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Alan H. Schoen

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Triply-periodic minimal surfaces
(TPMS)

This is an illustrated survey of TPMS, aimed at both the beginner and the specialist, that includes
links to the contemporary mathematical and scientific literature. It contains an account of how the
author happened to wander unexpectedly into this field in the first place and of the chance events
that led to three years of immersion in a study that was guided by a blend of both experiment and
theory. Help provided by several professional experts is acknowledged, and photographs of many
of these people are included.

The gyroid surface is given special attention here, because in recent years it has become an object
of some interest for materials scientists, who have found that it occasionally serves as a template
for a self-assembled surface that separates two inter-penetrating regions of matter, either organic
or inorganic, with a unit cell that ranges in size over several orders of magnitude, according to the
case.

The paired images — both computer drawings and photos — like the ones immediately below are
stereoscopic, and are arranged to be viewed with eyes crossed.


If you are very patient, you may be willing to spend a moment or two 'priming
the pump' by repeatedly running this brief time-lapse sequence (using the Page
Down and Page Up keys). After several such priming cycles, the animation will
have settled down into a more or less smooth movie that shows how a sequence
of symmetry operations — rotations and rotatory reflections — transform a single
Flächenstück ('asymmetric unit') of the gyroid TPMS into a hexagonal face that is
defined by the Coxeter map {6,4|4}. All of these images are in 'cross-eyed' stereo.



Two friends inside a sculpture
of the gyroid minimal surface at
the San Francisco Exploratorium


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)


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



The PGD family of associate minimal surfaces

Schwarz's P and D surfaces and their associate surface G (the gyroid) are the topologically
simplest examples
of embedded TPMS that have cubic lattice symmetry. They are related
by the continuous bending transformation described in 1853 by Ossian Bonnet. The mean
curvature
(which is zero at every point), the Gaussian curvature, and the orientation of the
tangent plane do not change anywhere throughout the bending. P and D are called adjoint
surfaces: straight lines in one surface are replaced by plane geodesics in the other, and vice
versa. G contains neither straight lines nor plane geodesics. Among the countable infinity
of TPMS related to P and D by Bonnet bending, G is the only example that is embedded,
i.e., free of self-intersections.


P                                               G                                              D
  {6,4|4}P                                      {6,4|4}G                                     {6,4|4}D
   s.c. lattice                                   b.c.c. lattice                                 f.c.c. lattice



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

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 inter-penetrating 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 encloses 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
those 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 now believe that the word 'reciprocal' would have been a better choice than 'dual' to characterize
the relation between the intertwined graphs of a pair of triply-periodic graphs like the simple cubic,
diamond, and Laves graphs (plus a variety of other examples of graphs — many of which define non-
congruent
pairs). That choice would have avoided confusion with the accepted meaning of 'dual' in
the expression 'dual graph', which customarily refers only to plane graphs (or to the graphs of edges
of the triply-periodic Coxeter-Petrie polyhedra). But in what follows I will continue to use the word
'dual' to refer to the rather vaguely defined but symmetrical relation between pairs of triply- periodic
graphs like the ones mentioned here.



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



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

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

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

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

One day in 1966, after making plastic models of a variety of soap films, I telephoned
J.C.C. Nitsche, the renowned authority on minimal surfaces, to ask him to identify for
me two intriguing examples of periodic surfaces I had constructed. He replied that my
descriptions suggested that they were almost certainly the two surfaces analyzed about
one hundred years earlier by H.A. Schwarz. He explained 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 1968, in the course of a telephone conversation with Blaine Lawson, I realized that the
gyroid is simply a hybrid of the two Schwarz surfaces D and P. The 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 simply a linear combination of the coordinates of the
corresponding points in D and P. For the gyroid the Bonnet angle is given by the equation
θG = ctn-1 (K′ ⁄ K)
≅ 38.01477398910808º,
where
K =K(1/2) ≅ 1.685750354812596,
K′ = K′(1/2) ≅ 2.156515647499643.

K and K′ are complete elliptic integrals
of the first kind, with modulus k = 1/2.

In contrast to D and P, G contains neither straight lines nor plane geodesics, and its labyrinths
are enantiomorphic (oppositely congruent). G can be classified as a kind of balanced surface.

Additional details concerning the history of G are
included in
'My first encounter with TPMS' (below).



How I derived the value of θG, the Bonnet angle for the gyroid

The boundary curves for the hex90 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 surface patch D_hex90 (blue) and its adjoint image P_hex90 (red)

A lattice fundamental domain of D or P can be tiled by eight differently oriented
replicas of a hexagonal surface patch. Outlines of these patches are shown here in
blue (D_hex90) and red (P_hex90). D_hex90 is inscribed in a cube of edge length
λD and P_hex90 is inscribed in a cube of edge length λP. The ratio λPλD has the
value required for D_hex90 and P_hex90 to be of equal 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 eight cube corners.
For the patches 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 upward).


If we fix point C at the center of D_hex90, then as D_hex90 morphs into P_hex90
by Bonnet bending, the point D1 on D_hex90 moves along an elliptical trajectory
r1(θ) that is centered at C. The equation for this ellipse is

                                       r1(θ) = D1 cos θ + P1 sin θ                                      (1)

|d1| and |p1| are the semi-minor and semi-major axes, respectively, of the ellipse.
For θ = π/2, the images under bending of D1 and D2 are P1 and P2, respectively.

If λD = 2, then
          D1 = (− 1,1,1)
  |D1| = √ 3
                        P1 = (λP λD) (0,− 1,1)
                                                        |P1| = √ 2 (λP λD).                                   (2)

Schwarz proved (cf. p. 88 of his Collected Papers, vol. I) that

                                         λPλD = K′ ⁄ K (≅ 1.2793).                                     (3)

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

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

For the gyroid, θ = θG . Hence

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

In September 1968, I at last recognized that the key to solving Equation 5 for θG lies
in the geometrical properties of a manifold I call M6. I had derived M6 from its dual,
M4, which had shown up unexpectedly in February while I was struggling to design
a recipe for associating saddle polyhedra with triply-periodic graphs. Photos of my
1968 models of these two manifolds are shown immediately below.

1968 models of M4 (left) and M6 (right), precursors of the gyroid
The hexagons of M6 are the duals of the quadrilaterals of M4.
Viewpoint is in the (1,1,1) direction.


The edges and vertices of M4 define a triply-periodic symmetric graph of degree
six. I called it BCC6, because when two additional edges are added at each vertex,
it is transformed into BCC8, 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.)

In 1966, I had tried unsuccessfully to construct a tiled manifold that partitions R3
into a pair of labyrinths that have enantiomorphic Laves graphs for their skeletal
graphs. What I found intriguing about M4 and M6 was that they are both examples
of just such a manifold! The tiling of M4 by skew quadrangles is described by the
Coxeter map {4,6|4}, and the tiling of M6 by skew hexgons is described by its dual
map {6,4|4}. Since I knew that these maps describe two of the three regular tilings
of Schwarz's D and P surfaces, I suspected (this was in February 1968) that I had at
last picked up the scent of the elusive gyroid. (But it wasn't until September of that
year that I at last understood that the gyroid is simply an associate surface of D and
P that — by an accident of geometry — is free of transverse self-intersections, i.e.,
is embedded. I had allowed myself to be misled by the fact that a physical model of
the pseudo-gyroid really does resemble a genuine triple-periodic minimal surface, as
I explain below.)

A consequence of vanishing coordinates

The edges of a hexagonal face of M6
The y-coordinate y1(θG) of the vertex G1 — and also
of the diametrically opposite vertex — is equal to zero.
This property of M6 is crucial for the derivation of the value of θG.

Each face of M6 is a straight-edged regular skew hexagon with face angle
= cos− 1(− 1/6) ≅ 99.594º. M6 has the same topology and global symmetry
(space group Ia3′d, No. 230) as the gyroid, which I had tried (but failed!)
to construct in summer 1966. The properties of M6 strongly suggested to
me the possibility of somehow constructing the gyroid out of hexagonal
faces whose vertices coincide with those of M6. Each face of M6, like the
hexagonal faces of D and P, is oriented in one of the eight (± 1, ± 1, ± 1)
directions. The y-coordinate y1(θG) of a vertex G1 that corresponds to D1
and P1 (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 M6 is equal to zero.

                                                                          From Eq. 5,
                                                                                                         y1(θG) = cos θG − (K′ ⁄ K) sin θG                             (6)
                                                                                                                     = 0.
                                                                          Hence
                                                                                                          cos θG = (K′ ⁄ K) sin θG .                                         (7)

                                                                         Therefore                        θG = ctn-1(K′ ⁄ K).                                             (8)



New models of M4 and M6 constructed in 2011

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


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


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

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


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

Tiling of M6 by skew hexagons
Viewpoint is in the (1,0,0) direction.



The relation between M6 and the gyroid
M6 is composed of regular skew hexagons with face angle = cos− 1(− 1/6) ≅ 99.594º.
M4 is composed of regular skew quadrilaterals with face angle =cos− 1(1/3) ≅ 70.53º.
Each of these manifolds is the dual of the other. An elementary calculation revealed
that the dihedral angle is smaller at the edges of M6 than it is at the edges of M4 (cf.
Infinite Periodic Minimal Surfaces Without Self-Intersections, Appendix II). These
angles are ~44.4° in M6 and 60° in M4. I anticipated that because of this difference,
M6 would bear a closer resemblance to a continuous minimal surface than M4 does.

My model of M6 proved to confirm this expectation. It showed that the edges of M6
define infinite straight-edged regular helical polygons, which are centered on lines
parallel to the rectangular coordinate axes. This suggested that if I were to replace the
edges of M6 by helical arcs, the dihedral angle of ~44.4° might be reduced 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 was also sure 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 below shows the difference.

I call the object that is tiled by skew hexagons with strictly helical edges the 'pseudo-
gyroid'. It is not a triply-perodic minimal surface. Even if all of its separate faces were
minimal surfaces, they would not define a 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 simply-connected minimal surface spanning a 'Schwarz chain'
of consecutive helical arcs.)

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

By early summer one of the progress reports I was required to write about my research
activity must have disturbed somebody in NASA Headquarters, because word reached
me that some officials there were concerned about my 'playing with soap bubbles'. I
was told that they were considering having me transferred to work on something more
closely related to NASA's mission, like the mechanical support structure of the Hubble
Space Telescope. This news induced in me a state of mild panic, and I decided to switch
my attention to the problem of collapsing graphs, because it appeared to me that NASA
might be at least mildly interested in the application of my ideas to portable spaceframes
that could be stored in a collapsed state and deployed into an expanded state as necessary.

A plastic model of the pseudo-gyroid (1968)
Click here for high resolution image.


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


The scales are finally lifted from my eyes

In early September 1968, I called Blaine on the government WATS telephone line. He
explained to me that he was going to be unable to work on the gyroid problem after all.
The details of what followed are described just below Fig. E2.69c. The point here is that
during our phone conversation, it suddenly dawned on me that the gyroid is just a surface
associate to P and D that happens to be free of self-intersections.. Immediately after we
hung up, I computed the angle of associativity of the gyroid. I was familiar with many of
the details of Schwarz's analysis of P and D, including what turned out to be his crucially
important analysis of λD and λP. In January 1968, out of curiosity I had asked Jim Wixson,
my colleague, to compute the numerical values of λD and λP. As I have described above,
the geometry of M6 provided all the rest of the required data (cf. 'How I derived the value
of θG, the Bonnet angle for the gyroid').


The cylindrical helix and the flattened cylindrical helix

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

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



The machined parts of the brass vacuum-forming tool

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

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

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

(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.)

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


Symmetrical lattice fundamental domains of the gyroid

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


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

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

high resolution image


Side view of lfd1(G)
stereo image

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

high resolution image


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

high resolution image


Side view of lfd2(G)
high resolution image


Two connected replicas of lfd2(G)
Front view

high resolution image


Top view of lfd2(G)
high resolution image


Bottom view of lfd2(G)
high resolution image


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

high resolution image


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

high resolution image


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

high resolution image


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

high resolution image


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

high resolution image


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

high resolution image


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

high resolution image



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

high resolution image


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

high resolution image


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

high resolution image


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

high resolution image


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

high resolution image


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

high resolution image


Polyhedral surrogates of TPMS

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

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

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

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

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


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

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

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


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

For the gyroid G, there exist no examples of either regular or quasi-regular ps,
but there is an infinite uniform skew polyhedron that is a ps of G. It is a 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
a single form. Its two labyrinths are enantiomorphic. It is vertex-transitive, but it
is neither edge-transitive nor face-transitive. It is called (6.32.4.3)-G. Two images
are shown below.

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

high resolution image


(6.32.4.3)-G,
including its skeletal graphs

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

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



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

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

The four faces incident at each vertex of (6.4)2 are arranged in cyclic order
6.4.6.4 — hence the name (6.4)2. In each labyrinth, there are two differently
oriented varieties of chambers. Chambers that share an open hexagonal face
belong to the same labyrinth and are oppositely oriented. Chambers that are
separated by a square face belong to different labyrinths and are identically
oriented.








The triply-periodic Coxeter-Petrie regular skew
polyhedron
{6,6|3}, which — like (6.4)2 — has
the same topology and symmetry as Schwarz's
diamond surface D. Here it is shown with just
one of its two skeletal graphs.


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


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





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


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

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

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

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




P                                               D                                              G
   {6,4|4}P                                    {6,4|4}D                                   {6,4|4}G
    s.c. lattice                                f.c.c. lattice                              b.c.c. lattice

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

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

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


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


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


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

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

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




Sequence of Bonnet bending stages from D to G to P (stereo)
The portion of the surface shown here is a lattice fundamental
domain plus one hexagonal face. The dashed lines are lattice
basis vectors
. But the surface is actually triply-periodic only
when the Bonnet angle θ satisfies the equation

θp,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.)


