.24a A cubic unit cell of I-WP The skeletal graphs are the bcc graph − aka the I graph −and the WP graph. Fig. E2.24b The bcc graph (Z=8) The edges of the tetragonal tetrahedron TT (cf. Fig. E2.25) are shown in blue. Fig. E2.24c The bcc graph (green vertices) and its dual, the WP graph (orange vertices) Fig. E2.25 The tetragonal tetrahedron TT, interstitial polyhedron of the bcc graph TT is the saddle polyhedron Peter Pearce constructed in an interstitial cavity of the bcc graph (cf. Fig. E2.24). Each face is a regular skew quadrangle with face angle θ = cos-1(1/3) =~70.13°. Fig. E2.26 The eight edges (purple) of TT For a sharper [pdf] version of this image, look here. Fig. E2.27 The expanded regular octahedron ERO, nodal polyhedron of the bcc graph Fig. E2.28 It appears that ERO is identical to the asymptotic limit surface suggested by this image from Ken Brakke's Surface Evolver sequence of surfaces of successively higher genus in the Neovius C(P) family. (I first observed this curious result in 1975 and described it in a letter to the physicist Tullio Regge in response to some questions from him about minimal surface soap film experiments. I had met Tullio a few weeks earlier at a Providence conference hosted by Tom Banchoff. Tullio was a formidable expert in differential geometry. He had studied the work of the pioneering 19th century Italian masters of the subject (Bianchi et al) when he was a young student.) Fig. E2.29 The twenty-four edges (black) of ERO For a sharper [pdf] version of this image, look here. Fig. E2.30 The 14-face Voronoi cell for the vertices of the b.c.c graph For a sharper [pdf] version of this image, look here. Fig. E2.31               Fig. E2.32 Fig. E2.33               Fig. E2.34 The pair of surfaces illustrated in Figs. E2.31 and E2.33 are portions of F—RD and I—WP, respectively. Note that each of the two surfaces shown in Figs. E2.32 and E2.34 can be regarded, approximately, as the image of the surface at its left after rotation through 45 degrees about a vertical axis. I obtained experimental confirmation of the existence of the surface in Fig. E2.32 in 1975, using a laser to measure the surface normal orientation for a set of hypothetical adjoint soap films near their boundaries. This method can be described as an extremely tedious (and far from precise) way to kill periods. In 2001, Ken Brakke accomplished the same task with enormously greater precision using his Surface Evolver and also confirmed the existence of the surface in Fig. E2.34. (These four images were all made by Ken Brakke.)   WP graph   dual graph   bcc graph − aka the I graph   nodal polyhedron   tetragonal tetrahedron TT   interstitial polyhedron   expanded regular octahedron ERO Fig. E2.35 Fig. E2.36 The WP graph (Z=4) The edges of ERO (cf. Figs. E2.27, E2.37) , interstitial polyhedron of this graph, are shown in blue. Fig. E2.37 The expanded regular octahedron ERO, interstitial polyhedron of the WP graph In 1966, while assembling the faces of this model, I discovered that if adjacent hexagons are related by rotation instead of reflection, the result is an infinite smooth surface — Schwarz's D surface. (That was my introduction to triply-periodic minimal surfaces. A few minutes later, I replaced the 90° hexagons by 60° hexagons and obtained Schwarz's P surface.) Fig. E2.38 The tetragonal tetrahedron TT, nodal polyhedron of the WP graph (cf. Fig. E2.36) Fig. E2.39 The expanded octahedron EO, the Voronoi cell for the vertices of WP For a sharper [pdf] version of this image, look here.   Laves graph   dual graph   enantiomorphic Laves graph   nodal polyhedron   trigonal trihedron T   interstitial polyhedron   trigonal trihedron T' (enantiomorph of T) Fig. E2.40 Fig. E2.41 A portion of the Laves graph (Z=3) The edges of the trigonal trihedron TT , interstitial polyhedron of the Laves graph, are shown in blue (cf. model in Figs. E2.43 and E2.44). Fig. E2.42 A view of the edges of TT from a direction different from that in Fig. E2.41 Fig. E2.43 The trigonal trihedron TT, nodal polyhedron for the Laves graph (cf. Figs. E2.41 and E2.42) is a skew decagon with 120° face angles. Fig. E2.44 Another view of TT       FCC6(I) graph   dual graph   a non-symmetric graph of degree 10   nodal polyhedron   pinwheel polyhedron PP   interstitial polyhedron   doubly expanded tetrahedron DET Fig. E2.45 Fig. E2.46 Four vertices' worth of the FCC6(I) graph (Z=6), a locally-centered deficient symmetric graph (LCDSG) The vertices of the FCC6(I) graph are those of the fcc lattice. At each vertex, six of the twelve edges of the standard fcc graph are omitted. FCC6(I) is described on pp. 47-48 of Infinite Periodic Minimal Surfaces Without Self-Intersections. Fig. E2.47 Four vertices' worth of the FCC6(I) graph inscribed on the surface of the polyhedron VPdiamond (Voronoi polyhedron for the diamond crystal structure) Fig. E2.48 Five VPdiamonds' worth of the FCC6(I) graph Fig. E2.49 Five VPdiamonds' worth of the FCC6(I) graph Fig. E2.50 Doubly expanded tetrahedron ('DET'), the interstitial polyhedron of the FCC6(I) graph Six faces are regular skew quadrangles, and four faces are regular skew hexagons. Fig. E2.51 Two views of DET (stereo) Fig. E2.52 The Doubly Expanded Tetrahedron is so named because its twenty-four edges are produced by reflecting each edge of every face of a regular tetrahedron in each of the two other edges of that face. (Every edge of the tetrahedron is reflected four times, since it is incident at two faces.) Fig. E2.53 When an infinite set of DETs is assembled by gluing hexagonal faces together in pairs, the quadrangular faces remain exposed and define Schwarz's D surface. Fig. E2.54 When an infinite set of DETs is assembled by gluing quadrangular faces together in pairs, the hexagonal faces remain exposed and define Schwarz's P surface. Fig. E2.55a Pinwheel polyhedron PP, the nodal polyhedron of the FCC6(I) graph Fig. E2.55b Pinwheel polyhedron PP, rendered in Mathematica Fig. E2.55c The red edges are the edges of the nodal polyhedron PP of the FCC6(I) graph. It has the same volume as the Voronoi polyhedron (rhombic dodecahedron). Fig. E2.55d Partial packings of PP, rendered in Mathematica (The quadrangular surface patch module is rendered here as a doubly-ruled surface. Although it resembles the actual minimal surface, it is only an approximation.) Below are four views of a portion of the compound [self-intersecting] surface composed of replicas of just one of the two enantiomorphous quadrangular surface patches that make up the pinwheel polyhedron PP (above). Three patches are incident at edges of type [111], but along edges of type [100], two patches are smoothly related by a half turn. I learned of the concept of compound surfaces of this type from the mathematician Dennis Johnson, who explained that intersections of three patches along directions of type [111] are analogous to the 120º intersections in froths of soap bubbles. Fig. E2.55e Viewed in [100] direction Click here for higher resolution image Fig. E2.55f Viewed in [110] direction Click here for higher resolution image Fig. E2.55g Viewed in [111] direction Click here for higher resolution image Fig. E2.55h Orthogonal projection on [111] plane Click here for higher resolution image       FCC graph   dual graph   fluorite graph   nodal polyhedron   rhombic dodecahedron   interstitial polyhedra   regular tetrahedron, octahedron Fig. E2.56a Fig. E2.56b                                              Fig. E2.56c Two views of F-RD Image at left courtesy of Ken Brakke Fig. E2.56d The nodal polyhedron of the FCC graph is the Voronoi polyhedron (rhombic dodecahedron). Fig. E2.57 