Back to GEOMETRY GARRET
Alan H. Schoen
Comments are welcome.
Triplyperiodic 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 threeyear immersion in this study, in which I was guided by both mathematics and
physical experimentation. I benefited greatly from discussions with several mathematicians, some of whom
appear in photos below. I owe special thanks to the architect Peter Pearce, who introduced me to his concept
of saddle polyhedron, and to the physicist Lester C. Van Atta, who in 1967 offered me a senior research position at
the NASA Electronics Research Center that I couldn't dream of refusing. The sculptor Harald Robinson provided
invaluable technical support during my time at NASA. 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. The
other people who assisted me in various ways in those days include John Brennan, Bob Davis, Jay Epstein, Bob
Fuller, Thad Heckman, Rick Kondrat, Randall Lundberg, Kenneth Paciulan, Charlie Strauss, and Jim Wixson. I am
extremely grateful to all of them. In September 1999, I began a long collaboration with Ken Brakke, who models
minimal surfaces with breathtaking precision with his Surface Evolver software. Many of the results of our
collaboration are displayed at Triply periodic minimal surfaces (just one of Ken's many informative web sites).
The gyroid surface receives special emphasis here, because in recent years it has become an object
of special interest for materials scientists. They have found, for example, that it serves as a template
for selfassembled periodic surfaces that separate two interpenetrating regions of matter, with unit
cells that vary in size over several orders of magnitude, depending on the constituent materials.
§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 timelapse 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 CoxeterPetrie map {6,44}. The entire surface
may be constructed by first attaching a replica of the hexagon to any open edge of
the initial hexagon and then repeating the attachment process at every open edge of
the developing assembly. Each attachment is implemented by applying a symmetry
of the gyroid: a halfturn about an axis normal to the surface through the midpoint
of each edge.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. Click here for larger image. }
_{ 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 form cylindrical helices centered on lines parallel to the x, y, and z axes. The helices in nearestneighbor parallel tunnels are of opposite handedness. }
_{ In 1969, I took this model to a math conference in Tbilisi, Georgia. To protect it from mechanical shock, I inserted rolledup tubes of newspaper into its three orthogonal sets of tunnels before packing it in a wooden box. 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.) }
_{ Two friends inside a gyroid sculpture at the San Francisco Exploratorium }
§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)
§3. Interface Focus summary article (2012)
Reflections Concerning TriplyPeriodic Minimal Surfaces,
a summary of my minimal surface research, which appeared in
October 2012 in Interface Focus, a journal of The Royal Society
§4. The P−G−D family of associate minimal surfacesSchwarz'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 selfintersections. P, D, and G are each of genus three, which is the minimum
possible genus for a triplyperiodic minimal surface.
_{ P } _{ G } _{ D }
_{ {6,44}P } _{ {6,44}G } _{ {6,44}D }
_{ sc lattice } _{ bcc lattice } _{ fcc lattice }
Each of the surfaces P, G, and D is shown here in a tiling pattern called {6,44} by Coxeter.
The prototile for each of these tilings is a specific variant of hex_{90}, a generic regular skew
6gon
with 90º face angles. There are
4 faces
incident at each vertex and holes with
4fold
symmetry.The three variants of hex_{90} for P, G, and D are called Phex_{90}, Ghex_{90}, and Dhex_{90}, respectively.
In a 1970 NASA Technical Note Infinite Periodic Minimal Surfaces Without SelfIntersections 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 (IWP) 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.
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'.)
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.
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.
I believe now that 'reciprocal' might have been a more suitable name than 'dual' for the relation between
the two triplyperiodic graphs — e.g., simple cubic, diamond, Laves, and others — of an intertwined pair.
(The two graphs in many of these pairs are noncongruent.) '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 triplyperiodic CoxeterPetrie polyhedra. In what
follows I will continue to use the word 'dual' to refer to the vaguely defined but symmetrical relation
between pairs of triplyperiodic graphs like those described here.
P and D were discovered and analyzed by H. A. Schwarz in 1865. He derived an
explicit EnneperWeierstrass parameterization for the surfaces, which morph into
each other via the Bonnet bending transformation. (Another well known example
of Bonnet bending is the helicoidcatenoid transformation.)P and D contain both embedded straight lines and plane geodesics. A straight line
in either surface morphs into a plane geodesic (a mirrorsymmetric plane line of
curvature) in the other surface. Because every straight line embedded in a minimal
surface is an axis of 2fold rotational symmetry, a halfturn about the line switches
the two sides of the surface and also switches the two interpenetrating 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.
_{ EnneperWeierstrass 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 triplyperiodic graphs.Using a toy vacuumforming 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 Dhex_{90} and Phex_{90} 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 a special kind of balanced surface.
Further details concerning my discovery of G are described in §3, §5, §7, and §51.
If λ_{D} = 2, then
§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 the surface under Bonnet bending is an ellipse. }
_{ The trajectories for the six vertices of each face are shown here. }
_{ 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. 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). }
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 (Dhex_{90}) and red (Phex_{90}). Dhex_{90} is inscribed in a cube of edge length
λ_{D} and Phex_{90} is inscribed in a cube of edge length λ_{P}. The ratio λ_{P} ⁄ λ_{D} is equal to
K(3/4)/K(1/4) ≅ 1.2792615711710064662 — the value required for Dhex_{90} and
Phex_{90} to have the same area. For both D and P, the normal vector at the center
of each of the eight hexagonal patches in a lattice fundamental domain is directed
toward a different one of the corners of the circumscribing cube. For the patches
Dhex_{90} and Phex_{90} shown above, the normal vectors are in the (1,1,1) direction
(x is positive toward the observer, y is positive to the right, and z is positive up).If we fix point O at the center of Dhex_{90}, then as Dhex_{90} morphs into Phex_{90}
by Bonnet bending, the point D_{1} on Dhex_{90} moves along an elliptical trajectory
r_{1}(θ) with center at O. The equation for this ellipse isr_{1}(θ) = d_{1} cos θ + p_{1} sin θ (1)
The vectors d_{1} and p_{1} are directed outward from the center of the ellipse along
its semiminor and semimajor axes, respectively.For θ = π/2, the images under bending of d_{1} and d_{2} are p_{1} and p_{2}, respectively.