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

The portion of the 1969 'Part 4'
movie that illustrates this bending

starts at 3min18sec after the beginning.


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


(image by Eric W. Weisstein,
Wolfram MathWorld)


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

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

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


{4,6|4}


{6,4|4}


{6,6|3}



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


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


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


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


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


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 b.c.c..

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



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

Each hex90 has 90º face angles.

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



(1)


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

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


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

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


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



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



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



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

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

High-resolution version



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



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




Gyroid sculpture
by Chaim Goodman-Strauss and Eugene Sargent

Click here for the authors' account of
how they implemented their design.

Here's their razzle-dazzle video
showing bits and pieces of its construction.


If you're unfamiliar with TPMS, one place to begin looking
is the set of Shapeways models made by

Ken Brakke

Alan Mackay

Bathsheba Grossman

and by others.

Shapeways displays a large collection
of gyroid and gyroid-related models here.


Bathsheba Grossman's model of
a kind of inversion of the gyroid
bounded by an ellipsoid


Below are a few of the TPMS models
produced by Shapeways
for Ken Brakke and for Alan Mackay.


There's much more information about TPMS at
one of Ken Brakke's webpages
,
including many illustrations he made with his Surface Evolver.



P                                                  C(P)
(models by Ken Brakke)


D                                                  C(D)
(models by Ken Brakke)


I-WP                                          F-RD
(models by Ken Brakke)


Batwing
(model by Ken Brakke)





      Gyroid                                  Double Gyroid
(models by Alan Mackay)


      Sven Lidin's Lidinoid               Sven Lidin's Lidinoid 222
(models by Alan Mackay)


NodalGyroid222a                             Fluoritesolid7
(models by Alan Mackay)


       SBA-1x50                                  Batwing5x2
(models by Alan Mackay)



The four founding fathers of triply-periodic minimal surfaces


      Georg Friedrich Bernhard Riemann      Karl Hermann Amandus Schwarz
                                        (1826-1866)                                            (1843-1921)                                     


               Alfred Enneper             Karl Theodor Wilhelm Weierstrass
                                     (1830-1885)                                    (1815-1897)                                     

(In §E7 there are photos of a few contemporary experts in this field.)


    INTRODUCTION

    Definition of minimal surface

    Mathematicians have studied minimal surfaces since 1762, when Lagrange derived the 'Euler-Lagrange equation', which is satisfied by the surface of least area spanned by a given closed curve. Aside from the plane, which defines a trivial solution of this equation, the first surfaces found as solutions of the Euler-Lagrange equation were the helicoid and catenoid, both of which were discovered by Meusnier in 1776. Meusnier also proved that the mean curvature of every solution surface is equal to zero. Since some closed curves span more than one surface with zero mean curvature everywhere, a minimal surface is conventionally defined as a surface with vanishing mean curvature at every point, rather than as a surface of least area. Of course every minimal surface is locally area-minimizing, i.e., the surface patch inside every sufficiently small closed curve enclosing any point of the surface has less area than any other surface bounded by that closed curve.

    Pioneering investigations of triply-periodic minimal surfaces (TPMS) were performed by Schwarz, Riemann, Weierstrass, Enneper, and Neovius in the middle of the 19th century. By the early 1960s, however, TPMS had almost faded from view in the mathematical literature. Since about 1970, there has been a revival of interest in TPMS as mathematical research on minimal surfaces of every kind has expanded. Now it is no longer just mathematicians who study TPMS. Materials scientists are also interested in them, because they have concluded that some of the few known examples of low genus — especially those on a cubic lattice — are useful as templates for the shapes of a variety of so-called self-assembled structures that are studied by physical, chemical, and biological scientists. Below are links to a tiny sample of the relevant mathematical and materials science studies, but the sample is neither comprehensive nor up-to-date.

    I have attempted here to summarize my own study of TPMS, which began quite unexpectedly in the spring of 1966. I have included an account — warts and all — of some of the events that led to my involvement in this study, during which I frequently wandered down bypaths that were well off the main route. (I have long believed that such bypaths sometimes offer a more rewarding view than the main route. John Horton Conway has explained that he finds it fruitful to juggle several ostensibly unrelated problems at the same time, because one problem may turn out to be the key to the solution of another. My discovery of a precursor of the gyroid minimal surface in 1968 was for me a validation of Conway's truism, as explained below.)


    My stint at NASA/ERC

    I am enormously indebted to the physicist Lester C. Van Atta, who created for me an unusual position as senior scientist, under his nominal supervision, at the NASA Electronics Research Center (ERC) in Cambridge, Masachusetts. Van Atta, who was both Associate Director and also Director of the Division of Electromagnetic Research, alowed me to indulge my new-found passion for TPMS, even though I lacked the credentials most employers would have considered a minimum requirement for such an undertaking. Because of his scientific reputation, he had sufficient clout to shield me from attacks both by local skeptics (of whom there were more than a few) and also by officials in NASA headquarters who wondered what on earth soap films might have to do with NASA's mission.


    Lester C. Van Atta
    1905-1994
    Associate Director, NASA Electronics Research Center

    But it was all too good to last! ERC was abruptly shuttered in July 1970 in what many of us concluded was probably an act of political malice directed by President Nixon against Senator Ted Kennedy of Massachusetts. I cannot avoid being somewhat sceptical of the purportedly objective history of the closing of ERC by the author of this contemporaneous account, in which no specific role is ascribed to Nixon.

    Until the announcement on December 29, 1969 by the Administrator of NASA that ERC would close on June 30, 1970, I felt quite free to decide what to investigate, with few strings attached. In retrospect, I believe that I would almost certainly have been unable to concentrate productively on my research at ERC if I had been aware of the turbulent political winds that were blowing about our heads.


    Peter Pearce's concept of saddle polyhedra

    Before arriving at ERC in the fall of 1967, I decided that I would concentrate there on two areas of research:

            (a)  symmetric triply-periodic graphs and their nodal and interstitial polyhedra (see explanation below), and
            (b)  a search for new examples of TPMS (even though I had not yet discovered any such examples).

    I had been strongly interested in connections between triply-periodic graphs and convex polyhedra since the mid-1950s, but before 1966 I knew nothing about minimal surfaces, aside from a nodding acquaintance with the helicoid and the catenoid. In April of that year, Konrad Wachsmann, chairman of the architecture department at the University of Southern California, suggested that I visit the North Hollywood architect/designer Peter Pearce, who had a grant from the Graham Foundation for a one-year study of polyhedra, crystal structure, and related topics. Although Peter did not claim to be an expert on the mathematics of minimal surfaces, he had developed a novel application of minimal surfaces to the design of periodic structures that led me to make a radical change in the direction of my research. Below I summarize how this happened.


    Peter Jon Pearce
    Architectural designer

    At Peter's studio I saw several elegantly crafted handmade models of crystal networks, including two that especially caught my eye, because they each contained an example of a novel interstitial object Peter had invented and named saddle polyhedron. These saddle polyhedra had straight edges, but each face was curved in the shape of a minimal surface.

    (All of the stereoscopic image pairs below are arranged for cross-eyed viewing.)

                        
                                 A portion of the diamond graph                                      The interstitial polyhedron
                                                                                                                                  of the diamond graph

                       
                                    A portion of the b.c.c. graph                                           The interstitial polyhedron
                                                                                                                                       of the b.c.c. graph


    Dual graphs and interstitial vs. nodal polyhedra

    I was thunderstruck by Peter's two saddle polyhedra, because I understood at a glance that they were the critical ingredient missing from a scheme I had tried to develop for illustrating the relation between the combinatorial and symmetry properties of crystal networks (triply-periodic graphs) and polyhedral packings. The intended purpose of my heuristic scheme was to represent every atomic site in a crystal strucure by a polyhedron with

            (i)   the same number f of faces as the number Z of edges incident at the corresponding node of the graph, and
            (ii)  the same symmetry as that node.

    For several lattices, the Voronoi polyhedron serves nicely for this purpose. For example, (i) the number f of faces of the cube, which is the Voronoi polyhedron for a vertex of the simple cubic (s.c.) lattice, is six, which is also the number Z of edges incident at each node (vertex) of a conventional ball-and-stick model of the lattice, and (ii) the cube also has the same symmetry as the node with respect to the surrounding lattice.



    A piece of the s.c. 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 s.c. graph

    A similar correspondence holds for the face-centered cubic (f.c.c) lattice if each vertex is enclosed by the rhombic dodecahedron, which is the Voronoi polyhedron for a vertex of this lattice.


    The nodal polyhedron of the f.c.c. graph is the
    rhombic dodecahedron, the Voronoi polyhedron of a vertex of the graph.


    The fluorite graph is the dual of the f.c.c. graph.
    Its nodal polyhedra are
    the regular tetrahedron and
    the regular octahedron.


    The f.c.c. graph and the fluorite graph
    It is convenient to define this pair of graphs as duals.


    For the body-centered cubic (b.c.c.) lattice, however, the combinatorial part of this correspondence breaks down. Although there are only eight nearest neighbors of each vertex in this lattice, the Voronoi polyhedron of a vertex is the truncated octahedron, which has fourteen faces. The reason for this numerical disparity is hardly profound. It's just that the second-nearest-neighbor sites in the b.c.c. lattice happen to be situated in directions and at distances that cause truncation of the six vertices of the regular octahedron, which — as a first approximation to the Voronoi polyhedron — takes only nearest neighbor sites into account. I had observed a similar mismatch for the diamond crystal structure: even though there are only four nearest neighbors for each site, the Voronoi polyhedron has sixteen faces.


    The 14-faced Voronoi cell for a vertex of the b.c.c. 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 b.c.c. lattice and the diamond crystal structure. I described to Peter a space-filling eight-faced saddle polyhedron, composed of regular skew hexagons with 90º corners, that would enclose each vertex of the b.c.c. graph. A few days later I dubbed it the expanded regular octahedron, or ERO (see stereo image below). I proposed calling vertex-enclosing polyhedra nodal polyhedra, irrespective of whether they turn out to be saddle polyhedra or convex polyhedra. For about ten years, I had been calling the triply-periodic graph whose edges correspond to the edges of a packing of expanded regular octahedra the WP graph, because it mimics the pattern of string tied around a wrapped cubic box (see image below).

    Curiously, the ERO was about to introduce me to TPMS!


    The expanded regular octahedron ERO,
    which is the nodal polyhedron of the b.c.c. graph
    and the interstitial polyhedron of the WP graph

    The eight faces of ERO match the number of edges incident at each vertex of the b.c.c. graph, and both the saddle polyhedron and a vertex of the graph have the same symmetry.


    The b.c.c. graph (Z=8)
    The edges of the tetragonal tetrahedron TT
    are shown in blue.


    The WP graph (Z=4)
    The edges of the expanded regular octahedron ERO
    are shown in blue.


    The b.c.c. graph (green vertices)
    and its dual,
    the WP graph (orange vertices)


    Soap film interlude

    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 span more than one shape of minimal surface and that there exist so-called unstable minimal surfaces that are not surfaces of least area spanned by their boundary frames. (On one of his web pages, Ken Brakke' illustrates some classical examples of these phenomena.)

    But as I learned more about minimal surfaces, I became uneasy about Peter Pearce's prescription of minimal surfaces for the faces of saddle polyhedra. I wondered what kinds of saddle polyhedra would result if there existed more than one shape of minimal surface spanned by a given circuit of edges in a triply-periodic graph.

    In June 1966, I undertook some soap film experiments in order to explore these questions. To my surprise I found that the boundary curve C0 (shown at left below), which has the shape of one of the several Hamilton cycles on the cuboctahedron, spans at least two disk-type soap films of different shape. One of these two surfaces, S1, is a surface of least area and is called stable. The other, S0, is not a surface of least area. It is called unstable, because it can be formed as a soap film on C0 only if one or more wires or threads are added to C0 along appropriate curves embedded in S0i.e., curves that partition S0 into an assembly of smaller surface patches each of whose boundaries spans a unique stable minimal surface.

    S1 can drape the boundary frame C0 in either of two positions. Let's call it S1a if it's in one of these positions and S1b if it's in the other. S1a and S1b are related by a halfturn about the axis A1A2. (See Ken Brakke's computed images of S1a and S1b below.) If a wire frame in the shape of C0 is withdrawn from a solution of soap and water, it will span a soap film in the shape of one or the other (but not both) of these surfaces.

    S0, which drapes the wire frame C0 in only one position, forms as a soap film if threads or wires are incorporated in C0 along one or both of the lines A1A2 or B1B2.


                                   C0, a curve that spans at least                                               Cuboctahedron
                                two soap films of different shape


    The three [orthogonal] c2 axes of the boundary curve C0.
    The two axes A1A2 and B1B2 each intersect C0;
    the vertical axis V1V2 does not.

    The soap film S1a, one of two
    congruent stable surfaces spanned by C0,
    is an area-minimizing ('least area') minimal surface.



    S1a                                                       S0                                                        S1b
    (Stable)                                   (Unstable, unless string                                   (Stable)
    or wire is added along
    either A1A2 or B1B2
    or
    both A1A2 and B1B2)

    Incorporating internal threads or wires along (a) either A1A2 or B1B2 or (b) both A1A2 and B1B2 in the wire frame C0, which has twelve edges, partitions C0 into an assembly of congruent skew polygon boundary frames, each with either seven edges [case (a)] or five edges [case (b)].

    The question of how many minimal surfaces are spanned by a given boundary curve is an extremely knotty one, but it is known that there are two properties of a simply-connected boundary curve C either of which guarantees that it spans only one minimal surface of disk type:

          (i) having a convex simple projection — whether central or parallel — onto a plane (Rado's 1932 theorem);
          (ii) having total curvature less than 4π (Nitsche's 1967 theorem).