The fluorite graph is the dual of the FCC graph. Its nodal polyhedra are the regular tetrahedron and the regular octahedron. Fig. E2.58 The FCC graph and the fluorite graph       sc graph   dual graph   sc graph   nodal polyhedron   cube   interstitial polyhedron   cube Fig. E2.59 Fig. E2.60 The sc graph, skeletal graph of one labyrinth of Schwarz's P surface Fig. E2.61 The sc graph, which is also the skeletal graph of the other labyrinth of Schwarz's P surface Fig. E2.62 The congruent skeletal graphs of the two disjoint labyrinths of Schwarz's P surface (stereo pair) A few days after I met Peter Pearce, I observed with astonishment that for certain shapes of saddle polygons spanned by a minimal surface, e.g., the 90° regular skew hexagon (a module for Schwarz's D surface) or the 60° regular skew hexagon (a module for Schwarz's P surface), if two specimens of the saddle polygon are related by a half-turn about a common edge, instead of by mirror reflection in a plane containing that edge (which is the arrangement in most, although not all, of the saddle polyhedra I had explored by then), not only does the junction between the two polygons appear to be perfectly smooth, but an endless sequence of these half-turns produces a single smooth, embedded infinite labyrinthine surface with the global topology and symmetry of a Coxeter-Petrie regular skew polyhedron! I had accidentally stumbled onto two examples of the application of Schwarz's reflection principle, his P and D surfaces − two objects that I had never heard of. I was unable to locate a reference to either of these surfaces in my books on geometry or differential geometry. Because I realized that some contemporary mathematicians must be familiar with these two surfaces, I paid a visit to the UCLA math department, where I showed my plastic models to the two faculty specialists in differential geometry. But neither of them recognized the two surfaces! Next: a visit to the UCLA science library, where I learned that Johannes C. C. Nitsche, a prolific mathematician on the faculty of the University of Minnesota, was a noted authority on minimal surfaces (cf. his summary of the field, "A Course in Minimal Surfaces"). When I telephoned him and described what I had been doing, he kindly explained that I had probably made models of the two TPMS for which H. A. Schwarz (and also Riemann and Weierstrass, as I was to learn later) had developed solutions in 1866. He referred me to Vol. 1 of Schwarz's Collected Works. From a quick perusal of this tome, I learned that Schwarz had also discovered two other examples of TPMS — H and CLP, both also of genus 3 — and I made plastic models of them too. I soon noticed that on p. 271 of Hilbert and Cohn-Vossen's 'Geometry and the Imagination'; the authors write that In this way, Neovius4 succeeded in constructing a minimal surface that extends over the entire space without singularity or self-intersection and has the same symmetry as the diamond lattice (italics added). 4E. R. Neovus, Bestimmung zweier speziellen periodischen Minimälflachen, Akad. Abhandlung, Helsingfors, 1883 I foolishly assumed that the authors had simply become confused here and were actually referring to Schwarz's D surface. Eighteen months later, I learned that Neovius had treated an entirely different surface of genus 9. I invented a naive scheme for identifying and labeling these surfaces, each of which I regarded as lying between the two triply-periodic graphs of a dual pair. I named these pairs of graphs 'skeletal graphs', because I thought of them as the skeletons of their respective hollow labyrinths. I found it helpful to regard the skeletal graph edges as thin hollow tubes that could be enlarged by inflating them until the whole graph was transformed into the TPMS. Then if the tubes were overinflated, the graph would eventually shrink down into the dual graph! I imagined that for at least a portion of this inflation cycle, the surface of the graph would define a triply-periodic surface of non-zero constant mean curvature. As I began my admittedly superficial study of the mathematical underpinnings of these surfaces, beginning with the two Schwarz reflection principles, I couldn't help wondering what other examples of embedded 'TPMS' might exist. In particular, I wondered whether there was a TPMS whose skeletal graphs were the enantiomorphous pair of Laves graphs I had learned about seven years earlier in 'Third Dimension in Chemistry', by Alexander F. Wells. A pair of enantiomorphic Laves graphs that are related by inversion has bcc. translation symmetry. It struck me as curious that of the three different cubic lattice symmetries — simple cubic (sc), face-centered cubic (fcc), and body-centered cubic (bcc) — bcc was missing from the inventory of cubic lattice symmetries for known examples of TPMS of ultimately simple topology (genus three). A second reason for my focus on the Laves graph was that it was apparently the only other example of a triply-periodic graph — besides the simple cubic and diamond graphs, which are the skeletal graphs of P and D, respectively — in which congruent regular polygons are incident at each edge. In the Laves graph, there are precisely two regular polygons incident at each edge. They happen to be infinite helical polygons, centered on lines parallel to two of the three coordinate axes. (I had been strongly influenced by Coxeter's 'Regular Polytopes', and I believed one should take regularity very seriously!) The Laves graph is not a reflexive regular polyhedron, however, and its lack of reflection symmetries made it impossible for me to imagine just how it could serve as the skeletal graph of the labyrinth of an embedded TPMS. But the most compelling reason for my conviction that there must exist an embedded TPMS whose skeletal graphs are enantiomorphic Laves graphs was that the simple cubic graph, the diamond graph, and the Laves graph were the only examples I could identify of symmetric triply-periodic graphs of cubic symmetry that are self-dual. Even though I knew of no theoretical justification for claiming that such graphs — regarded as skeletal graphs of embedded TPMS — play a unique role in defining embedded TPMS of cubic symmetry, I nevertheless believed that they must play such a role! I was aware of the fact that the concept of skeletal graph was itself somewhat ill-defined. It seemed to me to be a very 'natural' construct when applied to the then known examples of embedded TPMS, but I had no idea how to prove that for every possible example of an embedded TPMS there is a unique pair of skeletal graphs. All of these considerations at times seemed to me to smack more of theology than of mathematics. (I am reminded that the young Riemann, who was probably the first to solve the equations for what we now call Schwarz's P and D surfaces, as a young man abandoned the study of theology (his pastor father's choice) for a career in mathematics! I resolved to learn more about TPMS, which I recognized as far more interesting objects than saddle polyhedra, but in the meantime I was determined to continue exploring the relation between triply-periodic graphs and saddle polyhedra. On evenings and weekends throughout the spring and summer of 1966, I used a toy vacuum-forming machine and home-made moulds cast from polyester resin poured against a thin stretched rubber membrane to make dozens of saddle polyhedra of different shapes, all of which I shared with Peter Pearce. He preferred to make his saddle polygons by draw-forming— pushing a tool in the shape of a skew polygon outline against a transparent vinyl sheet that had been