d_{1} = (− 1,1,1) As stated above, H.A.Schwarz proved (cf. his Collected Papers, vol. I, p. 88) that
d_{1} = √3
p_{1} = (λ_{P} ⁄ λ_{D}) (0,− 1,1)
p_{1} = √2 (λ_{P} ⁄ λ_{D}). (2)
λ_{P} ⁄ λ_{D} = K′(1/4) ⁄ K(1/4) (≅ 1.2792615711710064662). (3)
If we substitute for d_{1} and p_{1} in Eq. 1 from Eqs. 2 and 3, we obtain
r_{1}(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
r_{1}(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 M_{6}, an infinite regular skew polyhedron whose faces are regular skew
hexagons. Below is an illustration of a hexagonal face of M_{6} with central normal
vector (not shown) oriented in the (1,1,1) direction.
_{ The edges of a hexagonal face of M6 with central normal vector in the (1,1,1) direction x is positive toward the observer, y is positive to the right, and z is positive up. The rectangular coordinates of the six vertices of M6, in CCW order from the top, are proportional to (−1,0,2), (0,−2,1), (2,−1,0), (1,0,−2), (0,2,−1), and (−2,1,0), respectively. }
_{ 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 M_{6} is replaced by a quarterpitch 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
G_{1}G_{2} y CW
G_{2}G_{3} x CCW
G_{3}G_{4} z CW
G_{4}G_{5} y CCW
G_{5}G_{6} x CW
G_{6}G_{1} z CCWLet us denote by Ghex_{90} the minimal surface that spans this modified hexagon
with helical edges G_{1}G_{2}, G_{2}G_{3}, ..., G_{6}G_{1}, because — like each of the hexagonal
faces Dhex_{90} and Phex_{90} of D and P— its face angles are also equal to 90º. Since
the directions of the tangents to the pairs of edge curves that intersect at D_{1}, P_{1},
and G_{1} are identical, the surface orientation is the same at these points.
Vector equation Eq. 5 is equivalent to the following three scalar equations, one
for each of the three components of the vector r_{1}(θ_{G}):x_{1}(θ_{G}) = −cos θ_{G} + (0) (K′ ⁄ K) sin θ_{G} (5')
y_{1}(θ_{G}) = cos θ_{G} − (K′ ⁄ K) sin θ_{G} (5'')
z_{1}(θ_{G}) = cos θ_{G} + (K′ ⁄ K) sin θ_{G} (5''')
But r_{1}(θ_{G}) = (−1,0,2).
Hence y_{1}(θ_{G}) = 0, (6)
If now we substitute for y_{1}(θ_{G}) from Eq. 6 in Eq. 5'', we obtain
cos θ_{G} = (K′ ⁄ K) sin θ_{G} . (7)
Therefore θ_{G} = ctn^{1}(K′ ⁄ K). (8)
Because of the 6fold rotatory reflection symmetry of Dhex_{90}, Phex_{90}, and Ghex_{90},
Eq. 8 could have been derived by considering any of the other five vertices of
Ghex_{90} instead of the vertex G_{1}.
_{ Models of the infinite regular skew polyhedra M4 (left) and M6 (right), precursors of the gyroid M4 was produced by executing an intermediate step of an empirically developed procedure that was aimed at deriving duals of triplyperiodic 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. }
Below are images of the Voronoi polyhedron of a vertex of M_{6}.
_{ Photos of 1968 models of the spacefilling Voronoi polyhedron of a vertex of the graph {6,44}, which is composed of the edges of M6 (cf. photo above) The Voronoi polyhedron of a vertex of the graph {4,64}, the dual of {6,44}, is the truncated octahedron, which has full cubic symmetry. }
A fortuitous failureOn February 14, 1968, I constructed a model (cf. photo at above left) of an infinite regular
skew polyhedron I call M_{4}. It appeared at the penultimate stage of a recipe I had been
testing for what I called the dual of a triplyperiodic graph. The graph I was examining is a
triplyperiodic symmetric graph of degree six that I called BCC_{6}, because when two
additional edges are added at each vertex, it is transformed into BCC_{8}, the ordinary bcc
graph of degree eight. 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 M_{4} 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
triplyperiodic 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 M_{4} was simply to be surprised that it was not finite. But I also
realized how naive I had been to imagine that anything as jerryrigged as my 'dual graph
recipe' would always perform the way I had planned. 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 'counterexample', and I had finally found one. At last I could stop searching.All these thoughts were swept aside as soon as I examined M_{4} 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 something, tiled by
regular skew quadrilaterals, whose faces, edges, and vertices seemed to have the same
overall 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 M_{4} into an actual minimal surface, but I judged by
eye that M_{6}, its dual, a tiling with four skew hexagons incident at each vertex, would
look somewhat less bumpy than M_{4}, because the dihedral angle at each edge would
be smaller. The face angles in M_{4} and M_{6} are cos^{− 1}(1/3) = ~70.529º and cos^{− 1}(−1/6)
= ~99.594º, respectively. A calculation showed that the dihedral angle in M_{4} is 60° but
only ~44.4° in M_{6}. I decided that this reduction in angle was large enough to justify
making a vacuumforming tool for the M_{6} skew hexagon.In three days, I finished assembling the model of M_{6} shown at the upper right. M_{6} really
did look somewhat smoother than M_{4}. More significantly, by this time I had come to
realize that M_{4} and M_{6} are described by Coxeter's dual regular maps {4,64} and {6,44}.