    Since the aforementioned 5-gons and 7-gons have total curvatures of only 216 π and 313 π, respectively, it follows from Nitsche's theorem that each of them spans only one minimal surface of disk type. This implies that incorporating a wire or thread along either or both of the axes A1A2 and B1B2 will convert the bare frame C0 into a frame that spans the surface S0.

    Note that the soap film in each of the photos below is a piece of S0.


             extra wire added along A1A2           extra wire added along B1B2      extra wires added along A1A2 and B1B2

    In October 1967, three months after I joined NASA/ERC, I was a self-invited guest at Hans Nitsche's home in Minneapolis. Although Hans showed considerable interest in my wire frame of C0 and in my plastic model of S1, he never mentioned that the ground-breaking 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 thanks to Ken Brakke for pointing out a serious elementary blunder in an earlier version of this discussion of the number of soap films spanned by the frame C0.


    Triply-periodic graphs

    Ever since 1954, when I began an informal study of polyhedral packings (triggered by my Ph. D. research on atomic diffusion in crystalline solids), I had ruminated from time to time over the relation between polyhedra and crystal structures. I became familiar with a variety of commonly known crystal structures, and I sawed wooden models of the Voronoi polyhedra that enclose the vertices of some of these structures.

    In 1956 I designed and ran a FORTRAN program that confirmed my hunch that for self-diffusion in f.c.c crystals, the isotope effect and the Bardeen-Hering correlation factor are numerically 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 of distinguishing between the interstitial and vacancy mechanisms of atomic self-diffusion.

    I modeled interstitial diffusion pathways (strictly random walk) by the edges of one triply-periodic graph and pathways for diffusion by the vacancy mechanism in the same crystal (correlated random walk) by the edges of a second triply-periodic graph intertwined with the first graph. I defined these two graphs as duals, and I attempted to discover whether it is possible to define which symmetry and combinatorial properties are required of a triply-periodic graph in order for it to have a unique dual, by analogy with the dual of a planar graph or the dual of a convex polyhedron.

    Using essentially ad hoc methods to identify dual pairs of triply-periodic graphs, I found that while the dual of the diamond graph is also a diamond graph, the dual of the f.c.c. graph is the fluorite graph and the dual of the b.c.c. 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.

    My first encounter with TPMS

    In April 1966, two days after meeting Peter Pearce, I made some examples of saddle polyhedra for myself, using the toy vacuum-forming machine I had bought for my children. My first model was the b.c.c. nodal polyhedron, the expanded regular octahedron ERO illustrated above. But afterwards out of curiosity I joined two of its skew hexagonal faces by rotation instead of reflection. To my great astonishment, I found that if I continued to add faces in this fashion, the infinite smooth labyrinthine structure shown below began to emerge. (This vinyl model, as well as those shown in the next three images, are new ones I made the following year, after I had purchased a larger vacuum-forming machine.)


    A piece of Schwarz's D surface

    Next I replaced the 90º skew hexagon by one with 60º corners, and a second such labyrinthine surface appeared!



    A portion of Schwarz's P surface
    tiled by 60º skew hexagons


    A transparent model of P

    I had unwittingly stumbled onto the two classical examples of adjoint (or conjugate) TPMS, which were discovered and analyzed in 1866 by H.A. Schwarz (and also — independently — by Riemann and Weierstrass). It took a telephone call to the minimal surface authority Hans Nitsche in Minnesota for me to identify these surfaces. I decided to name them D (for diamond) and P (for primitif), after the crystal structures with matching topology and symmetry. I recognized that the chambers in the two complementary labyrinths of P define the sites of the cesium and chlorine ions, respectively, in the ionic crystal Cs-Cl. Only after consulting a handbook of crystal structures did I learn that the atoms of sodium and thallium in the binary solid solution Na-Tl occupy sites that correspond to the symmetrical 'chambers' in the respective labyrinths of D. I began to study in earnest both differential geometry and the complex analysis used in investigations of minimal surfaces.

    With the benefit of hindsight, I later recognized that if — at some time during the year after I stumbled onto D and P in April 1966 — I had taken the time to read Schwarz's Collected Works more carefully, I might possibly have noticed the following theorem on p. 174:

    TRANSLATION:

    I didn't read that passage until September, 1968, when I understood at long last that the coordinates of every point on G are simply a linear combination of the coordinates of corresponding points of D and P, i.e. that G is associate to D and P. A few days later Blaine Lawson pointed out to me that if D and P are scaled so that the 90º hexagons D_hex90 and P_hex90 that tile the map {6,4|4} in these surfaces are inscribed in a cube of the same size, that linear combination becomes the arithmetic mean of the coordinates of corresponding points of D and P!

    It is perhaps surprising that Schwarz doesn't seem to have taken the trouble to sum the coordinates of at least a few pairs of corresponding points of D and P. It seems very likely that if he had done so, he would have discovered the gyroid.

    Dual graphs and skeletal graphs

    The concept of a dual relation for pairs of triply-periodic graphs had suddenly acquired new significance for me. I began to think of such graphs as potential skeletal graphs of the two labyrinths of an embedded TPMS. The geometry of such paired graphs would dictate the geometry of the TPMS. A literature search in the UCLA library indicated that besides D and P, only three other examples of embedded TPMS — H, CLP, and Neovius's surface — had been known since 1883. But I found it hard to imagine that there were not others!

    Early hints of the existence of the gyroid

    For the D and P surfaces, as well as for H, CLP, and Neovius's surface, both labyrinths of the surface are directly congruent, which implies that their skeletal graphs are also directly congruent. I wondered whether any other dual pairs of triply-periodic graphs I had identified — including the oppositely congruent Laves graphs — might also be skeletal graphs of the two labyrinths of an embedded TPMS. The Laves graphs were a troublesome case, because the absence of reflection symmetries made it impossible for me to imagine how such a surface could be generated. Sometimes the makeshift rule I had refined by exploiting the relation between triply-periodic graphs and saddle polyhedra yielded a dual pair of triply-periodic graphs that were neither directly nor oppositely congruent — for example, the f.c.c.fluorite pair and the b.c.c.WP pair. Did this mean that there exist examples of TPMS in which the two labyrinths are not congruent? I did not yet know. Discovering examples of such surfaces would have to wait until I was free to investigate TPMS as something more than an evenings-and-weekends hobby. (Further details of this story are described below.)

    I believed that Peter Pearce had made an inspired choice when he chose minimal surfaces for the faces of his diamond and b.c.c. interstitial saddle polyhedra, and I was becoming confident that the correspondence between polyhedra and the nodes of crystal structures was about to become much simpler. At the same time, however, I had a nagging feeling about certain loose ends that needed tidying up. By May 1966 I had devised an ad hoc recipe for constructing both interstitial and nodal polyhedra that I hoped would be effective for every possible example of a triply-periodic graph. Although the recipe worked without a hitch for every graph I tested, I felt distinctly uneasy, because I suspected that there must exist cases for which it would be ineffective. Although I modified the recipe several times during the next several months, I was never able to give it a solid theoretical foundation.

    Beginning in June 1966, as a spare-time hobby I set out to discover a 'counterexample' — a graph for which the recipe fails to produce either interstitial or nodal polyhedra. I continued to test a variety of graphs, gradually accumulating a diversified collection of vacuum-formed interstitial and nodal polyhedra. I had the additional goal of finding a way to construct a hypothetical TPMS I originally named L (for Laves). Here, however, I will refer to it as G (for gyroid), even though I didn't invent that name until almost two years later.

    It was obvious that G couldn't have any reflection symmetries, since the union of its two enantiomorphic skeletal graphs has no such symmetries. I also recognized that there could be no straight lines embedded in G, since by a theorem of H. A. Schwarz, a straight line embedded in a minimal surface is an axis of 2-fold rotational symmetry. This implies that a half-turn rotation of G about such an axis would interchange the two labyrinths of G — and therefore also interchange the two skeletal graphs of G. But that is impossible, since the two skeletal graphs are enantiomorphic. I couldn't imagine how to define the boundary curves of an elementary surface patch whose edges are neither straight line segments nor curved geodesics (mirror-symmetric plane lines of curvature). For each of the five examples of TPMS known before 1968, there exists a skew polygon surface patch with straight edges.

    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 what I believed to be an exceptionally rare property with the skeletal graphs of Schwarz's P and D surfaces: it is a symmetric graph. A second reason I believed G exists was based on purely visual evidence: when I compared the toy model of G that I had constructed out of stubby paper cylinders with the toy models I had made for P and D, I found that the G model was no less convincing than the other two. In all three models, when the cylinders are made as fat as they can possibly be (see the images just below), the total volume of the gaps between the two 'cylinder-ized' labyrinths is surprisingly small. I was startled to observe how snugly the cylinders of the intertwined labyrinths nestle against each other. When I compared the contours of my toy models of P and D with the smoother contours of my vacuum-formed models of P and D, it seemed entirely plausible that the junctions between cylinders in the toy model of G could be flared and filleted so that the envelopes of the two labyrinths would coalesce into one single surface — a TPMS — just as they do for P and D. I understood, of course, that vague intuitive arguments like these do not necessarily lead to rigorously demonstrable results, but on the other hand I was not prepared to dismiss the arguments as worthless.


    P


    D


    G

    'Toy models' of P, D, and G
    The edges of the dual skeletal graphs
    are represented as right circular cylinders.

    In each of the images at the right, the cylinder radius
    is the maximum possible consistent with the requirement
    that the cylinders intersect only at isolated points of tangency.


    On February 14, 1968, seven months after moving from Los Angeles to Cambridge, I reached two goals simultaneously. (a) I discovered a graph of degree six called 'BCC6' (see stereo image below) that provided the long-sought 'counterexample' to my empirical recipe for deriving the interstitial and nodal polyhedra of a triply-periodic graph. (b) Although this graph failed spectacularly to yield a finite interstitial polyhedron, it pointed toward something much more interesting — an infinite triply-periodic saddle polyhedron that I call M4. The symmetry and combinatorial structure of M4 strongly strongly suggested to me that the hypothetical G minimal surface might exist after all. I immediately lost almost all interest in saddle polyhedra and began to concentrate instead on confirming the existence and embeddedness of the G surface and on searching for other new examples of TPMS.



    A portion of the deficient symmetric graph BCC6 of degree six

    I define a deficient symmetric graph on a given set of vertices as a symmetric
    graph of degree less than the maximum possible for that set. (With
    two additional edges incident at each vertex, BCC6 would
    be transformed into BCC8, the bcc graph.)

    viewpoint: close to [100] direction



    30 quadrangles of M4 (stereo)
    view: [111] direction



    30 quadrangles of M4 (stereo)
    view: [110] direction



    30 quadrangles of M4 (stereo)
    view: [110] direction


    30 quadrangles of M4
    view: [111] direction, backlit by summer sky



    30 quadrangles of M4
    view: [100] direction, backlit by summer sky


    Abstract 658-30 submitted in summer 1968
    to the American Mathematical Society

    This was an awkwardly premature
    announcement of the existence of
    the gyroid, which I then called L.
    I had merely conjectured, not proved,
    that L is a minimal surface.
    (After I recognized that it is associate
    to P and D, I renamed it gyroid.)

    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 short-lived, however. On December 30, 1969, the director of NASA visited Cambridge to announce to a gathering of all employees that ERC would be permanently closed in exactly six months. We were of course startled — as well as disheartened — by this unexpected news. The six year-old ERC was by far the youngest of the eighteen NASA centers. It was the only federal research center with electronics research for its mission, a legacy inherited from the Kennedy presidency (though it was President Lyndon Johnson who presided over its development).

    ERC was famously top-heavy (or perhaps I should say bottom-heavy) with a bloated support infrastructure of low- and mid-level administrators, clerks, etc., many of whom were from the Boston area, hired in the early days before President Kennedy was assassinated. Soon after it opened, the hiring of scientists and engineers slowed down abruptly, and it appeared that the original plan to develop a well-rounded scientific and technical staff had been abandoned. It was our impression that 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 in-house colleagues who shared my scientific and mathematical interests. I would have benefited from having someone close by for chit-chat about — and even collaboration in — those areas of research in physics and mathematics in which I had a special interest. There were spectacular compensations for this deficiency, however. For one, I was acquainted with a few extremely bright young mathematicians in the greater Boston area who showed a friendly interest in my work, and I benefited greatly from my few conversations with them. If only I had shown more initiative, I could have benefited even more from knowing them than I actually did. They included Thomas Banchoff (differential geometry), Norman Johnson (convex polytopes), Nelson Max (computer graphics), and Charles Strauss (computer graphics).

    In 1966, a year before I joined NASA/ERC, Norman Johnson introduced me to the analysis by Coxeter and Moser of the infinite regular maps {6,4|4}, {4,6|4}, and {6,6|3} (cf. the book by these authors that is cited below, following Fig. E1.1k). These three regular maps describe the combinatorial structure of the flat-faced Coxeter-Petrie infinite regular skew polyhedra. But they also describe the combinatorial structure of H. A. Schwarz's P and D surfaces, the canonical 19th century examples of triply-periodic minimal surfaces, as well as that of their only embedded associate surface, the gyroid G (which I nearly discovered in February 1968, when by chance I found a doppelganger that is spookily similar).

    (Parenthetical note: In 1969 Donald Coxeter was my guest in Cambridge, Mass., where he presented a lecture at MIT. To my surprise, he told me that he had never heard of the Schwarz surfaces!)