softened by heating. I preferred vacuum-forming with solid moulds, but it was clear that Peter's method also worked well. It has the advantage of not requiring the extra labor involved in making a mould, but the disadvantage is that it cannot replicate the shape of a minimal surface as well as a carefully crafted mould can. During these months of experimenting, I found no counterexample to my improvised duality rule, even for triply-periodic graphs that are not symmetric. In May, I hit on the idea of what I rather lamely called a 'defective' symmetric graph (I decided later that 'deficient' might be a more appropriate name) — a symmetric graph A derived from a second symmetric graph B by omitting some of the edges but none of the vertices of B. In A, not every pair of nearest neighbor vertices is joined by an edge. I required that every deficient symmetric graph be locally-centered, i.e., that every vertex lie at the centroid of the vertices with which it shares an edge. For the simple cubic lattice, it's easy to prove — simply by enumerating each of the possible locally-centered subsets of edges that contains at least three edges — that it is impossible to construct a locally-centered deficient symmetric graph (LCDSG) on the vertices. I have no idea why I failed to ask myself in those days whether there exists a LCDSG on the vertices of the body-centered cubic lattice. The only example of a LCDSG that I examined in 1966 was a graph of degree six I call FCC6(I) (cf. Fgs. E2.45 - E2.55b). Its vertices are those of the face-centered cubic lattice. I used the name FCC for the familiar symmetric graph of degree twelve on the vertices of the fcc lattice, in which every pair of nearest neighbor vertices is joined by an edge. I derived the deficient graph FCC6(I) by removing a symmetrical set of six out-of-plane edges from each vertex, leaving behind a flat six-edge cluster that occurs in each of the four possible [111] orientations. Although FCC6(I) proved not to be a counterexample to the duality rule, I was not confident that the rule would always hold even if I were to restrict it to symmetric graphs only. As it happened, not only did the rule not fail in the case of FCC6(I) — it yielded an unexpectedly interesting pair of saddle polyhedra. I call the interstitial polyhedron — shown in Fig. E2.51 — the Doubly Expanded Tetrahedron ('DET'), and the corresponding nodal polyhedron — shown in Fig. E2.55a — the Pinwheel Polyhedron. The Doubly Expanded Tetrahedron is the first example I had encountered of a space-filling saddle polyhedron in which the faces are of two kinds — hexagons and quadrangles. As illustrated in Figs. E2.53 and E2.54, Schwarz's P and D surfaces can be formed from either the quadrangular or the hexagonal faces of an infinite 'porous packing' of DETs, according to whether neighboring DETs share quadrangular faces (P) or hexagonal faces (D). April 1967 − July 1970 In the spring of 1967, the physicist Lester C. Van Atta, who was Associate Director of the NASA Electronics Research Center ('ERC') in Cambridge, Massachusetts, came to Los Angeles for a few days to visit his physicist son Bill, a friend and colleague of mine who happened to be a whiz at solving combinatorial puzzles. A year earlier, stimulated by a Martin Gardner column in Scientific American that described Piet Hein's SOMA puzzle, I had become hooked on investigating the possible symmetries of complementary half-cube packings by the eight solid tetrominoes (cf. Fig. E1.13). After Bill lent his father a set of these puzzle pieces, his father told Bill that he 'wanted to meet the guy who had cost [him] a night's sleep'. At the end of a very long late evening visit to my home, Lester abruptly invited me to join the NASA/ERC research staff in Cambridge, Massachusetts. He explained that I would be required only to 'follow my nose'. I found it difficult to believe that he was making me a serious job offer, and I didn't make any response. A few days later, Van Atta phoned me from Cambridge and told me in no uncertain terms that I had only three or four days left to get the paper work (a few dozen pages of federal employment application forms) in the mail, because it would be impossible to keep the position open longer than that. This time I took him seriously, and in July 1967 I moved to Massachusetts. I wanted to immerse myself immediately in the study of TPMS. However, I believed that I should first make a better organized attack on my embryonic duality rule ('partitioning algorithm'). I knew it was unlikely that the rule would be applicable to every possible triply-periodic graph, but I had no idea how to characterize those graphs for which it worked and those for which it didn't. Inspired by Polya's rules for problem solving, I continued to emphasize symmetric graphs — those graphs for which there is a symmetry group transitive on both vertices and edges. I wondered whether my duality rule worked for every possible symmetric graph. If I could find a counterexample, I wanted it to be as simple as possible. I had been hired at NASA/ERC as a mathematician (chief of a special section Van Atta created for me, called the 'Office of Geometrical Applications'), even though I was at best an amateur mathematican. Curiously it was my forays into recreational mathematics that had led Van Atta to hire me. I was never told by him or anyone else what I should work on, but of course it was understood that if I saw possible applications of what I was doing that might be of interest to NASA, I should not fail to pursue those applications. From the start, I undertook to learn more about the mathematics used in the study of minimal surfaces. I pored over three books on differential geometry and Schwarz's Collected Works. I was especially curious to know whether there were additional examples of TPMS just waiting to be discovered, but when I began to read the published literature in the field (some of it in German, of course), I often felt overwhelmed. I believed that I didn't have time to become sufficiently knowledgeable about the deep foundations of the relevant branches of mathematics — differential geometry and complex analysis — to make any 'breakthrough' advances in the field. I had developed a special interest since 1958 in the properties of infinite periodic graphs, and this interest had eventually led me to papers and books by Donald Coxeter. My interest in these graphs had sprung from my research on atomic diffusion in crystalline solids and from the mathematics of correlated random walks on discrete lattices. Because I had discovered (in 1957) that the magnitude of the isotope effect for atomic diffusion in crystals could distinguish between interstitial diffusion and substitutional diffusion, I was curious to learn which elements and compounds were likely to be good candidates for measuring the isotope effect for diffusion. I tried — with only slight success, initially — to develop an algorithm ('duality rule') for deriving the infinite periodic graph whose vertices correspond to the principal interstitial sites of a crystalline solid. WARNING: Continue at your own risk. The text in the next few paragraphs has not yet been edited. Much of it duplicates other portions of text and will eventually be integrated into the main narrative. I was able to confirm my hunch that the duality rule was valid for many pairs of triply periodic graphs — one of the pair being called substitutional and the other interstitial. In every case I tested, both nodal and interstitial polyhedra exhibited the following properties: (a) the number of faces of the polyhedron is equal to