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 sure exactly what to make of all this.
After several years of searching for examples of triplyperiodic graphs on cubic lattices,
the only examples of symmetric graphs on cubic lattices I had found were the simple cubic,
diamond, and Laves graphs. This fact was one one of the main reasons for my conviction
that the gyroid exists (even though I knew of no theoretical justification for connecting
skeletal graphs with existence proofs for TPMS!).Encountering this 'exceptional case' induced me to change direction. I decided once and
for all to give up trying to formalize the concept of the dual of a triplyperiodic graph.
M_{4} and M_{6} 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 decided that the notion of duality for triplyperiodic 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 a NASA Technical
Report entitled Reflections Concerning TriplyPeriodic Minimal Surfaces.
The primary reason I so suddenly lost interest in trying to formalize the concept of duality
for triplyperiodic graphs is that I noticed that the skeletal graphs of the two intertwined
labyrinths of M_{4} are the Laves graphs. M_{4} and M_{6} both have the same topology and global
symmetry (space group Ia3′d, No. 230) as the gyroid, which I had unsuccessfully tried to
construct in the summer of 1966. The properties of M_{6} strongly suggested to me the
possibility of somehow constructing the gyroid out of hexagonal faces whose vertices
coincide with those of M_{6}. Each face of M_{6}, like the hexagonal faces of D and P, is oriented
in one of the eight (±1,±1,±1) directions. The ycoordinate y_{1}(θ_{G}) of a vertex G_{1} that
corresponds to D_{1} and P_{1} (and to the vertices that are diametrically opposite these vertices)
is equal to zero. For each of the other seven orientations of the faces, it is likewise true that
exactly one of the three rectangular components of each of the vectors that define the
positions of a pair of diametrically opposite vertices of M_{6} is equal to zero.My model of M_{6} showed that the straight edges of M_{6} define infinite regular helical
polygons, which are centered on lines parallel to the rectangular coordinate axes.
This suggested that if I were to replace the straight edges of M_{6} by helical arcs, the
dihedral angle of ~44.4° 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 that are centered on lines parallel
to the coordinate axes in the gyroid surface is quite small, it is not zero. The image in §10
shows the difference.)I call the object that is tiled by skew hexagons with strictly helical edges the 'pseudo
gyroid'. It is not a triplyperodic minimal surface. Even if all of its separate faces were
minimal surfaces, they would not define a global surface that is continuous across the
boundary between adjacent faces. (Incidentally, it is still not known how to derive an
analytic expression for a simplyconnected minimal surface spanning a 'Schwarz chain'
composed of consecutive helical arcs.)
§6. New models of M_{4} and M_{6} 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 }
_{ Tiling of M6 by skew hexagons Viewpoint is in the (1,0,0) direction. }
§7. The relation between M_{6} and the gyroidIn March 1968 I played my hunch about helical edges by sending a purchase order to a
machine shop for a customdesigned brass tool for vacuumforming plastic modules of
hexagons with edges in the shape of helical arcs of alternating handedness. Below is a
photo of the model of the pseudogyroid 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 he 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 pseudogyroid (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 would
be stored compactly in a collapsed state for eventual deployment in an expanded state.
§8. A long summer distractionDuring the summer of 1968, as I waited to hear from either Bob or Blaine, I analyzed
the collapse kinematics of several triplyperiodic 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) Onequarter 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 {6,44}G tiling of G. }
§10. The machined parts of the brass vacuumforming tool
_{ The components of the brass mould, shown before brazing and final machining, that I designed for vacuumforming the plastic hexagons of the 1968 model of the pseudogyroid. }
_{ 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 }
_{ (I naively imagined that the packing of the eight tetrominoes shown in this sketch might be unique. But 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, I would probably never 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 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
§11. Symmetrical lattice fundamental domains of the gyroidThe hexagonal faces of the gyroid CoxeterPetrie map {6,44} 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 (bodycentered cubic). There is a c2 axis (halfturn 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 counterclockwise 4fold 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 4fold rotationreflection 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 yz plane; the projection at the right is onto the xy 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 yz 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 yz 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 3fold 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 3fold 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
planefaced triplyperiodic 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
interpenetrating 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 {4,64} of Coxeter and Petrie is an example
of a ps of a TPMS. Since it corresponds to Schwarz's P surface, we call it
{4,64}_{P}. This is what it looks like:
_{ {4,64}P A regular ps of Schwarz's P surface }
A second example of a regular ps of P is {6,44}_{P}:
_{ {6,44}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 {6,63}_{D}:
_{ {6,63}D A regular ps of Schwarz's D surface }
Another ps of Schwarz's D surface is the quasiregular skew polyhedron
(6.4)^{2} (below). Quasiregular polyhedra are both edgetransitive and vertex
transitive, but they are not facetransitive.
_{ (6.4)2D A quasiregular 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 quasiregular 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 vertextransitive, but it
is neither edgetransitive nor facetransitive. It is called (6.3^{2}.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 }
(6.3^{2}.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 R^{3} and R^{4}.2.4.3)_{G} polyhedron was independently rediscovered several years ago by
John Horton Conway, who named it musnub 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.
(I plan to post additional links here from time to time, but this
_{ The triplyperiodic CoxeterPetrie regular skew polyhedron {6,63}, 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.