    H.S.M. "Donald" Coxeter



    Thomas Banchoff at Berkeley in 1973
    photo by George Bergman



    Charles Strauss (seated) and
    Thomas Banchoff

    at Brown University in 1979


    Norman Johnson



    Nelson Max


    MIT was directly across the street from ERC, and Harvard was only a 20-minute walk away. ERC staff members had unrestricted borrowing privileges at the MIT library — an enormous convenience. A few prominent members of the MIT math faculty indulged me now and then when I had a pesky mathematical question, but for the most part, I was hesitant about bothering them, partly because I held them in such awe but also because I knew that for them many of my questions would turn out to be extremely elementary, if not downright trivial.

    A few months after I arrived at ERC, I was visited by Harald ('Hal') Robinson, a sculptor, designer, master machinist, and model-maker who lived in a nearby suburb. We hit it off immediately. After examining my plastic minimal surface models, Hal easily convinced me that he could make more accurate and more durable vacuum-forming tools than I could. Dr. Van Atta was acquainted with Hal's father, an engineer who was president of High Voltage Engineering Corp., the manufacturer of Van de Graaff generators. I persuaded Dr. Van Atta to hire Hal in a flexible part-time arrangement so that he could fabricate vacuum-forming tools for me. Hal wan't interested in a fulltime job, since he had other clients, and in any event I expected to have only enough projects to keep him busy intermittently. From then until the end of my stay at ERC about thirty months later, Hal was my invaluable collaborator.

    Dr. Van Atta also arranged to hire — one or two at a time — part-time work-study students from area universities (Boston University, Northeastern University, Harvard, and MIT) to help with FORTRAN programming and the assembly of new minimal surface models. These young superstars were Kenneth Paciulan, Richard Kondrat, Randall Lundberg, Jay Epstein, and Dennis ____(?). I am grateful to them all.

    Dr. Van Atta also hired James Wixson, an experienced applied mathematician and computer programmer. Jim helped me with a variety of chores. One of his several accomplishments was the invention and programming of a computer algorithm for generating every possible skew quadrangle that serves as a module for a compound periodic minimal surface on a cubic lattice — an assembly of finite surface patch modules whose four straight edges include at least one edge along a [111] direction, coincident with an axis of 3-fold rotational symmetry. Decades later, these solutions have become of some interest as models for structures investigated by physicists and chemists who are soft matter specialists.

    I recall now with some embarrassment that during my first week at ERC, I visited the Harvard mathematics department and stopped by the offices of one after another member of the faculty to ask naive questions about the rather prosaic problem of how to go about enumerating those examples of triply-periodic graphs that are symmetric. Professors Zariski and Ahlfors were both polite, but it was clear that my questions held little interest for them, and the interviews were mercifully short.


    Andrew M. Gleason

    Andy Gleason was another matter, however. He cordially invited me into his office, where we spent the next ninety minutes or so discussing my problem. First he asked me why I was interested in this question. When I explained my still rather half-baked ideas about the connections to triply-periodic minimal surfaces, he showed considerable interest. Although he didn't provide me with definitive solutions for any of my problems, he did ask me a number of stimulating and provocative questions. I never met him again. It was only a few years ago that I learned of the great range of his highly original accomplishments in both 'pure' and 'applied' mathematics. He was a very kind person, and I shall never forget him.

    If I had known then that (a) Andy Gleason and I both graduated from high 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 long-distance telephone line. I made good use of it now and then, including having several fruitful conversations about the stability of minimal surfaces — beginning in 1968 — with Fred Almgren at Princeton. I first met him face-to-face in September 1969, when we arranged to have side-by-side seats on a flight to the USSR. For a week Fred was my roommate at the Hotel Iberia in Tbilisi, Georgia, while we were attending a conference on minimal surfaces. Afterward Fred went on to St. Petersburg for an extended sabbatical visit.


    Frederick J. Almgren, Jr.

    In April 1968, shortly after I discovered experimentally a remarkably close approximation to what I subsequently called the gyroid (but before I had any proof that such a minimal surface exists), I telephoned Robert Osserman at Stanford to ask for his help with a proof. I sent him a plastic model of the surface, and soon aferwards he asked his PhD student Blaine Lawson to investigate the problem. What followed is described below, just after Fig. E2.68c.9. From then on, I occasionally used the WATS telephone line to discuss some of my conjectures about minimal surfaces with Blaine, whom I found to be extremely knowledgeable about every conceivable aspect of minimal surface theory.


    Robert Osserman


    H. Blaine Lawson, Jr.

    I was free — within reason — to attend meetings of the American Mathematical Society, of which I was a member. In contributed 15-minute talks at one or two of those AMS meetings, I described my work and showed some of my minimal surface models. Once or twice someone in the audience would express interest in the mathematics, but more often it seemed that they were curious mostly about how I had constructed the models! After one of those AMS meetings in New York City, I visited the Courant Institute, where I had the enormous good luck to meet Stefan Hildebrandt, already one of the up-and-coming leaders in the mathematics of minimal surfaces. During the next few years, Stefan more than once saved me from making a serious blunder as I groped my way toward a fuller understanding of minimal surfaces.


    Stefan Hildebrandt at Berkeley (1979)
    photo by George M. Bergman
    ©George M. Bergman
    Source: Mathematisches Forschungsinstitut Oberwolfach gGmbH

    Stephen Hyde told me in 2011 that according to Stefan, the reason it fell to his lot to interview me during my 1968 visit to the Courant Institute was that he was at that time one of the youngest members of the research staff. It was the custom for junior members to be assigned the chore of hosting the cranks and crackpots who invited themselves to the Institute. Since I was self-invited, for all anyone could tell I — with my bizarre colored models of surfaces in tow — might turn out to be one of those unwelcome visitors. I was gratified to learn from Stephen that Stefan concluded — after listening to my spiel and examining my surfaces — that I was probably neither crank nor crackpot! He took some photos of me and my models on the roof of the Courant Institute. Here is one of me holding my plastic model of the gyroid.


    A.H.S. 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 about-to-be-formed California Institute of the Arts, in Valencia, California, so my distress over the demise of ERC was somewhat less acute than that of many of my colleagues.)

    In my last six months at ERC, I tried to record as much as possible of what I had learned about TPMS in a NASA technical note entitled 'Infinite Periodic Minimal Surfaces Without Self-Intersections'. Meanwhile, I had been commissioned to design and construct an 11-ft.-diameter model of the gyroid minimal surface for the Museum of Modern Art in New York City, where an Art and Mathematics exhibition was scheduled to open in mid-1970. Here's how the commission came about: Arthur Drexler, Director of the Department of Architecture and Design at MOMA, having heard about the gyroid from one of my colleagues, visited Cambridge in the late summer of 1969 to inspect my collection of minimal surface models. He immediately chose the gyroid as the surface he would like to see me sculpt for the exhibition.

    Dr.Van Atta then telephoned NASA headquarters and almost overnight obtained a grant of $25,000 to support the project. My friend Keto Soosaar, an expert structural engineer at MIT, introduced me to his colleague Jeannie Freiburghouse, an experienced Fortran programmer, and I immediately hired her to use the Weierstrass integrals to compute the coordinates of 8000 points on a hexagonal patch of the gyroid that corresponds to a face of the Coxeter-Petrie regular map {6,4|4}. I arranged for ERC to award a contract to the Gurnard Engineering Corporation of Beverly, Massachusetts to manufacture two CNC-milled aluminum dies for vacuum-forming two kinds of thin zinc-alloy modules — one in the shape of the hexagon of {6,4|4} and the other in the shape of the quadrilateral of {4,6|4}, its dual. These two kinds of module were to be joined by epoxy, in a face-to-face, overlapping arrangement, resulting in a design in which one side of the gyroid surface is tiled by hexagons while the other side is tiled by quadrilaterals. Such a design avoids the need for unsightly connectors along module edges. (I recognized that it would be more efficient to use a single module shape based on the hexagon of the self-dual map {6,6|3}, which is twice as large as that of {6,4|4}, but I didn't pursue this idea, because I knew from experience that the larger negative draft angle of such a big curvaceous module would probably prevent successful vacuum-forming, by introducing ugly ridge-like 'wrinkles' in each module.)

    In the spring of 1970, NASA funding for the project was abruptly subjected to a special kind of 'mid-course correction': it was cancelled. On investigation, I was informed that at a retirement party — presumably well lubricated — for the senior ERC comptroller responsible for my MoMA account, someone had 'accidentally hit the wrong key on his computer', with the result that the money still left in the account was sent back to Washington (i.e., NASA headquarters). The new heir to the comptroller's office told me that there was no way to recover this money. Having had earlier experience with bureaucracies, I recognized that the project was finished. (But see Figs. E1.18a-d below.)

    Dr. Van Atta resigned from ERC in the autumn of 1969 to become research vice-president of the University of Massachusetts/Amherst. Months later my colleagues and I guessed that he might have received early warning signals about the impending demise of ERC. His successor, Lou Roberts, an able electrical engineer and administrator, generously arranged for the last remaining technical typist in our division to be assigned the single task of typing my technical note, but the deadline was so tight that much of what I wrote was litle more than a first draft, since I had no opportunity for either proper editing or for review by another person. (Personal computers had not yet been invented. If computer work stations that allowed for some kind of word-processing existed in those days, I never heard of them.) Immediately after I submitted my manuscript to NASA, I handed in a list of typos and other errors for final corrections. Although I was promised that they would be dealt with, they were not. The one hundred complimentary copies I was promised turned out to be three copies. We all had the feeling that we were now ancient history, and nobody much cared. (Perhaps that is the way it always is with institutions that are in their death throes.)

    In August 1969, before I had any suspicion that my sojourn at NASA would soon end, I received an invitation — thanks to the kind intervention of the mathematicians Robert Osserman and Lipman Bers — to describe my research in a post-deadline presentation at a September conference in Tbilisi, Georgia, USSR on Optimal Control Theory, Partial Differential Equations, and Minimal Surfaces. The conference chairman was Revaz Gamkrelidze of the Steklov Institute.

    When I flew to the USSR consulate in Washington to apply for a visa, the apparatchik in charge at first turned me down, using the excuse that there wasn't enough time. Just at that moment, the distinguished UCLA plasma physicist Burton Fried happened to enter the office. Recognizing me (we had chatted at an APS January meeting a few years earlier), he instantly addressed me by my first name. The apparatchik, who somehow realized that Burt was an important personage, was clearly startled at this show of familiarity. He turned away from me and quickly processed Burt's visa (for an upcoming conference on plasma physics in Russia). Meanwhile, I had retired to a couch a few feet in front of the counter, determined not to give up my own quest for a visa.

    Once Burt's application was processed, he stopped by the couch for a brief chat and then departed. For the next several minutes, the apparatchik pretended to ignore me while he was shuffling papers at the counter. Finally he looked up and asked, "Why are you still here?" I replied that I expected him to change his mind about my visa application, since he had managed to grant Fried's request in spite of the fact that Fried's schedule was even tighter than mine. Perhaps he was impressed by my skill in what he may have perceived as Marxist dialectics. In any event, he appeared to have a sudden change of heart and grumbled, "Perhaps I can do something for you after all." (This was my first — but not my last — observation of obsequious behavior by a petty Soviet bureaucrat.)

    When I landed at Vnukovo International Airport in Moscow about ten days later, I was welcomed by Revaz Gamkrelidze, but I still had to get my bulky collection of plastic models of TPMS through customs. When a stolid Ukrainian customs agent showed signs of balking at the sheer number of boxes I had brought with me (I suppose he suspected they contained contraband), Gamkrelidze put on an impressive show of commanding authority. He announced in a magisterial voice (in Russian) that the boxes contained "mee'-nee-mal soor'-fa-cez". The agent echoed in a bewildered voice, "Mee'-nee-mal soor'-fa-cez?" Gamkrelidze replied with great emphasis that the conference would be impossible without them, and that was that.

    After the conference began, I recognized with dismay that it would have been more sensible to bring fewer models. A few of the very dignified Russian and Western European mathematicians at the conference appeared to be somewhat offended by the sheer quantity of models I had brought. In any event, by the end of the four-day conference, several of the larger models had magically disappeared from the locked auditorium storage room in which they were kept overnight after each day's session, thereby lightening my load on the trip home.

    The conference was nominally hosted by Lev Pontryagin, the giant of mathematics at the Steklov Institute, and Ilia N. Vekua, the amiable Georgian mathematician who was then Rector of Tbilisi State University. Gamkrelidze had been Pontryagin's doctoral student.


    Lev Pontryagin                                 Ilia N. Vekua

    Although I was introduced to Pontryagin, I felt far too intimidated to attempt conversation with him. With Vekua, who was a convivial sort of man, it was another matter. During a banquet at his home one evening, he told me that Southern California was one of his favorite places in the world. After I told him that I had lived in San Diego and L.A. for ten years, he spent the next hour showing me his color slides of the California landscape and telling me stories about his visits to California.

    Here is the program of the 1969 Tbilisi conference:

    page 1
    page 2
    page 3
    page 4
    page 5
    page 6
    page 7

    (Because I was invited at the last minute, I was not listed in the program.)

    Once I have digitized my stereoscopic Kodachrome slides of Tbilisi and the surrounding countryside, I will post images here.

    After the conference, Revaz Gamkrelidze generously arranged a special visit by four or five of us to a small local research institute, where his brother Tamaz, who is a distinguished orientalist, showed us a breathtakingly beautiful treasure that had recently been unearthed in Georgia. It was a tiny sculpture of a chariot and horses, composed entirely of thin gold wires (perhaps less than 1 mm. in diameter). I do not remember exactly how old it was estimated to be, but I vaguely recall hearing that it was about 4000 years old. (If one of my readers has information about this object, please share it with me.)