the number of edges of the associated periodic graph; (b) the symmetry of the polyhedron is identical to the symmetry of the associated vertex of the periodic graph. It didn't matter which graph was called substitutional and which was called interstitial. The dual relation between them is, after all, symmetrical. (Of course from a physical point of view, it would be absurd to call the large interstitial cavities in silicon — which can be occupied by smaller atoms like lithium, for example — substitutional and the silicon atomic sites interstitial!) I call the saddle polyhedron in Fig. E2.25 the interstitial polyhedron for the bcc graph of degree 8, because it is bounded by edges [of a periodic graph] that are the bars of a sort of interstitial cage. This same polyhedron is the nodal polyhedron for another triply periodic graph, which I named WP. The nodal polyhedron encloses a vertex of the periodic graph at its center, and the number of faces of the nodal polyhedron is equal to the number of edges incident at that vertex. The saddle polyhedron in Fig. E2.20 is both nodal polyhedron and interstitial polyhedron for the diamond graph. That means that in a space-filling array of these saddle polyhedra, each polyhedron can either (a) enclose at its center a vertex of the diamond graph or (b) occupy a single interstitial region bounded by the edges and vertices of the diamond graph. For some periodic graphs, either the nodal polyhedron or interstitial polyhedron (or both) may turn out to be a convex polyhedron with plane faces. For the simple cubic graph of degree six, for example, both polyhedra are cubes. In the simplest cases, the graph is unary, i.e., all the vertices of the graph are equivalent. But the duality rule works smoothly without requiring any ad hoc adjustments even for many non-unary graphs. (I plan eventually to post a picture or two of the polyhedra for such a graph.) As explained below, in early 1968 I searched for — and eventually found — a periodic graph for which the duality rule failed, and that failure led to the discovery of the pseudo-gyroid, which is composed of hexagons with perfectly helical boundary curves. Recall that FCC6(I) is a locally-centered deficient graph of degree six on the vertices of the fcclattice (cf. Fig. E2.48). In February, 1968, I realized that I had never searched for the obvious bcc. counterpart to FCC6(I): a locally-centered deficient graph on the vertices of the bcc. lattice. (It's easy to prove that no locally-centered deficient graph on the vertices of the sc lattice exists.) Once I started looking, it didn't take me long to discover BCC6, a symmetric graph of degree six on the vertices of the bcc. lattice. A portion of this graph is shown in Fig. E1.20d. BCC6 proved to be the long-sought counterexample to my 'duality rule'. Its edges are those of the infinite regular warped polyhedron ('IRWP') that I call M4 (cf. Fig. E1.16a and E1.20a to E1.20f). In the case of BCC6, the breakdown in the duality rule occurs at the very first step — the construction of the interstitial polyhedron. Instead of the finite interstitial polyhedron the duality rule was intended to generate, an infinite one — M4 — appeared. I shed no tears over this failure of the duality rule, because the two-labyrinth character of M4 suggested that something of potentially greater interest might be in the offing: an example of a previously unknown TPMS. I observed that the skeletal graphs of M4 were enantiomorphic Laves graphs. This was potentially exciting, because it suggested that M4 might somehow be transformed into the minimal surface (the gyroid) whose existence I had speculated about almost two years earlier. Appendix II of Infinite Periodic Minimal Surfaces Without Self-Intersections explains why I constructed M6 (in February 1968). M6 was the result of an attempt to improve on its predecessor, M4 (cf. Figs. E1.16a and E1.16b), which is composed of skew quadrangles spanned by minimal surfaces. I was looking for a way to 'smooth out the wrinkles' in M4. The hexagons in M6 are the duals of the quadrangles in M4. In M4, the bump across the edge shared by adjacent faces is 60°. In M6, it is only ~44.4°. I hoped that making the bump smaller by ~15.6° than the bump in M4 would enable M6 to look at least a little more like a continuous minimal surface than M4 did. As soon as I had constructed the model of M6 shown in Fig. E1.16b, I noticed that its edges define regular helical polygons with straight edges, centered on lines parallel to the rectangular coordinate axes (cf. Figs. E1.19b, c, e). I speculated that if I replaced the straight edges of M6 by helical arcs, the ~44.4° bump between adjacent faces might shrink to nearly zero. This was a wild and woolly guess, with no theoretical justification whatsoever, but the new physical model I constructed (cf. Fig. E1.17) was encouraging. The edge shape in the authentic gyroid just happens to differ so slightly from the shape of a helical arc that it is impossible to detect the difference betwen the two by eye. The glaringly obvious hints that I had missed from the outset stemmed from the identical combinatorial structure of each of the three regular tessellations of the three surfaces — P, D, and the pseudo-gyroid surface in Fig. E1.17. All three of these surfaces can be constructed of surface patches that correspond to faces of any of the three Coxeter-Petrie infinite regular skew polyhedra, with tangent planes identically oriented at corresponding vertices. Since I was already familiar with the details of how curves transform and how the tangent plane remains invariant at each point of the surface in the Bonnet bending of the catenoid into the helicoid, it should have occurred to me (but didn't!) that the almost perfectly circular lines of curvature in the coordinate planes of Schwarz's P surface would be transformed by Bonnet bending into almost perfectly helical lines of curvature before they finally became the linear asymptotics in Schwarz's D surface parallel to the coordinate axes. For the previous several months, I had begun to feel pressure to do something 'useful'. Even though Dr. Van Atta himself never once hinted that he was less than satisfied about how I chose to spend my time, there were growing signs that I could not afford to ignore indefinitely NASA's expectations that my work suggest at least the possibility of some 'practical' offshoots. Since I had no idea how to obtain an analytic solution for a minimal surface patch bounded by six helical arcs, I decided to give up trying to prove that the pseudo-gyroid is a minimal surface. Instead I sent a physical model of the pseudo-gyroid to Bob Osserman, who passed it along to his PhD student Blaine Lawson at Stanford. Blaine agreed to think about the problem, but he warned me that his dissertation would be keeping him extremely busy. I delved more deeply into the analysis of the 'continuous transformation on vertices and edges' mentioned in the abstract of Fig. E2.10. I attempted to identify every possible example of non-self-intersecting 'infinite regular warped polyhedra' (and 'infinite quasi-regular warped polyhedra'), whose faces are regular skew polygons. (cf. Figs. E1.35b and E1.35c.). At the same time, I analyzed the geometry of what I called the 'graph collapse' transformation, which is diagrammed for the 2-dimensional square graph in Fig. E2.68c. To the exclusion of almost everything else, I concentrated for several weeks on the engineering requirements for the application of this transformation to the design of expandable space frames. Finally I wrote a patent application with the help of