Below is a partial list of online sources (not limited to periodic minimal surfaces): mathematics, images, videos, applications in physics, chemistry, biology, engineering, etc.
 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. 103107) of H.A. Schwarz's then almost brandnew 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 he who called special attention to the central importance of such a proof.
 Tomonari Dotera, Masakiyo Kimoto, and Junichi Matsuzawa
Hard Spheres on the Gyroid Surface
 EPINET:
Triply Periodic Minimal Surfaces
An encyclopedia of triplyperiodic 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:
GergonneSchwarz 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 IWP 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 GoodmanStrauss and John Sullivan:
Cubic Polyhedra
 Wojciech Gòzdz and Robert Holyst:
High Genus Periodic Gyroid Surfaces of NonPositive 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, 291357 (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
3periodic minimal surfaces at the '[Details]' hyperlink in their Mathematical Crystallography
 Elke Koch and Werner Fischer:
3periodic surfaces without selfintersections
 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:
PetrieCoxeter Maps Revisited
 Alan H. Schoen:
Infinite Periodic Minimal Surfaces Without SelfIntersections,
NASA TN D5541 (May 1970)
 Alan H. Schoen
Reflections Concerning TriplyPeriodic Minimal Surfaces
A summary of my minimal surface research, published in October 2012 in Interface Focus, a journal of The Royal Society
 Gerd SchröderTurk:
The bicontinuous fish tank
 Gerd SchröderTurk:
Bonnet transformation between the D, G and P minimal surfaces (video)
 Gerd SchröderTurk, Stuart Ramsden, Andrew Christy, and Stephen Hyde:
Medial Surfaces of Hyperbolic Structures
 Gerd SchröderTurk, Andrew Fogden, and Stephen Hyde:
Local v/a variations as a measure of structural packing frustration in bicontinuous mesophases, and geometric arguments for an alternating Im3m (IWP) phase in blockcopolymers with polydispersity
 Gerd SchröderTurk, Andrew Fogden, and Stephen Hyde:
Bicontinuous geometries and molecular selfassembly: comparison of local curvature and global packing variations in genus three cubic, tetragonal, and rhombohedral surfaces
 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
 3DXplorMathJ Applets
 Wikipedia:
Calculus of Variations
list will never be more than a tiny sample of what's out there.)
§14. Regular warped polyhedra based on the gyroid
_{ Abstract 65830, submitted in summer 1968 to the American Mathematical Society for the Madison meeting in August }
_{ A summary of my enumeration of the quasiregular tessellations of the {6,4} family (abstract published by the American Mathematical Society in 1969) }
_{ (6.4)2 front view high resolution image (64mesh quadgrid) high resolution image (1024mesh quadgrid) }
_{ (6.4)2 top view high resolution image (64mesh quadgrid) high resolution image (1024mesh quadgrid) }
(6.4)^{2} — alternatively written (6.4.6.4)_{G} — is a triplyperiodic
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 triplyperiodic 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.
§15. Regular tilings of P, D, and G
P D G
{6,44}_{P} {6,44}_{D} {6,44}_{G}
sc lattice fcc lattice bcc lattice
These three TPMS are tiled here in a pattern
called {6,44}. The prototile for each surface
is a particular variant of hex_{90}, a regular skew
6gon
with 90º face angles. In this pattern, there are
4 faces
incident at each vertex and 'holes' with
4fold
symmetry.Alternatively, these surfaces can be tiled in the
patterns {4,64} and {6,63}. The prototiles for
{4,64} are variants of quad_{60}, a regular skew
4gon with 60º face angles. In this tiling, there
are 6 faces incident at each vertex and holes
with 4fold symmetry.The prototiles for {6,63} are variants of hex_{60},
a regular skew 6gon with 60º face angles. In
this tiling, there are 6 faces incident at each
vertex and holes with symmetry of order 3.
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 {6,44}, {4,64}, and {6,63} 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. {6,44} and
{4,64} are homeomorphic to P, while
{6,63} 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:hex_{90}(D) ↔ hex_{90}(G) ↔ hex_{90}(P).
quad_{60}(D) ↔ quad_{60}(G) ↔ quad_{60}(P).
hex_{60}(D) ↔ hex_{60}(G) ↔ hex_{60}(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 hex_{90}(D) ↔ hex_{90}(G) ↔ hex_{90}(P).
§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 selfintersecting. } The portion of the surface shown here is a lattice fundamental domain plus oneThe 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 PDP11 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.
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 S_{1} into S_{2}, not all
pairs of faces in S_{1} that share a common edge preserve this connection also in S_{2}.
The portion of the
1969 'Part 4'
movie that illustrates this bending
starts at 3^{min}18^{sec} after the beginning.
§17. Poincaré's hyperbolic disk model of uniform tilings
The combinatorial structure of the three tiling patterns
{6,44}, {4,64}, and {6,63} is illustrated by
Poincaré's hyperbolic disk model of
uniform tilings in the hyperbolic plane.
(image by Eric W. Weisstein,
Wolfram MathWorld)
{4,64}
{6,44}
{6,63}
(images from Wikipedia)
§18. The three CoxeterPetrie infinite regular skew polyhedra
Coxeter and his friend Petrie long ago discovered the three
triply
periodic regular skew polyhedra called {p,qr}.
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.
{4,64}
{6,44}
{6,63}
{4,64}, including the skeletal graph of labyrinth A
{4,64}, including the skeletal graphs of both labyrinth A and labyrinth B
{6,44}, including the skeletal graph of labyrinth A
{6,44}, including the skeletal graph of labyrinth B
{6,63}, including the skeletal graph of labyrinth A
§19. Views of G tiled by hex_{90}
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 hex_{90} ({6,44} tiling)
or by
(ii) regular skew quadrilaterals quad_{60} ({4,64} tiling)
or by
(iii) regular skew hexagons hex_{60} ({6,63} tiling).