    Gamkrelidze invited those of us who were planning to be in Moscow during the week after the conference to attend a party at his Moscow apartment. Since I had a Moscow appointment scheduled at just the right time with V.A. Koptsik, the Lomonosov University specialist in ShubnikovBelov color symmetry theory, I was able to attend the party. Koptsik graciously arranged for the 78 year-old Nikolai Belov, who had long since retired, to make a special trip to the university so that I could meet him.

    Belov was one of the truly memorable people I met during my two weeks in the USSR. I regret that I did not have the opportunity to spend more time with him. In his booming voice and heavily accented English, while loudly thumping his chest, he told me that he was "not a Communist, but a Russian!" (I thought that he was being somewhat indiscreet, but perhaps he was confident that he was too distinguished for Brezhnev to bother him.) I shot a stunning pair of stereoscopic photographs of Belov that I am still trying to locate in my cluttered files, because I would like to post them here. Since I am not a particularly skillful photographer, it is all the more remarkable that his portrait looks almost as if it had been taken by Yousuf Karsh.

    I first met Prof. Koptsik in 1968 at a geometry conference at the Ledgemont Laboratory of Kennecott Copper Co. in Massachusetts. It was organized and hosted by a metallurgical physicist, the late Arthur Loeb, whose specialty was crystallography.

    Here is the program of the Ledgemont conference:
    page 1
    page 2
    page 3
    page 4

    Prof. Koptsik showed special interest in my model of the minimal surface C(H). He explained that the distribution of the six colors in the model reminded him of a certain color symmetry group. An image of this model is shown below in Figs. E3.3 - E3.5, He invited me to visit him in Moscow, if I ever visited the USSR. (One year later, I did visit him there.)


    Kenneth Brakke

    In 1999, thirty years to the day after the opening day of the Tbilisi conference, I telephoned Ken Brakke, whom I had met in 1991 at a University of Minnesota conference on minimal surfaces, to ask if he would like to collaborate on an illustrated book about TPMS. He replied that he would be interested in doing the illustrations for such a book, but not in writing the text. The book never materialized, because I never got around to writing it. Instead, over the next few years I occasionally sent Ken adjoint surface data derived mainly from soap film experiments carried out between 1969 and 1974. Using his powerful Surface Evolver program to 'kill periods', Ken quickly produced and posted online images of each conjectured surface. In a small fraction of the cases, he found that the hypothetical embedded surface does not exist.

    But my proposal to write a book with Ken evaporated. That wasn't his fault! I just decided that examining Ken's beautiful computer graphics images and reading his commentary was much more enjoyable than writing a book would have been.

    In 2010, at the instigation of my friend the mathematician Jerzy Kocik, I started this website, which will probably continue to grow for a while. With Ken's permission, I have included here a few examples of his images of TPMS, but I encourage you to visit Ken's set of websites. They're vastly more orderly than the collection of oddments below, and his ilustrations are supplemented by all sorts of information about a variety of other topics in geometry.


    Stephen T. Hyde and Gerd E. Schröder-Turk have effectively summarized the state of our understanding in 2012 of the role of triply-periodic minimal surfaces in chemistry and biology in their introductory review article, Geometry of interfaces: topological complexity in biology and materials, which was published in the Royal Society's Interface Focus (2012) 2, 529-538. Here is the Table of Contents for that journal volume.

    In October 2012, my wife Reiko and I attended the Primosten, Croatia conference, organized by Hyde and Schröder-Turk, on which these conference papers are based. In my own presentation, entitled Reflections concerning triply-periodic minimal surfaces, I described how I came to be involved in the investigation of minimal surfaces, beginning in 1966. (There is considerable overlap between some parts of this account and the material on this website.)


    I recommend the following authoritative on-line introductions to the sprawling subject of minimal surfaces, including triply-periodic ones:

    Hermann Karcher and Konrad Polthier's
    "Touching Soap Films, An Introduction to Minimal Surfaces"

    Elke Koch and Werner Fischer's
    3-periodic surfaces without self-intersections

    Eric Lord and Alan Mackay's
    Periodic minimal surfaces of cubic symmetry

    Matthias Weber's
    Bloomington's Virtual Minimal Surface Museum

    Also of interest is
    Ken Brakke's 2005 translation of the monumental 1873 treatise by J. Plateau:
    Experimental and Theoretical Statics of Liquids Subject to Molecular Forces Only

    Additional on-line references, profusely illustrated, are listed in §E6 (below).

    I recommend especially the book by Stefan Hildebrandt and Anthony Tromba entitled
    The Parsimonious Universe: Shape and Form in the Natural World


    E1. Examples of dual skeletal graphs

    Skeletal graphs of the two inter-penetrating labyrinths of a TPMS
    and Voronoi polyhedra that enclose the vertices of the graph are useful
    for representing the symmetry and topology of some examples of TPMS,
    especially when the graph edges for each labyrinth are symmetrically equivalent.
    Of course these geometrical constructions do not yield analytic solutions for the surfaces.



    (All stereoscopic image pairs are arranged for 'cross-eyed' viewing.)




    Fig. E1.0a
    A dual pair of diamond skeletal graphs


    Fig. E1.0b
    A dual pair of primitive cubic skeletal graphs

    Curiously, the set of vertices of a dual pair of diamond (D) graphs
    and the set of vertices of a dual pair of primitive cubic (P) graphs
    are identical.
    The lattice for this set is b.c.c (body-centered cubic).

    The Voronoi polyhedron for a vertex in this set is the truncated octahedron.

    I know of no other example of a pair of
    directly congruent dual graphs with cubic lattice symmetry.



             1. Dual diamond graphs (skeletal graphs of the D surface)


    A triply-periodic minimal surface (TPMS) that is embedded, i.e., free of self-intersections, partitions space into a pair of disjoint labyrinths. If the labyrinths are congruent — either directly or oppositely — the surface is called balanced. The D (diamond) surface of H.A. Schwarz, which is a balanced TPMS, is shown in Fig. E1.1a.


    Fig. E1.1a
    Four translation fundamental domains of Schwarz's D surface
    (genus 3)

    For some purposes it is convenient to represent a TPMS of low genus by a surrogate with plane faces — a triply-periodic polyhedron with the same symmetry and topology as the surface. The two labyrinths of the TPMS and of its surrogate may be represented by a pair of triply-periodic skeletal graphs that have the same symmetry as the TPMS and its surrogate. Every edge in these graphs joins a pair of vertices that lie at centers of symmetry of the TPMS.

    A simple example of a surrogate of the D surface is the triply-periodic quasi-regular skew polyhedron (6.4)2 (cf. Fig. 1.1b), which is derived from the Coxeter-Petrie regular skew polyhedron {6,4|4} (cf. Fig. E1.35c). Quasi-regular polyhedra are edge-transitive, but not face-transitive.


    Fig. E1.1b
    (6.4)2, a surrogate of Schwarz's D surface (cf. Fig. E1.1a).

    The red and green skeletal graphs are both replicas of the diamond graph.
    Its edges join the sites of adjacent carbon atoms in diamond.

    The Voronoi polyhedron of a vertex of the union of the two dual skeletal graphs is the truncated octahedron. The vertices of the dual skeletal graphs in Fig. E1.1b lie at the centers of the chambers of the respective labyrinths. Each chamber in (6.4)2 is a truncated octahedron from which a tetrahedrally arranged subset of four hexagons has been removed. Hence the boundary of (6.4)2 is composed of four regular hexagons and six squares (cf. Fig. E1.1c).



    Fig. E1.1c
    The two differently oriented chambers in (6.4)2 (cf. Fig. E1.1b).

    The four faces incident at each vertex of (6.4)2 are arranged in cyclic order 6.4.6.4 — hence the name (6.4)2. In each of the two labyrinths, there are two differently oriented varieties of chambers. They are related by a quarter-turn about any one of the three Cartesian axes. In each labyrinth, adjacent chambers related by a translation of type [111] are of opposite variety.



    Fig. E1.1d


    Fig. E1.1e


    Fig. E1.1f


    Fig. E1.1g


    Like (6.4)2, the Coxeter-Petrie triply-periodic regular skew
    polyhedron
    {6,6|3} (cf. Figs. E1.2a,c) has the same topology
    and symmetry as Schwarz's diamond surface D (cf. Fig. E1.2b).
    In this example, in contrast to the case of (6.4)2, the
    unit cell has only one orientation (cf. Fig. E1.2c),
    since it is a translation fundamental region.


    Fig. E1.2a
    A lattice fundamental region of {6,6|3}
    The lattice is f.c.c.


    Fig. E1.2b
    A lattice fundamental region of Schwarz's diamond surface D


    Fig. E1.2c
    Thirteen lattice fundamental regions of {6,6|3}


    Dual pairs of diamond graphs are shown below in both 'medium thick' and 'thick' versions. In the thick version (cf. Figs. E1.3e-h), the diameter d of the cylindrical tubes is the largest possible, consistent with the requirement that the dual graphs not overlap. Overlap occurs when the ratio d/e ≥ 21/2/2 (~.707), where e is the edge length of an ideally thin skeletal graph.

    (For enlarged views, select the hyperlinks just below the images.)


    Medium thick diamond graphs
    view along ~(100) direction


    Fig. E1.3a                        Fig. E1.3b                        Fig. E1.3c
          graph 1                              graph 2                       graphs 1 and 2


    Fig. E1.3d
    Orthogonal projection of graphs 1 and 2 on [111] plane



    Thick diamond graphs
    view along ~(111) direction


    Fig. E1.3e                        Fig. E1.3f                        Fig. E1.3g
          graph 1                              graph 2                       graphs 1 and 2


    Fig. E1.3h
    Orthogonal projection of graphs 1 and 2 on [111] plane



    Fig. E1.3i
    (6.4)2 — a triply-periodic quasi-regular polyhedron
    that has the same topology and symmetry as
    Schwarz's diamond surface D



    Fig. E1.3j
    (6.4)2 with embedded [skinny] dual graphs


    Fig. E1.3k
    (6.4)2 with embedded [fat] dual graphs



             2. Dual simple cubic graphs (skeletal graphs of the P surface)


    Dual pairs of simple cubic graphs are shown below in both 'medium thick' and 'thick' versions. In the thick version (cf. Figs. E1.4e-h), the diameter d of the cylindrical tubes is the largest possible, consistent with the requirement that the dual graphs not overlap. Overlap occurs when the ratio d/e ≥ 1/2, where e is the edge length of an ideally thin skeletal graph.

    Medium thick simple cubic graphs
    oblique view


    Fig. E1.4a                        Fig. E1.4b                        Fig. E1.4c
          graph 1                              graph 2                       graphs 1 and 2


    Fig. E1.4d
    Orthogonal projection of graphs 1 and 2 on [100] plane



    Thick simple cubic graphs
    oblique view


    Fig. E1.4e                        Fig. E1.4f                        Fig. E1.4g
          graph 1                              graph 2                       graphs 1 and 2


    Fig. E1.4h
    Orthogonal projection of graphs 1 and 2 on [100] plane



    Fig. E1.4i
    The Coxeter-Petrie triply-periodic regular skew polyhedron {6,4|4},
    which has the same topology and symmetry as
    Schwarz's primitive surface P



    Fig. E1.4j
    {6,4|4} with embedded [skinny] dual graphs


    Fig. E1.4k
    {6,4|4} with embedded [fat] dual graphs

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

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


The two intertwined skeletal graphs of the gyroid,
in a Shapeways 3D printed version designed by virtox.

Click here for a video of 'Bones',
an animated view of these graphs,
also made at Shapeways by virtox.

Dual pairs of Laves graphs are shown below in 'thin', 'medium thick', and 'thick' versions. In the thick version (cf. Figs. E1.5i-l), the diameter d of the cylindrical tubes is the largest possible, consistent with the requirement that the dual pair of tubular graphs not overlap. Overlap occurs when the ratio d/e ≥ 31/2/2 (~.866), where e is the edge length of an ideally thin skeletal graph.

The fact that the d/e ratio is significantly larger for the pair of thick Laves graphs than it is for the thick simple cubic and thick diamond graph pairs suggests (but does not prove) that the thick Laves graphs occupy a larger fraction of space. In order to make the comparison precise, it would be necessary to take into account the detailed geometry in the neighborhood of the intersections of the cyclindrical tubes. I have not done this.

In order to display the pairs of intertwined graphs as clearly as possible, views are shown for each of the three principal 'crystallographic' directions: [100], [111], and [110].