two NASA patent attorneys who for two weeks flew up to Cambridge every morning from Washington. Here is a synopsis of the expandable space-frame patent, which was issued in 1975, and here is the complete text of the NASA patent, from which an illustration is shown in Fig. E2.68a. I estimated that with realistically designed struts, a value of 80:1 was feasible for the ratio of the expanded to collapsed volume of the space frame. With the help of Charles Strauss, Randy Lundberg, Bob Davis, Ken Paciulan, and Jay Epstein, I made an animated film of the collapse of the Laves graph and of three other symmetric graphs. The portion of the video '1969 'Part 4' that shows examples of the graph collapse transformation begins at 7min00sec after the beginning of the video. A few single frames from the film that illustrate the geometry of the collapse transformation applied to the Laves graph are on pp. 86-88 of Infinite Periodic Minimal Surfaces Without Self-Intersections. In the fully collapsed state, the vertices and edges of the [infinite] Laves graph are mapped onto the four vertices and six edges, respectively, of a single regular tetrahedron (cf. the tetrahedron AOBC in Fig. E2.68b.2). If one vertex (vertex O in Figs. E2.68b.0, E2.68b.1, and E2.68b.2) is fixed, the collapse trajectories of all the other vertices are ellipses centered on that vertex. For every vertex V, the major radius of the ellipse is equal to the initial distance of V from O. The minor radius of the ellipse is equal to the edge length of the graph if V is related to vertex a, b, or c by a translation that is a symmetry of the Laves graph — i.e., if V is red, green, or blue. The minor radius is equal to zero if V is related to vertex O by a translation that is a symmetry of the Laves graph — i.e., if V is yellow. Collapse onto tetrahedron AOBC occurs twice in each period of the transformation: at the two moments when either one-quarter or three-quarters of each elliptical trajectory has been traversed. Each of the vertices a, b, c rotates on a circular trajectory in one of the three orthogonal coordinate planes. Because of the screw isometries of the Laves graph, edges collide only at the two instants of collapse in each period. In an actual physical space-frame, however, struts are of finite thickness, and this causes edge collisions to occur well before collapse. (Precisely how early the collisions occur in each period depends on the thickness of the struts.) The analogous transformations for those regular graphs derived from Coxetrie-Petrie maps that contain reflection isometries are not physically realizable, because edges collide early in the transformation even though they are of zero thickness. Fig. E2.68a A hinged joint in the expandable space frame The collapse of the Laves graph is readily depicted by regarding the graph as initially embedded in the D surface (cf. Fig. E2.69b.0) and then allowing every vertex to be translated along a linear trajectory in a direction normal to the surface. Vertices related by a translational symmetry of the graph are colored the same. If the two sides of the D surface are labeled A and B, with motion in the direction from A to B defined as positive and motion in the direction from B to A defined as negative, then the two vertices incident on each edge move along normals of opposite sense. The computed positions of the vertices at each stage of the collapse are scaled by the requirement that edge lengths remain invariant, thereby causing the graph to shrink continuously, with all of its edges finally collapsing onto the six edges of a single regular tetrahedron — the tetrahedron with vertices O, A, B, C in Fig. E2.68b.2. Fig. E2.68b.0 The Laves graph embedded in the D surface The arrows indicate the initial directions of the curvilinear displacements of the vertices. Green diplacements are called positive, and red diplacements are called negative. Fig. E2.68b.1 The yellow vertex at O is now fixed. The arrows here indicate the initial directions of the curvilinear displacements in a coordinate system in which the yellow vertex O is at the origin. Fig. E2.68b.2 The circular trajectories of vertices a, b, and c and the elliptical trajectory of vertex d, in Fig. E2.68b.1 One-fourth of a complete trajectory period is shown here for these four vertices. Perhaps the easiest way to illustrate the collapse is to depict the circular trajectories, in orthogonal coordinate planes, of the three vertices that are nearest neighbors of any vertex V if V is regarded as fixed.. The next four images illustrate these trajectories for V = a yellow, red, green, or blue vertex. Fig. E2.68b.3 Rotation of R, G, and B vertices around Y vertex stereo view Fig. E2.68b.4 Rotation of G, B, and Y vertices around R vertex stereo view Fig. E2.68b.5 Rotation of B, Y, and R vertices around G vertex stereo view Fig. E2.68b.6 Rotation of Y, R, and G vertices around B vertex stereo view The stereo mages in Fig. E2.68c illustrate the application of the graph collapse transformation to the square graph, which is initially embedded in a minimal surface — the plane. In this 2-dimensional example, all the edges of the graph coalesce into a single vertical edge. The images below are parametrized by the value of θ, the angle of rotation of each edge of the graph out of the horizontal plane. The discussion below applies to a 16-vertex square portion of the infinite graph, but in principle the analysis applies to any finite [connected] piece of the graph, however large. In Fig. E2.68c.0, the vertex at A is held fixed. The horizontal positions of the vertex at A and the seven vertices (C, F, H, I, K, N, P) to which downward-pointing red arrows are attached are maintained at a constant level. (Ignore the red arrows. I shouldn't have included them in this image, which I plan to replace.) The vertices to which green arrows are attached are displaced upward, but they are constrained to move sideways because the [identical] links that connect the vertices are inextensible. The net result is that the links all rotate, moving on elliptical trajectories centered on A (cf. Fig. E2.68c.9). The ellipses for vertices F, K, and P degenerate into straight lines, and those for vertices B and E are circles. Collapse of all the links onto a single vertical edge occurs at θ=90° and θ=270° (cf. Figs. E2.68c.4 and Figs. E2.68c.5). (I'll soon make an animated sequence showing a full period of the motion.) Fig. E2.68c.0 Square grid graph before the start of the collapse transformation θ=0° Fig. E2.68c.1 θ=18° Fig. E2.68c.2 θ=52.2° Fig. E2.68c.3 θ=81° Fig. E2.68c.4 θ=90° Fig. E2.68c.5 θ=279° Fig. E2.68c.6 θ=307.8° Fig. E2.68c.7 θ=345° Fig. E2.68c.8 θ=360° Fig. E2.68c.9 Some additional history Inserting handles into a TPMS (See also Figs. E2.83 to E2.85.)   It occurred to me one day in the spring of 1969 that for some pairs S1 and S2 of moderately low-genus embedded TPMS for which the respective      adjoint surfaces S1† and S2† have elementary patches that are bounded by simply-connected straight-edged polygons P1 and P2, there must exist an embedded hybrid TPMS Sh whose adjoint surface Sh† has an elementary   patch with boundary polygon equal to a linear combination of P1 and P2.    (I defined a linear combination of two polygons P1 and P2 to be a polygon        interpolated between P1 and P2. The idea of constructing a linear combination of two polygons occurred to me after I recalled a description of linear       combinations of convex polyhedra that I had read in a Russian book on         polyhedra. I no longer recall the name of the book's author, but it may have been Aleksandr Aleksandroff. I have been unable to trace the book.)        