Below are views of G tiled by hex_{90}
in the CoxeterPetrie {6,44} map.
Each hex_{90} has 90º face angles.
Every hex_{90} face is
related to each of six faces with
which it shares an edge by a halfturn 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
§20. Views of G tiled by hex_{60}
Below are views of G tiled by the regular
skew hexagon hex_{60}, the prototile of
the CoxeterPetrie {6,63} map.
hex_{60} is so named because it has 60º face angles.
It is related to six other hex_{60} faces, with each of
which it shares an edge, by a halfturn about an
axis of type (110) perpendicular to the surface at
the midpoint of the shared edge.
The midpoints of the six edges of
hex_{60} coincide with vertices
of the cuboctahedron.
(111) viewpoint
§21. Views of G tiled by quad_{60}
§22. Gyring Gyroid — a sculpture by GoodmanStrauss and Sargent
Gyroid sculpture
Click
here for the authors' account of
Here's their razzledazzle video
§23. Shapeways 3Dprinted models of TPMS is the set of Shapeways models made by and by others.
Shapeways displays a large collection
Bathsheba Grossman's model of
Below are a few of the TPMS models
There's much more information about TPMS at
Gyroid Double Gyroid (models by Alan Mackay)
§24. The four founding fathers of triplyperiodic minimal surfaces
(In §E7 there are photos of a few contemporary experts in this field.)
Mathematicians have studied minimal surfaces since 1762, when Lagrange derived the 'EulerLagrange 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 EulerLagrange 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 areaminimizing, 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 triplyperiodic minimal surfaces (TPMS) were performed by Schwarz, Riemann, Weierstrass, Enneper, and Neovius in the middle of the 19^{th} 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 socalled selfassembled 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 uptodate. 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.)
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 newfound 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 19051994 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.
(a) symmetric triplyperiodic graphs and their nodal and interstitial
polyhedra (see explanation below), and
I had been strongly interested in connections between triplyperiodic graphs and convex polyhedra since the mid1950s, 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 oneyear 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
(All of the stereoscopic image pairs below are arranged for crosseyed viewing.)
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 (triplyperiodic 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
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 ballandstick 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
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 bodycentered 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 secondnearestneighbor 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 14faced 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 spacefilling eightfaced 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 vertexenclosing 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 triplyperiodic 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 bcc graph (Z=8) The edges of the tetragonal tetrahedron TT are shown in blue.
The bcc graph (green vertices) and its dual, the WP graph (orange vertices)
When I started playing with soap films and minimal surfaces in May 1967, I was ignorant of the extensive literature on these subjects. At first I didn't know that there are boundary frames that are spanned by more than one shape of minimal surface and that there exist socalled unstable minimal surfaces that are not surfaces of least area spanning their boundary frames. (On one of his web pages, Ken Brakke illustrates some classical examples of these phenomena.) I recall poring over James Clerk Maxwell's classical article, in the legendary Eleventh Edition of the Encyclopedia Britannica I had inherited from my father, in which the author analyzed the stable and unstable versions of the catenoid. As I learned more about minimal surfaces, I began to realize 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 be area minimizing — and therefore stable. (In fairness to Peter, I believe it is very likely 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 triplyperiodic graph. In June 1966, I undertook some soap film experiments in order to explore these questions. To my surprise I found that the boundary curve C_{0} (shown at left below), which has the shape of one of the several Hamilton cycles on the cuboctahedron, is spanned by at least two disktype soap films of different shape. One of these two surfaces, S_{1}, is a surface of least area and is called stable. The other, S_{0}, is not a surface of least area. It is called unstable, because it can be formed as a soap film on C_{0} only if one or more wires or threads are added to C_{0} along appropriate curves embedded in S_{0} — i.e., curves that partition S_{0} into an assembly of smaller surface patches, each of whose boundary curves is spanned by a unique stable minimal surface. S_{1} can drape the boundary frame C_{0} in either of two positions. Let's call it S_{1a} if it's in one of these positions and S_{1b} if it's in the other. S_{1a} and S_{1b} are related by a halfturn about the axis A_{1}A_{2}. (See Ken Brakke's computed images of S_{1a} and S_{1b} below.) If a wire frame in the shape of C_{0} 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. S_{0}, which drapes the wire frame C_{0} in only one position, forms as a soap film if threads or wires are incorporated in C_{0} along one or both of the lines A_{1}A_{2} or B_{1}B_{2}.
The three [orthogonal] c2 axes of the boundary curve C_{0}.
The soap film S_{1a} is an areaminimizing
('least area')
S_{1a} S_{0} S_{1b} (Stable) (Unstable, unless string (Stable) or wire is added along either A_{1}A_{2} or B_{1}B_{2} or both A_{1}A_{2} and B_{1}B_{2})
The question of how many minimal surfaces span a given boundary curve has been found to be an extremely knotty one, but it is known that there are two properties of a simplyconnected 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);
Since the aforementioned 5gons and 7gons have total curvatures of only 2^{1}⁄_{6} π and 3^{1}⁄_{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 A_{1}A_{2} and B_{1}B_{2} will convert the bare frame C_{0} into a frame that is spanned by the surface S_{0}.
In October 1967, three months after I joined NASA/ERC, I was a selfinvited guest at Hans Nitsche's home in Minneapolis. Although Hans showed considerable interest in my wire frame of C_{0} and in my plastic model of S_{1}, he never mentioned that the groundbreaking paper in which he introduced and proved the 4π theorem was about to be published! Since I 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 sthanks 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 C_{0}.