Thin Laves graphs
[100] view


Fig. E1.5a                        Fig. E1.5b                        Fig. E1.5c
      graph 1                              graph 2                       graphs 1 and 2


Fig. E1.5d
Orthogonal projection of graphs 1 and 2
[100] view


Medium thick Laves graphs
[100] view


Fig. E1.5e                        Fig. E1.5f                        Fig. E1.5g
      graph 1                              graph 2                       graphs 1 and 2


Fig. E1.5h
Orthogonal projection of graphs 1 and 2
[100] view


Thick Laves graphs
[100] view


Fig. E1.5i                        Fig. E1.5j                        Fig. E1.5k
      graph 1                            graph 2                       graphs 1 and 2


Fig. E1.5l
Orthogonal projection of graphs 1 and 2
[100] view


Thin Laves graphs
[111] view


Fig. E1.5m                Fig. E1.5n                Fig. E1.5o
      graph 1                      graph 2               graphs 1 and 2


Fig. E1.5p
Orthogonal projection of graphs 1 and 2
view: [111]


Medium thick Laves graphs
[111] view


Fig. E1.5q                Fig. E1.5r                Fig. E1.5s
      graph 1                      graph 2               graphs 1 and 2


Fig. E1.5t
Orthogonal projection of graphs 1 and 2
[111] view


Thick Laves skeletal graphs
[111] view


Fig. E1.5u                Fig. E1.5v                Fig. E1.5w
      graph 1                      graph 2               graphs 1 and 2


Fig. E1.5x
Orthogonal projection of graphs 1 and 2
view: [111]


Thin Laves skeletal graphs
[110] view


Fig. E1.6a                     Fig. E1.6b                     Fig. E1.6c
      graph 1                           graph 2                      graphs 1 and 2


Fig. E1.6d
Orthogonal projection of graphs 1 and 2
view: [110]


Medium thick Laves skeletal graphs
[110] view


Fig. E1.6e                Fig. E1.6f                Fig. E1.6g
      graph 1                      graph 2               graphs 1 and 2


Fig. E1.6h
Orthogonal projection of graphs 1 and 2
view: [110]


Thick Laves skeletal graphs
[110] view


Fig. E1.6i                    Fig. E1.6j                    Fig. E1.6k
      graph 1                           graph 2                   graphs 1 and 2


Fig. E1.6l
Orthogonal projection of graphs 1 and 2
view: [110]



Fig. E1.7
Straw model of the pair of dual Laves skeletal graphs (1960)
view: [100]


Additional stereo images of the Laves graph


Fig. E1.8
The 'clockwise' Laves graph,
skeletal graph of one labyrinth of the G surface



Fig. E1.9
The 'counter-clockwise' Laves graph,
skeletal graph of the other labyrinth of the G surface



Fig. E1.10
The enantiomorphic skeletal graphs of the two disjoint labyrinths of the G surface



Fig. E1.11
Orthogonal projection on [100] plane of the enantiomorphic Laves graphs



Fig. E1.12
Another view of the 'counter-clockwise' Laves graph



              4. Other pairs of dual skeletal graphs



Fig. E1.13a


Fig. E1.13b

The dual skeletal graphs of a hypothetical but
nonexistent embedded TPMS called TO-TD

(TO stands for truncated octahedron,
the interstitial cage of the blue graph.
TD stands for tetragonal disphenoid,
the interstitial cage of the orange graph.)

blue graph: degree 4
orange graph: degree 14
The blue graph is symmetric.
The orange graph is regular but not symmetric.

The edges of the blue graph are all symmetrically equivalent, but the edges of the orange graph are clearly not all symmetrically eqivalent. They are not even all of the same length. If TO-TD existed, it would be a non-balanced TPMS— i.e., its two labyrinths would be non-congruent. Hence there could be no straight lines embedded in the surface, since such lines are c2 axes and would have the effect of interchanging the two labyrinths. Instead, the surface would be tiled by replicas of a patch S bounded by curved geodesics that — because of the reflection symmetries of the union of the blue and orange graphs — are mirror-symmetric plane lines of curvature.

In 1974, I tested for the existence of TO-TD by using a laser-goniometer method I had devised in 1968 at NASA/ERC. This extremely tedious method requires the construction of a set of several straight-edged boundary frames of various proportions. The laser is used to measure the orientation of the normal to the surface of a [long-lasting] polyoxyethylene soap film S' bounded by each of these frames at many points that are as close as possible to the edges of the frame. Each S' is a candidate for the surface adjoint to S. The adjoint curves computed for the edges of S demonstrated that it is impossible to 'kill the periods' and therefore that TO-TD does not exist.

In 2001, Ken Brakke used his Surface Evolver program to confirm this conclusion with enormously greater speed and accuracy than is possible with the soap film-laser technique.




Fig. E1.13c
The dual pair of skeletal graphs for another
hypothetical but nonexistent embedded TPMS

The green vertices define the sites of the Cu atoms,
and the blue vertices define the sites of the Mg atoms
in the binary alloy Cu2Mg, which has the structure called
Cubic Laves phase C15.


I call the Cu graph FCC6(II). The Mg graph is the diamond graph (cf. Fig. E1.3d). Both graphs are symmetric. The interstitial cavities in the Cu graph are of two kinds: small tetrahedral cages and large truncated tetrahedral cages. All the interstitial cavities in the Mg graph are identical: the expanded regular tetrahedron (ERT) (cf. Fig. E2.19c, E2.20).

In 2001, Ken Brakke used his Surface Evolver program to demonstrate that it is impossible to kill periods for this hypothetical surface. Hence it is almost certainly safe to conclude that the surface does not exist.

Note that in this example, in contrast to other pairs of dual graphs treated here, it is not true that for both graphs, every interstitial cavity of the graph contains a vertex of the dual graph. (The small tetrahedral cages of the Cu graph do not contain any vertex of the Mg graph.)


E2. Mathematical Preliminaries

The Gauss map


Fig. E2.1
(stereo image)
Curved triangular Flächenstück ABC
of Schwarz's Diamond surface


Fig. E2.2
(stereo image)
Riemann sphere
(unit sphere)

The elementary minimal surface Flächenstück ABC shown in Fig. E2.1 is mapped
onto the spherical triangle ABC on the Riemann sphere shown in Fig. E2.2 by the
Gauss map.
Each point on the minimal surface is mapped onto a point on the
Riemann sphere that has the same normal vector.
The red arrows at points A, B, and C indicate
the directions of the surface normal vectors.

There are twelve replicas of the Flächenstück ABC in the skew
hexagonal face EDAE'D'A' of Schwarz's D surface (cf. Fig. E2.1), but
there are only six corresponding spherical triangles in the large spherical triangle
AED on the Riemann sphere (cf. Fig. E2.2). The two Flächenstücke ABC and A'B'C,
for example, are both mapped onto the same spherical triangle. An entire lattice fundamental
region covers the Riemann sphere twice. As a consequence the mapping defines a two-sheeted
Riemann surface, with branch points at the eight 'cube corner' points like C.


Fig. E2.3a
Stereographic projection onto the complex plane
of the elementary triangular Flächenstück ABC
of Fig. E2.2



Fig. E2.3b


Triply periodic minimal surfaces are infinitely-multiply-connected, but it is nevertheless easy to characterize the topological complexity of every example of such a surface by computing the genus p of a single lattice fundamental domain. Except where it is specifically stated to the contrary, it will be assumed in all that follows that TPMS refers to an embedded surface, i.e., one that is free of transverse self-intersections.

Since the smallest posssible value for the genus is three, Schwarz's P and D surfaces are members of a very small select group of topologically simplest examples of TPMS. Below is a recipe for computing the genus of a TPMS. It is based on one of Gauss 's most astounding discoveries, the Gauss-Bonnet theorem, which links the topology and the geometry of a surface. One can very crudely express the essence of the Gauss-Bonnet theorem in this context by saying that the larger the value of the integrated Gaussian curvature for one lattice fundamental region of the surface, the steeper the saddle-like surface contours, and — therefore — the larger the number of tubular 'handles' in the surface as it 'bends around' this way and that.

On p. 233 of the 13th edition of 'Mathematical Recreations and Essays' by W.W. Rouse Ball and H.S.M. Coxeter, the authors use Euler's formula

FE + V = 2,

which relates the number F of faces, the number E of edges, and the number V of vertices of a convex polyhedron to prove that adding a handle to an orientable surface reduces the Euler-Poincaré characteristic Χ = 2 − 2p by 2 and therefore increases the genus p by 1. The proof simply updates the values of F, E, and V after two different n-gons of a map on the surface are joined by a 'bent prism' (which is a convenient device for representing a handle). F is increased by n − 2, E is increased by n, and V remains unchanged. Since

X = FE + V,

Χ is reduced by 2.




Fig. E2.4
Recipe for calculating the genus
of one lattice fundamental domain of a TPMS,
applied to Schwarz's P surface

Another way to calculate the genus is to
substitute for dG from Eq. 5 in the equation p =1− dG.
|dG| is equal to the number of times the Gauss map
of the minimal surface ('Gauss image') covers the Riemann sphere.
For Schwarz's P surface, dG= − 2.
The sign of dG for surfaces of negative Gaussian curvature,
like minimal surfaces, is negative because the sense
of a geodesic edge-circuit on the surface is opposite
to that of its Gauss image on the Riemann sphere.

If you're not familiar with the Gauss map, look here.

For discussion of the Euler-Poincaré characteristic Χ=2 − 2p, look here.

For information about the Gauss-Bonnet theorem, look here.



E3. The D-G-P family of minimal surfaces



Fig. E3.1a
A page from Schwarz's Collected Works



      
(E3a) H. A. Schwarz's diamond surface D

In 1966 I named this surface 'diamond' because both of its interwined labyrinths, which are congruent, have the shape of an inflated tubular version of the familiar diamond graph (cf. Figs. E1.3d to E1.3k).
Below are three of H. A. Schwarz's illustrations of the D surface in in his Gesammelte Mathematische Abhandlungen, Springer Verlag, 1890.


                       Fig. E3.1b                                            Fig. E3.1c


Fig. E3.1d


Fig. E3.1e
Stereo view of the linear asymptotics
embedded in the 'crossed triangles D-catenoid'
wire-frame of Schwarz's diamond surface D
(cf. Fig. E3.1d)
The ratio 2h/λ of the triangle separation 2h
to the triangle edge length λ is equal to √ 6 / 6 (~.408).

On p. 105 of Part I of his Collected Works
(published in 1890),
Schwarz comments as follows:

It appears that for arbitrary values of the
separation of the bounding triangles,
the equations of these surfaces [D and P]
cannot be expressed as elliptic functions of the coordinates.


Fig. E3.1f
Orthogonal projection of the linear asymptotics
embedded in the 'crossed triangles D-catenoid'.


Fig. E3.1g
Four translation fundamental domains of Schwarz's D surface
Each face is one of the hex90 faces shown below in Fig. E3.1h.


The lattice for D is face-centered cubic (f.c.c.), and the translation fundamental domain has genus 3. The edges of the skeletal graph of degree 4 for each of the two congruent labyrinths correspond to nearest-neighbor links in the diamond crystal structure.



Fig. E3.1h
The hexagonal face hex90 of D is
defined by the Coxeter map {6,4|4}.
Its face angles are 90º, and its area is
half the area of the hexagonal face
hex60 defined by the Coxeter map {6,6|3},
shown below in Fig. 3.1i.


Fig. E3.1i
The hexagonal face hex60 of D is
defined by the Coxeter map {6,6|3}.
Its face angles are 60º, and its area is
twice the area of the hexagonal face
hex90 defined by the Coxeter map {6,4|4},
shown above in Fig. 3.1h.


Fig. E3.1j
A rhombic dodecahedral translation fundamental domain
of Schwarz's diamond triply periodic minimal surface D

As a toy model for generating the 'pipejoint' module of D in Figs. E3.1j, k, l,
imagine that you are inside a spherical soap bubble at the center of a rhombic
dodecahedron. Deform the bubble by blowing toward its interior surface in the
four tetrahedral directions [1,1,1], [-1,-1,1], [-1,1,-1], [1,-1,-1] simultaneously,
forming four cylindrical tubules attached symmetrically to the inside faces of
the rhombic dodecahedron around four of its eight trigonal corners.



Fig. E3.1k
A stereo image of the
translation fundamental domain of D in Fig. 3.1j
larger image


Fig. E3.1l
A translation fundamental domain of D on which
approximations to closed geodesics (the red curves)
are inscribed. These geodesics are not plane curves.
larger image


Fig. E3.1m
A regular skew curvilinear hexagon of D,
which is a face of the regular map with holes {6,6|3}
larger image

The inscribed regular skew hexagon with straight edges
is a face of {6,4|4},
one of the three regular maps with holes described in
Generators and Relations for Discrete Groups,
H.S.M. Coxeter and W.O.J. Moser, Springer-Verlag, New York, 1965
and in
Infinite Periodic Minimal Surfaces Without Self-Intersections,
NASA TN D-5541, p. 49.


      
(E3b) H. A. Schwarz's primitive surface P

In 1966 I named this surface primitive because its two interwined labyrinths,
which are congruent, each have the shape of an inflated tubular version
of the familiar primitive (or simple cubic) graph (cf. Figs. E3.3d to E3.3k).
D and P are adjoint surfaces: each surface can be mapped into the other by an isometry
(the Bonnet transformation). Straight lines in one surface are mirror-symmetric
plane lines of curvature (plane geodesics) in the other.



Fig. E3.2a
Stereoscopic image of
the linear asymptotics (blue)
and plane geodesic curves (green)
in the 'square catenoid' of P (cf. Fig. 1.2d, e, f)



Fig. E3.2b
Stereoscopic image of the linear asymptotics
embedded in the 'crossed triangles P-catenoid'
wire-frame of Schwarz's primitive surface P
(cf. Figs. 1.2c, d, e)

Here the triangles are only half as far apart as
the crossed triangles in Schwarz's D surface
(cf. Figs. 1.1d, e, f). The ratio h/λ of the triangle
separation h to the triangle edge length λ is equal
to √ 6 / 12 (~.204).

The P and D surfaces are related by a dilatation
along any of the four [111] directions.

An annular 'crossed-triangles catenoid' (CTC) minimal surface exists
for every value of h/λ less than some allowed maximum value (h/λ)max,
but there are embedded straight lines only in the CTCs of D and P. The
proof depends in part on Schoenflies's theorem (cf. Fig. E2.1), which
proves that there are only six skew quadrilaterals, spanned by minimal
surfaces, that generate TPMS by half-turn rotations about their edges,
i.e., by repeated applications of Schwarz's reflection principle.


Fig. E3.2c
Orthogonal projection of the linear asymptotics
in the 'crossed triangles P-catenoid' of Fig. E3.2b


Fig. E3.2d
A cubically symmetrical translation fundamental domain
of the primitive triply periodic minimal surface P
discovered and analyzed by H. A. Schwarz
in 1866 together with its adjoint surface
D (cf. Fig. E3.1h).


Fig. E3.2e
A translation fundamental domain of P
stereo image



Fig. E3.2f
Six translation fundamental domains of Schwarz's P surface



The lattice for P is simple cubic (s.c.), and the translation fundamental domain has genus 3. If each pair of opposite holes were joined by a hollow tube, the translation fundamental domain would be transformed into an object that is homeomorphic to a sphere with three 'handles'.