Fig. E2.70 shows a 1969 sketch illustrating my scheme for constructing the hybrid of P and C(P). Fig. E.2.70 1969 proposal for a genus-14 embedded hybrid of P (genus 3) and C(P) (genus 9) The two scribbled captions 'Schwarz's "D" ' are erroneous. They should read 'Schwarz's "P" '. The arrows indicate the directions of the local surface normals. Consecutive edges of the quadrangle P1 at the upper right are 12, 34, 45, 51. Consecutive edges of the quadrangle P2 at the lower right are 12, 23, 34, 41. The transformation of P1 into P2 can be described as follows:   1. Edge 12 remains fixed in place. 2. Edges 34 and 45 are translated    along a linear trajectory in the [101] direction.                             3. New edge 23 grows at a steady rate.                         4. Old edge 51 is reduced to zero at a steady rate. The relative weights assigned to the adjoint polygons P1 and P2       require trial-and-error adjustment to make arcs 23 and 15 coplanar. The cognoscenti call this process 'killing periods'.                               After I sketched the notes shown in Fig. E.2.70, I phoned Blaine Lawson, who was already an expert on minimal surfaces. He was then approaching the last stages of his PhD dissertation research at Stanford under Bob Osserman. I asked him if he thought it was plausible that a hybrid derived from these two straight-edged polygons would define an embedded surface. Blaine replied that it was not an unreasonable idea, because the intermediate value theorem guarantees successful 'period killing' — or words to that effect. I'm sure 'period killing' wasn't the exact expression he used. I recall first hearing those words several years later, in a telephone conversation with David Hoffman who — in collaboration with Bill Meeks— derived the spectacular family of Costa-Hoffman-Meeks surfaces. The original surface from that family is shown below, in a stereo image due to Hermann Karcher. The original Costa-Hoffman-Meeks surface I mailed (faxed?) Blaine a copy of my Fig. E.2.70 sketch, but then I abandoned the P—C(P) hybrid, because I realized that vacuum-forming a surface patch with a severe undercut — like the one shown at left center in Fig. E.2.70 — would be difficult or impossible. Determined to build a physical model of some hybrid surface, I turned instead to a topologically simpler case — the hybrid of P and I-WP that I call O,C-TO (cf. Figs. E2.79 to E2.81). Its genus is only 10. Then for the next forty-two years, I forgot all about the P—C(P) hybrid! This process of hybridization is equivalent to attaching a handle to a minimal surface. During the years 1969-1975, I employed an extremely laborious experimental technique I had devised, based on a method of successive approximations, to hybridize several examples of TPMS. With this technique, a laser is used to measure the orientation of the tangent plane of a long-lasting polyoxyethylene soap film S† at a sequence of closely spaced points near its boundary. S† is spanned by a straight-edged skew polygon P†, which is a candidate for the boundary of the adjoint of the curved-edge boundary polygon P of an elementary patch S of the hybrid (the surface containing the added handles). The relative lengths of the straight edges of the first of several successive candidates for P† are just rough estimates. When the shapes of the curved edges of the corresponding version of P are numerically derived (using a very simple computer program) from the measurements of this first P† candidate, it is inevitably found that P fails to close. Corrections are then applied to the relative lengths of the edges of P†, and an improved version of P — i.e., one that is more nearly a closed polygon — is obtained. After four or five iterations of this procedure, a satisfactory approximation to P is obtained (unless, of course, there is no such surface as P). The laser measurements and the accompanying numerical calculations are the least tedious steps in this procedure. The most time-consuming step by far is fabricating an accurate physical model of P†. By 1991, Ken Brakke's Surface Evolver had rendered this oppressively tedious method obsolete. The laser spectrometer Hal Robinson assembled in 1968 for measuring the surface orientation of a soap film I no longer have any photos of the similar instrument I constructed in 1975. Many other people subsequently discovered the idea of handle attachment and applied it to a variety of minimal surfaces, not just periodic ones. (I have been publicly scolded by several mathematicians — most often by J.C.C. Nitsche — for failing to publish my work in refereed journals. Mea culpa.) In May, 2011, I discovered in my files the long-forgotten sketch shown in Fig. E.2.70 and emailed a copy to Ken Brakke. He quickly confirmed (with his Surface Evolver) that the P—C(P) hybrid is embedded. Ken's pictures of this surface, which he dubbed 'N14', are shown in Figs. E2.71, E2.72, and E2.73 and also at his Triply Periodic Minimal Surfaces web site, under the name N14. Fig. E2.71 Ken Brakke's May 2011 Surface Evolver solution for an elementary patch of N14, the embedded hybrid of P and C(P) (cf. Fig. E2.70) (image courtesy of Ken Brakke) Fig. E2.72                                                    E2.73 A cubic unit cell of the P—C(P) hybrid (genus 14) The unit cells in the two images are displaced with respect to each other by one-half of the body diagonal of an enclosing cube. (cf. Fig. E2.71) (images courtesy of Ken Brakke) Fig. E2.74 1969 sketch showing how to generate O,C-TO The 'bcc labyrinth surface' referred to at the top of the page is the surface I later renamed I-WP. Fig. E2.75 O,C-TO (genus 10), a hybrid of I-WP (genus 4) and P (genus 3) cubic unit cell view: [100] direction Fig. E2.76 O,C-TO unit cell view: [111] direction Fig. E2.77 O,C-TO 1.5 unit cells oblique view Fig. E2.78 shows Ken's picture of the Manta surface, which is one of many examples of hypothetical minimal surfaces whose existence I conjectured in 1971. Some of them were inspired by experiments with soap films blown inside a kaledioscopic cell. Others were inspired by considering the structure of various highly symmetrical inorganic crystals. Manta is a balanced surface; the P—C(P) hybrid is non-balanced. If you compare Fig. E2.73 and Fig. E2.74, you will see that the P—C(P) hybrid has a simpler topology than Manta. Manta has [100] tunnels, while the P—C(P) hybrid does not. A few days after Ken sent me his image of the P—C(P) hybrid, which is shown in Fig. E2.73, he sent me images of two slightly more complicated surfaces he told me he had obtained by 'poking holes' in the P—C(P) hybrid. Images of this new pair — N26 and N38 — can be seen at his website, together with several other hybrids. Of course, purists rightly claim that the existence of all of these hypothetical minimal surfaces is somewhat suspect, since it has not been established by rigorous mathematical proof. A diagram of the unit cell of the cubic phase of the compound BaTiO3 is shown in Fig. E2.75 for comparison with Manta. One can try to match the sizes and positions of the ions in BaTiO3 to symmetrical cavities in the labyrinths of the surface. At my request, Ken produced the three orthogonal projections of Manta shown in Figs. E2.76, E2.77, and E2.78, together with the following numerical data on the radii of spheres that fit snugly against the surface in three classes of symmetrical cavities: Radii of tangent spheres in 1x1x1 unit cell of manta genus 19 surface:        corner sphere radius: 0.18560130        center sphere radius: 0.18560130        