In 1956 I designed and ran a FORTRAN program that confirmed my hunch that for selfdiffusion in fcc crystals, the isotope effect and the BardeenHering 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 selfdiffusion in crystals. I modeled interstitial diffusion pathways (strictly random walk) by the edges of one triplyperiodic graph and pathways for diffusion by the vacancy mechanism in the same crystal (correlated random walk) by the edges of a second triplyperiodic 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 triplyperiodic 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 triplyperiodic 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.
In April 1966, two days after meeting Peter Pearce, I made some examples of saddle polyhedra for myself, using the toy vacuumforming 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 vacuumforming 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 CsCl. Only after consulting a handbook of crystal structures did I learn that the atoms of sodium and thallium in the binary solid solution NaTl 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:
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_hex_{90} and P_hex_{90} that tile the map {6,44} 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.
The concept of a dual relation for pairs of triplyperiodic 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!
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 triplyperiodic 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 triplyperiodic graphs and saddle polyhedra yielded a dual pair of triplyperiodic 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 eveningsandweekends hobby. (Further details of this story are described below.) I believed 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 triplyperiodic 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 sparetime 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 vacuumformed 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 2fold rotational symmetry. This implies that a halfturn 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 (mirrorsymmetric 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 'cylinderized' 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 vacuumformed 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
'Toy models' of P, D, and G
In each of the images at the right, the cylinder radius
§36. BCC_{6} and M_{4} — 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 'BCC_{6}' (see stereo image below) that provided the longsought 'counterexample' to my empirical recipe for deriving the interstitial and nodal polyhedra of a triplyperiodic graph. (b) Although this graph failed spectacularly to yield a finite interstitial polyhedron, it pointed toward something much more interesting — an infinite triplyperiodic saddle polyhedron that I call M_{4}. The symmetry and combinatorial structure of M_{4} 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 BCC_{6} of degree six
I define a deficient symmetric graph on a given set of
vertices as a symmetric
viewpoint: close to [100] direction
30 quadrangles of M_{4} view: [111] direction, backlit by summer sky
30 quadrangles of M_{4} view: [100] direction, backlit by summer sky
Abstract 65830 submitted in summer 1968
This was an awkwardly premature
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.)
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 shortlived, 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 yearold 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 topheavy (or perhaps I should say bottomheavy) with a bloated support infrastructure of low and midlevel 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 wellrounded 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.
My own position there was relatively comfortable, however, with one exception: the absence of inhouse colleagues who shared my scientific and mathematical interests. I would have benefited from having someone close by for chitchat 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 {6,44}, {4,64}, and {6,63} (cf. the book by these authors that is cited below, following Fig. E1.1k). These three regular maps describe the combinatorial structure of the flatfaced CoxeterPetrie 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 triplyperiodic 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
Nelson Max
A few months after I arrived at ERC, I was visited by Harald ('Hal') Robinson, a sculptor, designer, master machinist, and modelmaker 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 vacuumforming 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 parttime arrangement so that he could fabricate vacuumforming 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 — parttime workstudy 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 3fold 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 triplyperiodic 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 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 halfbaked ideas about the connections to triplyperiodic 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 schools 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 longdistance 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 facetoface in September 1969, when we arranged to have sidebyside 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.
I was free — within reason — to attend meetings of the American Mathematical Society, of which I was a member. In contributed 15minute 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 upandcoming 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 selfinvited, 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. and the gyroid at the Courant Institute, 1968 Photo by Stefan Hildebrandt
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 that day at the end of 1969 when we were informed about the impending shutdown, Richard Nixon had been president for almost a year. According to a contemporary news account, 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. During the late winter and early spring of 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, 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 abouttobeformed 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.)
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 SelfIntersections'. Meanwhile, I had been commissioned to design and construct an 11ft.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 mid1970. 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 CoxeterPetrie regular map {6,44}. I arranged for ERC to award a contract to the Gurnard Engineering Corporation of Beverly, Massachusetts to manufacture two CNCmilled aluminum dies for vacuumforming two kinds of thin zincalloy modules — one in the shape of the hexagon of {6,44} and the other in the shape of the quadrilateral of {4,64}, its dual. These two kinds of module were to be joined by epoxy, in a facetoface, 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 selfdual map {6,63}, which is twice as large as that of {6,44}, 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 vacuumforming, by introducing ugly ridgelike 'wrinkles' in each module.) In the spring of 1970, NASA funding for the project was abruptly subjected to a special kind of 'midcourse 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. E1.18ad below.) Dr. Van Atta resigned from ERC in the autumn of 1969 to become research vicepresident 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 wordprocessing 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.)
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 postdeadline 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'neemal soor'facez". The agent echoed in a bewildered voice, "Mee'neemal soor'facez?" 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 fourday 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
Here is the program of the 1969 Tbilisi conference: 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 yearold 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. page 1 page 2 page 3 page 4
Kenneth Brakke
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
In October 2012, my wife Reiko and I attended the Primosten, Croatia conference, organized by
Hyde and SchröderTurk, on which these conference papers are based. In my own presentation, entitled
Reflections concerning triplyperiodic 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.)

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.

oblique view
Fig. E1.4a Fig. E1.4b Fig. E1.4c graph 1 graph 2 graphs 1 and 2
Thick simple cubic graphs oblique view
Fig. E1.4e Fig. E1.4f Fig. E1.4g graph 1 graph 2 graphs 1 and 2
Fig. E1.4i The CoxeterPetrie triplyperiodic regular skew polyhedron {6,44}, which has the same topology and symmetry as Schwarz's primitive surface P
Fig. E1.4j {6,44} with embedded [skinny] dual graphs
Fig. E1.4k {6,44} with embedded [fat] dual graphs

{6,44} 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. E1.4ah. 