P is the unstable stationary state of an inflated jungle-gym-like soap film. Any finite portion of such a soap film can be made stable if threads are stretched along a sufficient number of the embedded straight lines ('linear asymptotics').

As a sort of metaphor for the 'pipejoint' module of P in Figs. E3.2d, e, imagine that you are inside a spherical soap bubble at the center of a cube. Now deform the bubble by blowing against its interior surface in the six directions
x, −x,   y, −yz, −z simultaneously, forming six cylindrical tubules attached symmetrically to the inside of the cube faces at their centers. P partitions R3 into two congruent inter-penetrating labyrinths. The skeletal graph (NASA TN D-5541, pp. 38-39) of each labyrinth is the graph of degree 6 whose edges are those of a packing of congruent cubes.

In Figs. E1.4a-h below are images of tubular simple cubic graphs shown in both 'medium thick' and 'thick' versions. In the thick version, the diameter d of the cylindrical tubes is the largest possible, consistent with the requirement that the dual pair of tubular graphs not intersect. Intersection occurs when the ratio d/e ≥ 1/2, where e is the edge length of the [thin] skeletal graph.



The three Coxeter-Petrie infinite regular skew polyhedra

It dawned on me one day in 1966 that
each of these polyhedra is homemorphic to
— and has the same symmetry as —
either the Schwarz surface D
or the Schwarz surface P.

{6,6|3} is homeomorphic to D, while
{6,4|4} and {4,6|4} are homeomorphic to P.


Regular skew polyhedron {6,6|3}
(homeomorphic to D)


Fig. E3.2g
Translation fundamental domain
The lattice of the oriented surface is f.c.c.


Fig. E3.2h
Assembly of thirteen
translation fundamental domains


Regular skew polyhedron {6,4|4}
(homeomorphic to P)


Fig. E3.2i
Translation fundamental domain
The lattice of the oriented surface is s.c.


Fig. E3.2j
Assembly of twenty-seven
translation fundamental domains


Regular skew polyhedron {4,6|4}
(homeomorphic to P)


Fig. E3.2k
translation fundamental domain
The lattice of the oriented surface is s.c.


Fig. E3.2l
Assembly of twenty-seven
translation fundamental domains



For a comprehensive discussion
of the Coxeter-Petrie regular skew polyhedra,
which includes animated graphics, see this
Wikipedia article.



       (E3c) The gyroid surface G


Fig. E3.3a
A lattice fundamental domain of the gyroid G
(100) viewpoint

The Coxeter-Petrie {6,4|4} map defines the
arrangement of the hex90 hexagonal faces.
G has the same symmetry as that of the union
of its two enantiomorphic skeletal Laves graphs.
The lattice is b.c.c..

G is the only embedded surface among the countable infinity of surfaces
that are associates (cf. Fig. E1.2m) of Schwarz's P and D surfaces.
G contains neither straight lines nor plane geodesics.

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


Fig. E3.3b
Stereo view of the lattice
fundamental domain of
G shown in Fig. E3.3a
higher resolution image

(100) tunnels in G


Fig. E3.3c
A ninth hex90 face has been added here at the
top of the piece of G shown in Figs. E3.3a,b.

This orthogonal projection of G onto the (100) plane shows that the
projected outline S of the spiralling geodesic that bounds each
(100)-type tunnel in G is approximately circular.

For a higher-resolution version of this image, look here.
For a stereoscopic perspective view, look here.

(111) tunnels in G


Fig. E3.3d
This orthogonal projection onto the [111] plane of the piece
of G in Fig. E3.3a demonstrates that the (111) tunnels are
fatter than the (100) tunnels and not so nearly circular.

For a high-resolution view, look here.
For a stereoscopic perspective view, look here.



Fig. E3.3e
The hexagonal tile hex90 of G
(front view)

The regular skew hexagon hex90 is a face of the
Coxeter-Petrie {6,4|4} map. Its face angles
are 90º, and its area is one-half the area
of the hexagonal face hex60 defined
by the Coxeter-Petrie {6,6|3} map.

(The hexagon hex60 is shown in Figs. 3.3m, n.)


Fig. E3.3f
The hexagonal face hex90 of G
(back view)
The front and back surfaces are not the same!


Fig. E3.3g
The hexagonal face hex90 of G
(side view)



Fig. E3.3h
This image suggests that hex90 of G can be inscribed
in a truncated octahedron, but that is impossible.
Although the vertices of hex90 coincide with six
vertices of the truncated octahedron, its edges are
not plane curves. Each edge approximates the shape
of a quarter-pitch of a helix. One half of each edge
lies inside the truncated octahedron, and the other
half lies outside. Alternate edges are curves of
opposite handedness.

G contains one replica of hex90 in three of every four
truncated octahedra in a packing of truncated octahedra.



Fig. E3.3i
The quadrangular tile quad60 of G,
a regular skew polygon
(front view)


Fig. E3.3j
The quadrangular tile quad60 of G
(back view)


Fig. E3.3k
The quadrangular tile quad60 of G
(side view)

The tile quad60 is a face of the Coxeter-Petrie {4,6|4}
map, which is the dual of {6,4|4}. Its face angles are
60º, and its area is equal to two-thirds that of hex90.

Every quad60 face is related to each of four faces with
which it shares an edge by a half-turn about an axis of
type (110) perpendicular to G at the midpoint of the
shared edge.

Each of these midpoints is also the midpoint of an
edge of a dual hex90 face (cf. text below Fig. E3.3a).

(Fig. E3.3l illustrates a well-known property of dual regular tilings
of the plane: the midpoints of edges of dual polygons coincide. Not
surprisingly, this property holds for dual regular polyhedra as well.)



Fig. E3.3l
A pair of dual regular graphs in the plane
Points like P lie at the coincident midpoints of a
pair of triangle and hexagon edges that intersect.


Fig. E3.3m
A semi-regular skew 12-gon composed
of six replicas of quad60 of G
Its face angle sequence is
..., 60º, 120º, 60º, 120º,...

Unlike the three regular skew polygons
quad60, hex90, and hex60,
this 12-gon does not tile G.

For a high-resolution view, look here.




Fig. E3.3n
The hexagonal face hex60 of G is defined
by the self-dual Coxeter map {6,6|3}.
Its face angles are 60º, and its area is
twice the area of the hexagonal face
hex90 shown above in Figs. 3.3e,f,g.



Fig. E3.3o
The hexagonal face hex60 of G
(side view)

Every hex60 tile is related to each of six faces with
which it shares an edge by a half-turn about an axis
of type (110) perpendicular to G at the midpoint of the
shared edge.



Fig. E3.3p
Assembly of approximately octahedral shape
tiled by the hexagonal faces hex90 of G
view: (100) direction


Fig. E3.3q
Another view of the model of G shown in Fig. E3.3p
view: (111) direction

      
How I got started (1966-1970)

In the spring of 1966, I accidentally 'discovered' the Schwarz surfaces P and D and then observed their close connection to the Coxeter-Petrie regular skew polyhedra. In order to represent the symmetry and combinatorial structure of both the surfaces and their flat-faced relatives, the Coxeter-Petrie polyhedra, I employed the metaphorical device of dual skeletal graphs, which I'll call g1 and g2. These are triply-periodic graphs regarded as lying centered in the interiors of the two intertwined labyrinths of these structures. In the discussion that follows, it is assumed (although not stated!) that g1 and g2 are either directly or oppositely congruent. If this restriction is dropped, the assumption (stated below) that the two labyrinths each contain exactly half of space in their interiors must also be dropped.

Now imagine that every edge of g1 is replaced by an infinitely thin hollow tube with walls composed of some soap-film-like material, and that the space inside the entire connected network of these tubes defines a single hollow — but shrunken — labyrinth t1. (Assume that t1 has no self-intersections. Remember: this is a metaphorical concept, not a rigorous mathematical construction.)

Here is how I described the relation between g1 and g2, the two skeletal graphs of a TPMS in 1970, on p. 79 of Infinite Periodic Minimal Surfaces Without Self-Intersections:

"Assume that the skeletal graph is given for one labyrinth of a given intersection-free TPMS. Let each edge of the skeletal graph be replaced by a thin open tube, and let these tubes be smoothly joined (without [self]-intersections) around each vertex so that the whole tubular graph forms a single infinitely multiply-connected surface, which contains the skeletal graph in its interior. Such a tubular graph is globally homeomorphic to the corresponding minimal surface. If the tubular graph is sufficiently "inflated", it becomes deformed into a dual tubular graph which contains in its interior the skeletal graph of the other labyrinth of the surface. The "outside" of the first tubular graph is the "inside" of the second tubular graph. The two tubular graphs of a given TPMS are required to have the same space group as the TPMS, and to correspond, respectively, to two tubular graphs which are globally homeomorphic to the TPMS."

Now imagine inflating t1 so that at its summit,
        (i)    it contains exactly half of space in its interior,
        (ii)   its surface has zero mean curvature everywhere,
        (iii)  it has the same symmetry as the configuration of the two dual skeletal graphs g1 and g2, and
        (iv)  it has no self-intersections.
At the inflation summit, the surface t1 is an embedded TPMS. Until it reaches the summit, t1 exhibits the symmetry of g1. At the summit, it has the symmetry of the union of the [intertwined] g1 and g2. As the inflation proceeds beyond the summit, the surface exhibits the symmetry of g2. It eventually shrinks down to the thin tubular graph t2, which has the symmetry of g2.

Curiously, the outside of t1 is transformed into the inside of t2.

In early 1969, the distinguished topologist Dennis Sullivan


Fig. E1.2m
Dennis Sullivan

was sharing an office at MIT (just across the street from NASA/ERC) with the mathematics professor Dirk Struik,


Fig. E1.2n
Dirk J. Struik
1894-2000
In 1967, I had the rare privilege of becoming acquainted with Dirk Struik.
We both enjoyed hiking along the nature trails in Concord near my home.
When Prof. Struik was more than 100 years old, I attended his lecture
on the history of mathematics at an AMS meeting in Cincinnati.
He was in tip-top form.

who had shown him a draft copy of my Infinite Periodic Minimal Surfaces Without Self-Intersections. Dennis invited me to his office to explain that the transformation of the tubular graphs g1 and g2 is an example of the classical Alexander-Pontryagin duality (which I had never heard of before).

For the Schwarz surface P, the skeletal graphs g1 and g2 are identical to the 6-valent simple cubic graph defined by the edges of an ordinary packing of cubes. For the Schwarz surface D, both g1 and g2 are copies of the 4-valent diamond graph, whose edges correspond to the nearest neighbor links in the diamond crystal structure. Both of these graphs are symmetric, i.e., there is a group of symmetries that is transitive on all the edges and all the vertices.

I knew of only one other example of a symmetric triply-periodic graph on a cubic lattice — the Laves graph. In the spring of 1966, I was seized by the notion that there must exist a TPMS with [enantiomorphic] Laves graphs for its skeletal graphs. I called it the Laves surface L. Unlike the skeletal graphs of P and D, however, the configuration of two dual Laves graphs has no reflection symmetries, and its axes of rotational symmetry lie in directions that I determined could not possibly correspond to lines embedded in the surface I was seeking. As a consequence, I had no idea how to generate a surface patch bounded by either straight line segments or plane geodesics a ("Schwarz chain").

By the time I moved to NASA in July 1967, I had made a reasonably thorough study of Schwarz's writings on periodic minimal surfaces, and I understood the Bonnet associate surface transformation that defines the relation between P and D (and also the relation between the catenoid and the helicoid). The brightly colored plastic models of P and D I had constructed were almost literally screaming out to me that I should explore the territory between these two surfaces (where one surface is bent continuously into the shape of the other), but I did not hear their screams!

In February 1968, I stumbled accidentally on a very close approximation of the gyroid. I'll call it the pseudo-gyroid. The models illustrated in Figs. E1.2k, E1.2l, and E1.17 show the final steps in the procedure that led to this pseudo-gyroid. The resemblance between this virtual doppelgänger and the true gyroid is so close that with the naked eye it is impossible to tell them apart. I still had no idea yet that the gyroid is just a surface associate to P and D that happens to be free of self-intersections. In those days there were not yet any known examples of embedded TPMS derived by examining intermediate stages of the 'morphing' transformation that bends one minimal surface into its adjoint surface via the associate surface transformation, and I didn't have the imagination to think of that possibility.

In 1990, it occurred to Sven Lidin and Stefan Larsson to look for an embedded surface among the surfaces associate to Schwarz's [embedded] surface H and its self-intersecting adjoint surface, and they found exactly one — the lidinoid:


Fig. E1.2o
The lidinoid
(which was originally dubbed 'the HG surface'
by its Swedish discoverer Sven Lidin)


Fig. E1.2p
Schwarz's H surface

To return to the gyroid story, in May 1966 — as mentioned above — I had already begun to suspect that a minimal surface with the symmetry and topology of the gyroid might exist. My suspicions were based on the fact that the Laves graph — like the skeletal graph of each labyrinth in Schwarz's P and D surfaces — is not merely regular (all vertices are of the same degree), but also symmetric (it is both vertex-transitive and edge-transitive). In H and CLP, the two other examples of Schwarz's TPMS, which — like P and D — are of genus 3, the skeletal graphs are merely regular and not symmetric. My intuition suggested that a symmetric graph is so homogeneous that it is very likely to be the skeletal graph of a labyrinth of some embedded TPMS. (I eventually discovered that although some of the few known examples of symmetric triply-periodic graphs are skeletal graphs of labyrinths of such surfaces, by no means all of them are.)