midedge sphere radius: 0.23553163 (Here's a useful Wikipedia article about ionic radii.) Fig. E2.74 Manta (genus 19) (image courtesy of Ken Brakke) Fig. E2.75 Unit cell of cubic phase of barium titanate (BaTiO3) Ba2+ red Ti4+ green O2− blue Fig. E2.76 Orthogonal projection of Manta unit cell on [100] plane (image courtesy of Ken Brakke) Fig. E2.77 Orthogonal projection of Manta unit cell on [110] plane (image courtesy of Ken Brakke) Fig. E2.78 Orthogonal projection of Manta unit cell on [111] plane (image courtesy of Ken Brakke) 'Notched adjoints': grafting handles onto embedded surfaces Fig. E2.82a A note by Ernst Eduard Kummer (H.A. Schwarz's father-in-law) that is included in Schwarz's Gesammelte Werke The illustration shows eight triangular Flächenstücke of Schwarz's diamond surface D inside a tetragonal disphenoid. Fig. E2.82b Free translation of the note by Kummer in Fig. E2.82a One weekend in 1968, while I was reading p. 150 of Schwarz's Collected Works (cf. Fig. E2.82a), it occurred to me that one might be able to model a small portion of a triply-periodic minimal surface, like the portion of Schwarz's D surface shown in Fig. E2.82, by means of a soap film in the interior of the appropriate polyhedral cell. (Schwarz and Plateau had a very active correspondence for many years about soap films and minimal surfaces. I'm surprised that they seem not to have performed experiments of this type.) I quickly constructed a transparent model of the tetragonal disphenoid from four vinyl triangles, stretching cotton threads along the two internal symmetry axes, which are clearly visible in Kummer's Fig. E2.82 sketch. I cut a hole in one of the faces of the cell large enough to provide access to the interior with a soda straw. Using a soap solution containing some glycerine, I discovered that the modeling of the minimal surface is quite easy! One of its elegant features is that by blowing through the straw on one face or the other of the soap film, you can toggle back and forth between two stationary states: a triangular patch of D and a quadrangular patch of C19(D) (cf. Figs. E2.4b, c, d). You stretch the soap film by blowing one corner of it right up to a corner of the cell, and then with a light puff of air through the straw, you push that part of the film just beyond the cell corner. At that point it automatically slides down into its other equilibrium position. Fig. 2.82c Soap film model of an elementary patch of Schwarz's H surface (1968) (Photograph copied from an article about Harald Robinson, on p. 55 of the Nov. 1969 issue of Innovation, published by The Innovation Group of Technology Communication, Inc., Saint Louis, Missouri) My colleague Hal Robinson constructed this triangular-prism-shaped Coxeter cell. The nylon thread that is stretched horizontally across the interior of the cell lies on   an axis of 2-fold rotational symmetry of the surface and serves to stabilize the soap     film. Without the thread, the film would be in an unstable stationary state and would quickly slide away from its equilibrium position and collapse. Many of the vacuum-formed plastic models of TPMS I constructed after Hal began to work with me were made from modules whose boundary curves were derived by tracing the curved edges of soap films like this one. My model of the tetragonal disphenoid was a flimsy one. When I arrived at NASA on the following Monday morning, I phoned Hal Robinson, the sculptor and model-maker who had recently started to work with me part-time and asked him to make a more physically rugged tetragonal disphenoid. Within a couple of days or so, he produced a lucite model with highly accurate proportions, using monofilament nylon instead of cotton threads for the internal symmetry axes. Next I asked Hal to make me a lucite model of another Coxeter cell relevant for cubic TPMS — the quadrirectangular tetrahedron, which is one-quarter of the tetragonal disphenoid. It contains only one internal axis of two-fold rotational syrmmetry, not two. With this cell, you can toggle back and forth between a patch of P and a patch of the Neovius surface C9(P) (cf. Figs. E2.2a, b), again by blowing on the soap film to stretch it over one corner of the cell. Just as with the tetragonal disphenoid, from there the soap film slides into its other stable stationary state automatically. Not until the spring of 1970 did it occur to me that perhaps Schwarz's P surface and Neovius's surface C9(P) are merely the topologically simplest members of a countably infinite sequence of embedded surfaces of progressively higher genus: P, C9(P), C15(P), C21(P), ... I obtained experimental evidence for the existence of C21(P) by producing the four-sided Flächenstück of C21(P) as a soap film in a stationary — but unstable — state inside the quadrirectangular tetrahedron. This was a more difficult soap film experiment than toggling back and forth between P and C9(P), which are in stable equilibrium. Although the C21(P) soap film corresponds to a [mathematical] stationary state, its area is larger than that of nearby lying [non-minimal-surface] soap films, and I had to struggle to maneuver the film into its unstable equilibrium position long enough for a camera to capture it. In 1992 I made an impromptu video about minimal surfaces in which I attempted to demonstrate the art of blowing these unstable soap films inside Coxeter cells, but I had run out of glycerine that day. After many tries, I succeeded for a fleeting moment in capturing the gracefully curved Flächenstück of C21(P). I plan to post here a snapshot or two from these videos, but the images of the soap films are somewhat obscured by the transparent tape I used to join the faces of the vinyl tetrahedra. I plan to obtain clearer photos of soap films inside glass tetrahedra I have recently made. Fig. E2.83a Twelve elementary triangular Flächenstücke of Neovius's embedded surface C9(P) Fig. E2.83b Twelve elementary triangular Flächenstücke of Neovius's self-intersecting surface C9(P)† In both C9(P) and C9(P)†, the triangular Flächenstück abc is analytically continued by reflection in its edges. The normal vectors (red arrows) at corresponding points of the two adjoint surfaces C9(P) and C9(P)† have the same directions. In 1971, with the assistance of my Cal Arts students John Brennan and Bob Fuller, I performed additional soap film experiments aimed at modeling the elementary Flächenstücke for higher-genus variants of the P and D surfaces and a variety of non-cubic TPMS. All of these experiments involved modifying the shapes of stationary-state soap films inside transparent plastic models of Coxeter cells by blowing on them. Fig. E2.84 shows some of Ken Brakke's Surface Evolver for some of these higher-genus variant surfaces. I was unable to produce the genus-15 soap film, but occasionally I succeeded with the genus-21 case. genus=9 genus=15 genus=21 genus=27 genus=33 Fig. E2.84 Ken Brakke's Surface Evolver solutions for the first few high-genus variants of Neovius's C9(P) 'NOTCHED' ADJOINT SURFACES genus=3 genus=9 genus=15 genus=21 genus=27 genus=33 Fig. E2.85 Stereo images of the sequence of 'notched' variants of the adjoints of P, C9(P), ... (left) and Ken Brakke's images of the corresponding embedded surfaces (right) The relative lengths of the line segments in the serrated edges that make up the notched outlines of the adjoint surfaces that are illustrated here (in stereo) only roughly approximate the actual values, which Ken Brakke derived with high precision when he used his Surface Evolver software to kill periods, thereby generating each of the embedded surfaces (shown at the right