The two intertwined skeletal graphs of the gyroid, in a Shapeways 3D printed version designed by virtox.
Click here for a video of 'Bones',
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].

[100] view
Fig. E1.5a Fig. E1.5b Fig. E1.5c graph 1 graph 2 graphs 1 and 2
[100] view
Fig. E1.5e Fig. E1.5f Fig. E1.5g graph 1 graph 2 graphs 1 and 2
[100] view
Fig. E1.5i Fig. E1.5j Fig. E1.5k graph 1 graph 2 graphs 1 and 2
[111] view
Fig. E1.5m Fig. E1.5n Fig. E1.5o graph 1 graph 2 graphs 1 and 2
[111] view
Fig. E1.5q Fig. E1.5r Fig. E1.5s graph 1 graph 2 graphs 1 and 2
[111] view
Fig. E1.5u Fig. E1.5v Fig. E1.5w graph 1 graph 2 graphs 1 and 2
[110] view
Fig. E1.6a Fig. E1.6b Fig. E1.6c graph 1 graph 2 graphs 1 and 2
[110] view
Fig. E1.6e Fig. E1.6f Fig. E1.6g graph 1 graph 2 graphs 1 and 2
[110] view
Fig. E1.6i Fig. E1.6j Fig. E1.6k graph 1 graph 2 graphs 1 and 2
Fig. E1.7 Straw model of the pair of dual Laves skeletal graphs (1960) view: [100]
Fig. E1.8 The 'clockwise' Laves graph, skeletal graph of one labyrinth of the G surface
Fig. E1.9 The 'counterclockwise' 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 'counterclockwise' Laves graph
Fig. E1.13a
The dual skeletal graphs of a hypothetical but
(TO stands for truncated octahedron,
blue graph: degree 4
In 1974, I tested for the existence of TOTD by using a lasergoniometer method I had devised in 1968 at NASA/ERC. This extremely tedious method requires the construction of a set of several straightedged boundary frames of various proportions. The laser is used to measure the orientation of the normal to the surface of a [longlasting] 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 TOTD 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 filmlaser 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,

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.)
The elementary minimal surface Flächenstück
ABC shown in Fig. E2.1 is mapped
There are twelve replicas of the Flächenstück
ABC in the skew

Triply periodic minimal surfaces are infinitelymultiplyconnected,
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 selfintersections.
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 GaussBonnet theorem, which links the topology and the geometry of a surface. One can very crudely express the essence of the GaussBonnet 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 saddlelike 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 13^{th} edition of 'Mathematical Recreations and Essays' by W.W. Rouse Ball and H.S.M. Coxeter,
the authors use Euler's
formula
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 EulerPoincaré 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 ngons 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
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
If you're not familiar with the Gauss map, look here. For discussion of the EulerPoincaré characteristic Χ=2 − 2p, look here. For information about the GaussBonnet theorem, look here.


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.
On p. 105 of Part I of his Collected Works
It appears that for arbitrary values of the
Fig. E3.1f Orthogonal projection of the linear asymptotics embedded in the 'crossed triangles Dcatenoid'.
Fig. E3.1g Four translation fundamental domains of Schwarz's D surface Each face is one of the hex_{90} faces shown below in Fig. E3.1h. 
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,
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 {6,63} larger image
The inscribed regular skew hexagon with straight edges
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 mirrorsymmetric 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 Pcatenoid' wireframe of Schwarz's primitive surface P (cf. Figs. 1.2c, d, e)
Here the triangles are only half as far apart as
The P and D surfaces are related by a dilatation
An annular 'crossedtriangles catenoid' (CTC) minimal surface exists
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. E3.2f 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 junglegymlike 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
In Figs. E1.4ah 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.
Fig. E3.3a A lattice fundamental domain of the gyroid G (100) viewpoint
The CoxeterPetrie {6,44} map defines the
G is the only embedded surface among
the countable infinity of surfaces
Every hex_{90} face is
related to each of six faces with
Fig. E3.3c A ninth hex_{90} 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
For a higherresolution version of this image, look
here.
(111) tunnels in G
For a highresolution view, look
here.
Fig. E3.3e The hexagonal tile hex_{90} of G (front view)
The regular skew hexagon hex_{90}
is a face of the
(The hexagon hex_{60} is shown in Figs. 3.3m, n.)
Fig. E3.3f The hexagonal face hex_{90} of G (back view) The front and back surfaces are not the same!
Fig. E3.3g The hexagonal face hex_{90} of G (side view)
Fig. E3.3h This image suggests that hex_{90} of G can be inscribed in a truncated octahedron, but that is impossible. Although the vertices of hex_{90} coincide with six vertices of the truncated octahedron, its edges are not plane curves. Each edge approximates the shape of a quarterpitch 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 hex_{90}
in three of every four
Fig. E3.3i The quadrangular tile quad_{60} of G, a regular skew polygon (front view)
Fig. E3.3j The quadrangular tile quad_{60} of G (back view)
The tile quad_{60} is a face of the CoxeterPetrie
{4,64}
Every quad_{60} face is
related to each of four faces with
Each of these midpoints is also the midpoint of an
(Fig. E3.3l illustrates a wellknown property of dual regular tilings
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 semiregular skew 12gon composed of six replicas of quad_{60} of G Its face angle sequence is ..., 60º, 120º, 60º, 120º,...
Unlike the three regular skew polygons
For a highresolution view, look
here.