of the corresponding adjoint surface outlines). The genus pk of the kth surface Mk in the family {Mk} is defined as pk = p0 + k gap (k = 0, 1, 2, ...); the values of p0 and the positive integer gap are characteristic of the family. These families include — but are not limited to — high-genus complements of P and D. For the P and D families, for example, p0=3 and gap=6. Hence the surfaces in these two families are of genus 3, 9, 15, 21, ... . On that day in 1971, John and Bob and I blew a large variety of "finely filigreed" soap films in a variety of Coxeter cells, and we made detailed drawings of our results. Most — but not all — of these soap films included one or two nylon threads stretched along 2-fold symmetry axes of the enclosing polyhedral cell. Each curved boundary edge of such a soap film lies in a face plane of the cell and is a 'mirror-symmetric plane line of curvature'. Every face plane is a plane of reflection symmetry for both the assembly of cells and the soap films in their interiors. The soap films meet the enclosing face planes orthogonally. Each of these soap films is a physical model of the stationary state of the adjoint of a minimal surface bounded both by straight line segments and by either one or two curved edges — according to whether the number k of rotational symmetry axes through the cell is one or two. Films with k=0 or 1 are in stable equilibrium. If k=2, the film is in a delicate state of unstable equilibrium. I found it impossible to produce films for k>2. Some skill is required to arrest a film for k=2 in the neighborhood of its equilibrium position long enough to confirm the existence of the equilibrium. (The films were composed of a mixture of distilled water, detergent, and glycerine and were thick and viscous enough for both gravity and capillarity effects to impose some limits on the accuracy of the modeling.) In 1999, I began sending Ken Brakke data from these 1971 experiments as well as some additional data for surfaces whose existence I conjectured during the following three years, for authentication with his Surface Evolver computer program. Many of these authenticated surfaces are illustrated on his Triply Periodic Minimal Surfaces web site. In this work, Ken uses Surface Evolver to 'kill periods' — i.e., to derive the unique values for relative edge lengths that allow the surface to be embedded (cf. the brief description of this problem on pp. 45-46 of Infinite Periodic Minimal Surfaces Without Self-Intersections). In Infinite Periodic Minimal Surfaces Without Self-Intersections, I included only two examples of hybrid surfaces — C(H) (genus 7) and O,C-TO (genus 10), because at the time of writing, these were the only examples of such surfaces for which I had already constructed and photographed vacuum-formed plastic models. (In a footnote on p. 46, I mentioned a third example, of genus 5, that I called g-g'. I soon renamed that surface g-W, after I confirmed that its dual skeletal graphs are related to the structures of hexagonal graphite and of wurtzite.) Figs. E3.7, E3.8, and E3.9 show three views of g-W. E3. Triangle lattice surfaces All of the minimal surfaces described in this section are named according to the conventions in Infinite Periodic Minimal Surfaces Without Self-Intersections. Fig. E3.1a Schwarz's H surface (genus 3) Fig. E3.1b A smaller piece of Schwarz's H surface Fig. E3.2 C(H) (genus 7) The [first-order] complement of Schwarz's H surface unit cell view: c-axis Fig. E3.3 C(H) view along c-axis Fig. E3.4 C(H) view: c2 axis (intersection of horizontal and vertical mirror planes) in basal plane Fig. E3.5 C(H) view: line in basal plane that is below and parallel to a linear asymptotic (2-fold axis embedded in the surface) Note the infinitely long straight tunnels with pointy oval cross-section Fig. E3.6 C(H) view: c-axis, silhouetted by bright summer sky backlighting Note the infinitely long straight tunnels. (The trigonal symmetry of the surface would be more apparent if the image were rotated 30º in the image plane, as in Fig. 3.3!) Fig. E3.7 g-W ("graphite-wurtzite") (genus 5) oblique view Fig. E3.8 g-W oblique view Fig. E3.9 g-W and C(H) oblique view Fig. E3.10 H''-R (genus 5) view: c-axis Fig. E3.11 H''-R (genus 5) view: c-axis, silhouetted by bright summer sky backlighting Fig. E3.12 H''-R oblique view Fig. E3.13 H''-R view: c2 axis (intersection of horizontal and vertical mirror planes) in basal plane Fig. E3.14 H''-R view: line in basal plane that is below and parallel to a linear asymptotic (2-fold axis embedded in the surface) Note the infinitely long straight tunnels with pointy oval cross-section Fig. E3.15 H'-T (genus 4) view (stereo): c-axis E4. Surfaces on other lattices Fig. E4.1 S'-S'' genus 4 view: oblique E5. A few minimal surface people from around the world Christian Bär (right), and Hermann Karcher Ken Brakke at Selinsgrove, Pennsylvania Tomonari Dotera and Junichi Matsuzawa at Carbondale, Illinois (November, 2013) higher resolution image Shoichi Fujimori at Bloomington, Indiana (2008) Wojciech Góźdź Karsten Grosse-Brauckmann Bathsheba Grossman at Santa Cruz Stefan Hildebrandt at Berkeley (1979) photo by George M. Bergman ©George M. Bergman Source: Mathematisches Forschungsinstitut Oberwolfach gGmbH David Hoffman at the University of Granada Minimal Surface Conference, June 17, 2013 higher resolution image Stephen Hyde at Canberra Hermann Karcher (left), David Hoffman (center), and Manfredo Perdigão do Carmo (right) at Granada (1991) photo by Dirk Ferus ©Dirk Ferus Source: Mathematisches Forschungsinstitut Oberwolfach gGmbH Katsuei Kenmotsu at Oberwohlfach (2009) photo by Renate Schmid Source: Mathematisches Forschungsinstitut Oberwohlfach gGmbH Rafael López Camino (right) and me at the University of Granada, June 11, 2013 higher resolution image Blaine Lawson (left) and Bill Meeks (right) at Rio (1980) photo by Dirk Ferus ©Dirk Ferus Source: Mathematisches Forschungsinstitut Oberwolfach gGmbH Bill Meeks at the University of Granada Minimal Surface Conference, June 17, 2013 higher resolution image Johannes C. C. Nitsche (1925-2006) photo by Ludwig Danzer ©Ludwig Danzer Source: Mathematisches Forschungsinstitut Oberwolfach gGmbH Robert Osserman (1926-2011) at Berkeley in 1979 photo by George M. Bergman ©Mathematisches Forschungsinstitut Oberwolfach gGmbH Raymond Redheffer (1921-2005) Markus Rissanen (right) and me (left) in Helsinki (2012) Markus is a Finnish artist/architect/mathematician expert on quasi-periodic tilings. Here's a link to the article he wrote in 2016 with the mathematician Jarkko Kari describing his 'SUB ROSA' tilings of the plane. A light moment during a break at the University of Granada Minimal Surface Conference, June 17, 2013 higher resolution image Magdalena Rodriguez, Matthias Weber, and Bill Meeks at the University of Granada Minimal Surface Conference, June 17, 2013 higher resolution image Antonio Ros Mulero, me, and my wife Reiko Takasawa at the University of Granada Minimal Surface Conference, June 17, 2013 high resolution image Harold Rosenberg and Magdalena Rodriguez at the University of Granada Minimal Surface Conference, June 17, 2013 high resolution image Gerd Schröder-Turk at Erlangen-Nürnberg In 2015, Gerd joined the faculty of the maths department of Murdoch University in Perth, Australia. Isaac Van Houten at Carbondale (2008) photo by the author Matthias Weber at Oberwohlfach (2009) photo by Renate Schmid ©Mathematisches Forschungsinstitut Oberwohlfach gGmbH Adam Weyhaupt (right) at Edwardsville with two of his students — Darren Garbuz and Caroline Coggeshall photo by the author Back to GEOMETRY GARRET