Fig. E3.3n The hexagonal face hex_{60} of G is defined by the selfdual Coxeter map {6,63}. Its face angles are 60º, and its area is twice the area of the hexagonal face hex_{90} shown above in Figs. 3.3e,f,g.
Every hex_{60} tile is
related to each of six faces with
Fig. E3.3p Assembly of approximately octahedral shape tiled by the hexagonal faces hex_{90} of G view: (100) direction
Fig. E3.3q Another view of the model of G shown in Fig. E3.3p view: (111) direction
After I met Norman Johnson in June, 1966,
I realized
Regular skew polyhedron {6,63}
(homeomorphic to P)
(homeomorphic to P)
polyhedra, which includes animated graphics, see this Wikipedia article.
§51. My early interest in TPMS (19661970) In the spring of 1966, I accidentally 'discovered' the Schwarz surfaces P and D and then observed their close connection to the CoxeterPetrie regular skew polyhedra. In order to represent the symmetry and combinatorial structure of both the surfaces and their flatfaced relatives, the CoxeterPetrie polyhedra, I employed the metaphorical device of dual skeletal graphs, which I'll call g_{1} and g_{2}. These are triplyperiodic 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 g_{1} and g_{2} 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 g_{1} is replaced by an infinitely thin hollow tube with walls composed of some soapfilmlike material, and that the space inside the entire connected network of these tubes defines a single hollow — but shrunken — labyrinth t_{1}. (Assume that t_{1} has no selfintersections. Remember: this is a metaphorical concept, not a rigorous mathematical construction.)
Here is how I described the relation between g_{1} and g_{2},
the two skeletal graphs of a TPMS in 1970, on p. 79 of
Infinite Periodic Minimal Surfaces Without SelfIntersections:
"Assume that the skeletal graph is given for one labyrinth of a given intersectionfree 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 multiplyconnected 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 t_{1} so that at its
summit,
Curiously, the outside of t_{1} is transformed into the inside of t_{2}. In early 1969, the distinguished topologist Dennis Sullivan
was sharing an office at MIT (just across the street from NASA/ERC) with the mathematics professor
Dirk Struik,
_{ Dirk J. Struik 18942000 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 SelfIntersections technical note. Dennis invited me to his office to explain that the transformation of the tubular graphs g_{1} and g_{2} is an example of the classical AlexanderPontryagin duality (which I had never heard of before). For the Schwarz surface P, the skeletal graphs g_{1} and g_{2} are identical to the 6valent simple cubic graph defined by the edges of an ordinary packing of cubes. For the Schwarz surface D, both g_{1} and g_{2} are copies of the 4valent 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 triplyperiodic 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 pseudogyroid. The models illustrated in Figs. E1.2k, E1.2l, and E1.17 show the final steps in the procedure that led to this pseudogyroid. 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 selfintersections. 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 selfintersecting 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 vertextransitive and edgetransitive). 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 triplyperiodic 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 socalled preliminary report (an internal NASA document, not intended for general circulation), entitled "ExpansionCollapse Transformations on Infinite Periodic Graphs", NASA/Electronics Research Center Technical Note PM98 (September 1969), draft versions of two patent applications, and computer graphics animations of collapsing graphs. The considerably less timeconsuming 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 pseudogyroid (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 pseudogyroid, 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
In early September 1968, I returned to Cambridge from an AMS summer meeting at Madison, Wisconsin, where I had used the pseudogyroid model shown in Fig. E1.17 to illustrate my 15minute 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.)
H. Blaine Lawson, Jr.
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 spaceframes. 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 triplyperiodic 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 spaceframes because of pairwise collisions of edges (called webs or struts by spaceframe engineers) that occur early in the collapse. In contrast to this behavior, for all of the twisted graphs derived from the pseudogyroid, 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 selfintersections). 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 CoxeterPetrie map {6,44} on the pseudogyroid 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 {6,44} 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 soapfilm 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 quasihelical image S_{G} in G of the equatorial geodesic S_{P} in P. I found that this projection of S_{G} (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 quasihelical geodesic S_{G} 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 S_{P} 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 outoftown trip. He had been following my struggles with the pseudogyroid for months. As soon as I told him my exciting news, including my plan to copublish 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 M_{4} and M_{6}, 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 quasicircular edges of a {6,44} 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 PDP11 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 quasicircular 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 quasihelical curves in the pseudogyroid (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 'levelset' gyroid (cf. Fig. E1.31.)
In late 1968, I decided that I must somehow force a nodal polyhedron for BCC_{6} into being, and by trialanderror I produced the spacefilling saddle polyhedron shown In Figs. E2.70a, b, and c. It is described in Infinite Periodic Minimal Surfaces Without SelfIntersections.
Fig. E2.70a BCC_{6} Pinwheel polyhedron: the 6faced nodal polyhedron of the deficient symmetric graph BCC_{6} of degree 6 (stereo pair) The vertices of the graph are the complete set of vertices of the bcc lattice. BCC_{6} is described on p. 82 of Infinite Periodic Minimal Surfaces Without SelfIntersections.
Fig. E2.70c The BCC_{6} graph (orange) and its dual graph (black) (stereo pair)
The edges of the black graph are the edges
A property of the black graph that
I regard as not strictly kosher is that
The rectangular coordinates of G, defined by Schwarz's solution (WeierstrassEnneper parametrization) for the entire family of surfaces associate to P and D If θ_{G} in the term e^{iθG} is replaced by zero, the coordinates are those of D. If θ_{G} is replaced by π/2, the coordinates are those of P — the adjoint of D.
(The value given above for θ_{G} agrees up to eight significant figures
Fig. E1.2p The D surface
Fig. E1.2q The G surface
Fig. E1.2r The P surface
For highresolution pdf versions of these three images, look