During the spring and summer of 1968, I concentrated on the writing of a so-called preliminary report (an internal NASA document, not intended for general circulation), entitled "Expansion-Collapse Transformations on Infinite Periodic Graphs", NASA/Electronics Research Center Technical Note PM-98 (September 1969), draft versions of two patent applications, and computer graphics animations of collapsing graphs. The considerably less time-consuming one of the two patent drafts was eventually entitled, "Honeycomb Panels Formed of Minimal Surface Periodic Tubule Layers".

I had discovered no useful ideas about how to prove that the pseudo-gyroid (cf. Fig. E1.17) was the basis for a bona fide minimal surface. Blaine Lawson told me in early August that he too had made no progress toward figuring out how to prove that a skew hexagonal face of the pseudo-gyroid, with its strictly helical edges, could somehow be analytically continued to generate an embedded periodic minimal surface.

But ever since my first phone conversation with Blaine in late spring, I had found it extremely helpful to discuss with him a variety of questions concerning minimal surfaces other than the gyroid. I used him as a sounding board on some of my still tentative ideas about how to derive new examples of embedded TPMS by
       (a) enumerating all the ways of constructing a Schwarz chain as a connected sequence of arcs on the n faces (one arc on each face) of what is now called a 'Coxeter cell' — a convex polyhedral space-filler related to each of its neighbors by reflection in a face, and, somewhat later,
       (b) 'hybridizing' two TPMS (cf. Fig. E2.71).

In early September 1968, I returned to Cambridge from an AMS summer meeting at Madison, Wisconsin, where I had used the pseudo-gyroid model shown in Fig. E1.17 to illustrate my 15-minute talk (cf. Fig. E2.10). I was still calling the surface the 'Laves surface' in those days.


Fig. E2.69a
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.)


Fig. E2.69b
Wolfgang Wasow (left) and Magnus Hestenes (right)


Fig. E2.69c
H. Blaine Lawson, Jr.

About a week after my return to Cambridge, I used the government WATS line to phone Blaine to ask him whether he had made any progress toward a proof that the gyroid is a minimal surface. He replied that he hadn't, because finishing his dissertation had left him little time to think about other matters. He said he was going to have to abandon work on the problem. I begged him not to give up, because I felt certain the solution was close at hand (even though I had no rational grounds for believing that to be the case!).

To change the subject, I told Blaine about the graph collapse transformation I had discovered, and how it could be 'run backwards' to provide the basis for the design of expandable space-frames. I had investigated the transformation for graphs associated with the P, G, and D surfaces (cf. Figs. E2.68b.0, E2.68b.1, and E2.68b.2, for example). I described what I called 'just a coincidence' (or words to that effect): that the trajectory of every graph vertex is an ellipse not only in the associate surface transformation of Bonnet but also in the totally unrelated graph collapse transformation. I emphasized that there is no fundamental connection between these two transformations. I described how I had found that of the triply-periodic graphs that are associated with P and D, either as embedded graphs or skeletal graphs, those that have reflection symmetries are not candidates for expandable space-frames because of pairwise collisions of edges (called webs or struts in the space-frame) that occur early in the collapse. In contrast to this behavior, for all of the twisted graphs derived from the pseudo-gyroid, including the Laves graph, no such collisons of edges occur. The only collisions are the ones that would occur in actual physical spaceframes, in which struts collide somewhat before the 'complete collapse' stage because of their finite thickness.

I had not previously even mentioned graph collapse to Blaine, and it's hardly surprising that he didn't seem to understand the details of what I said to him. It was obvious that I hadn't explained the elliptical trajectories coincidence very well, because Blaine's response was something like:

"Are you saying that the gyroid is associate to Schwarz's P and D surfaces?"

I hadn't said that at all, but it hardly mattered, because at that instant, everything suddenly fell into place. The fog had finally lifted! Thanks to Blaine's question, I finally understood that the gyroid is just a surface associate to P and D that happens to be embedded (free of self-intersections). It is the only such surface, as I was soon able to confirm by means of simple 'morphing' sketches similar to the computer drawings in Fig. E1.21.

Because I had spent the summer analysing the details of graph collapse transformations on P, D, and G, I was aware that the surface orientation at the vertices of the hexagonal faces of the Coxeter-Petrie map {6,4|4} on the pseudo-gyroid is identical to the surface orientation at the corresponding vertices of P and D. That was a powerful hint pointing to the Bonnet transformation that had been 'staring me in the face' every day since March, when I assembled my first plastic model of the gyroid. I was hugely embarrassed, realizing how obvious it should have been to me that the gyroid is associate to P and D! After all, I was familiar with the properties of the Bonnet transformation. I had long since traced out the geometrical relation between the equatorial circle in the catenoid and the central axis of the helicoid, which I had found illustrated in Dirk Struik's marvellous Lectures on Classical Differential Geometry. I had also sketched the corresponding curves in P and D countless times. Those relations should have been the clue. I had also spent days studying not only H. A. Schwarz's Collected Works, but also Erwin Kreyszig's Differential Geometry, Luther Pfahler Eisenhart's A Treatise on the Differential Geometry of Curves and Surfaces, and Barrett O'Neill's Elementary Differential Geometry.

Although I knew from experience that ideas that should be obvious are sometimes anything but obvious, I nevertheless felt stupid when I realized that I had posed the wrong question to Bob Osserman back in March. when I asked him whether there might be a way to derive the Weierstrass parametrization for a Schwarz chain composed of six helical arcs. I had mistakenly assumed that the edges of the hexagonal faces of the {6,4|4} map on the gyroid were perfect helices.

Immediately after Jim Wixson joined NASA/ERC in January 1968, I asked him to write a FORTRAN program for calculating — from Schwarz's equations — the coordinates of a set of closely spaced points on the equatorial geodesic of the 'square catenoid' in P. A simple soap-film demonstration suggests that although this curve appears to be approximately circular, it cannot be a circle. Consider its shape in the limit of very small separation of the boundary squares of the 'square catenoid'. In that limit it can be described roughly as a square with slightly rounded corners. As the separation of the boundary squares is increased, the curve looks more and more like a circle, but I found it impossible to imagine that it becomes exactly circular when the separation becomes equal to its value in the P surface. When I plotted the points computed by Jim, I found that the equatorial geodesic departs from perfectly circular shape by slightly less than 0.5%.

Eight months later, after understanding at last that the coordinates of every point in a lattice fundamental region of G are a simple linear combination of the coordinates of a pair of corresponding points in D and P, I plotted a graph of the orthogonal projection of the quasi-helical image SG in G of the equatorial geodesic SP in P. I found that this projection of SG (cf. Fig. E3.2b) also departs from perfectly circular shape by slightly less than 0.5%.


Fig. E3.2a
Stereoscopic view of the linear asymptotics (blue) and plane geodesic curves (green)
in the 'square catenoid' of P (cf. Fig. 1.2d, e, f)


Fig. E3.2b
Orthogonal projection on the [100] plane
of the quasi-helical geodesic SG in G (cf. Fig. E3.3c)

(I didn't learn about Björling's Strip Theorem until several years after I left NASA/ERC. This theorem proves very simply that the equatorial geodesic SP in Schwarz's P surface cannot be circular.)

I felt only slightly less stupid when I discovered that Blaine's response to my harangue about the elliptical trajectories of the vertices of collapsing graphs was not actually the result of his understanding that the gyroid was associate to P and D. He had been justifiably confused by my rambling description of those irrelevant elliptical trajectories. When I explained to him the evidence for the associate surface relationship, he agreed that it was a reasonable idea. I immediately proposed that we publish together an announcement about the gyroid. He courteously refused, explaining that his crucial question to me was prompted by a misunderstanding of what I was saying. But I insisted that if he had not asked me that question in precisely those words, it would have been impossible to say how long it might have taken me to understand what was going on. He then reluctantly agreed to collaborate on a paper about the gyroid.

Two days later, Dr. Van Atta returned to ERC from an out-of-town trip. He had been following my struggles with the pseudo-gyroid for months. As soon as I told him my exciting news, including my plan to co-publish with Blaine Lawson, he scolded me in no uncertain terms! He insisted that I phone Blaine and explain that I had made a serious error, and that I must publish alone. (It was the only time Dr. Van Atta ever displayed anger or impatience toward me.) Blaine was courteous when I relayed my new message to him, but I realized that my vacillation must have offended him.

Gradually I succeeded in feeling very slightly less stupid than I had at first, after reflecting on the fact that neither Schwarz, Riemanm, Weierstrass, nor any of their successors seem to have suspected the existence of an embedded surface associate to P and D, in spite of the fact that they were all experts on the Bonnet transformation. On the other hand, I realized that it had been pure dumb luck for me to stumble onto M4 and M6, the precursors of the gyroid.

I was able to derive the angle of associativity (cf. Fig. E1.23) easily, because I had already made a detailed study of the geometrical calculations Schwarz carried out in his analysis of the P and D surfaces. In January 1968, because I was curious about the precise shape and arc length of the quasi-circular edges of a {6,4|4} hexagon of P (cf. Fig. E1.2c), I sketched the outline of a computer program for getting answers to these questions. My colleague Jim Wixson coded the program in FORTRAN and ran it on ERC's PDP-11 minicomputer. The output of Jim's program, combined with Schwarz's analysis, provided the required clues to the value of the angle of associativity of G (cf. Fig. E1.23). These results demonstrated that the departure from perfect circularity of the quasi-circular holes in a pipejoint unit cell of P is in the range of approximately ± 0.5% of the hole's mean radius, implying a comparably small departure from perfect helicity of their image curves in the gyroid.

Not only did the quasi-helical curves in the pseudo-gyroid (cf. Fig. E1.17) turn out to be very close approximations to the corresponding curves in the true gyroid, but these curves are also extremely close to — but not quite the same as — the corresponding curves in the 'level-set' gyroid (cf. Fig. E1.31.)


In late 1968, I decided that I must somehow force a nodal polyhedron for BCC6 into being, and by trial-and-error I produced the space-filling saddle polyhedron shown In Figs. E2.70a, b, and c. It is described in Infinite Periodic Minimal Surfaces Without Self-Intersections.



Fig. E2.70a
BCC6 Pinwheel polyhedron:
the 6-faced nodal polyhedron
of the deficient symmetric graph BCC6 of degree 6
(stereo pair)
The vertices of the graph are the complete set of vertices of the b.c.c. lattice.
BCC6 is described on p. 82 of Infinite Periodic Minimal Surfaces Without Self-Intersections.



Fig. E2.70b
An oblique view of the BCC6 Pinwheel polyhedron
(stereo pair)



Fig. E2.70c
The BCC6 graph (orange) and its dual graph (black)
(stereo pair)

The edges of the black graph are the edges
of the BCC6 Pinwheel polyhedron.

A property of the black graph that
is not strictly kosher is that it
intersects the orange graph.
(Perhaps one can find a 'nicer' example of
an improvised nodal polyhedron
for the BCC6 graph.)



Fig. E2.7
The six components of the brass tool, shown before assembly
and final machining, that I designed for vacuum-forming the plastic
hexagons of the 1968 model of the pseudo-gyroid shown in Fig. E1.4c.

Two years earlier, I discovered an arrangement of two sets of the eight
solid tetrominoes in enantiomorphic trigonal packings of a half-cube, shown
in Figs. E2.8a and b. These two arrangements of the tetrominoes pack the cube.
The shapes of these packings suggested the design of the tool parts shown above.

(I naively imagined that the packing of the eight tetrominoes shown in
Figs. E2.8a and b might be unique. But no sooner did I ask my friend
George Bell to investigate than he replied (same day!) that there are 36
solutions. He emailed me, on 2/18/2014, the one shown in Fig. E2.8c.)


Fig. E2.8a
Mirror-symmetric triskelion packings
by the eight solid tetrominoes


Fig. E2.8b


Fig. E2.8c
One of George Bell's solutions for the
packing of the triskelion by eight solid tetrominoes


My addiction to recreational mathematics, which worsened considerably
once I started playing with these 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.



Fig. E2.9a
Martin Gardner
10 October 1914 — 5 May 2010



Fig. E2.9a
Solomon Golomb



Fig. E2.10
Abstract 658-30 submitted to the American Mathematical Society in 1968
announcing the discovery of the gyroid
I referred to the gyroid here as 'L' (for 'Laves').
A few weeks later I renamed it 'gyroid'.


This announcement was slightly premature! I mailed in the abstract in Fig. E2.10 a month or so before the Madison summer meeting of the AMS, even though I had not yet succeeded in proving that the surface represented by the pseudo-gyroid (cf. Fig. E1.4c) is a single continuous minimal surface. I had naively assumed that any expert on minimal surfaces would be able to construct such a proof. A few months earlier, I had sent a plastic model of the pseudo-gyroid to Bob Osserman, who passed it along to his PhD student Blaine Lawson, Blaine promised to think about the problem in his spare time, even though he was already fully occupied with his dissertation research.

It wasn't until about ten days after the Madison meeting that at last I understood — thanks to a fruitful phone conversation with Blaine — that the gyroid is an associate surface in the Schwarz PD family. (The details are described just after Fig. E2.68c.9.) By means of drawings based on hand calculations, I confirmed that there are no other intersection-free associate surfaces between P and D. A few months later, the differential geometer Tom Banchoff introduced me to Charlie Strauss, the mathematician/computer graphics expert who is his friend and collaborator. I hired Charlie to write a computer graphics program for producing stereoscopic perspective animations, and I used one such animation sequence (cf. Fig E1.21) to strengthen the evidence that every other associate surface is self-intersecting. Twenty-eight years later, this claim was at last proved rigorously by Karsten Grosse-Brauckmann and Meinhard Wohlgemuth, in their article, 'The gyroid is embedded and has constant mean curvature companions', Calc. Var. Partial Differential Equations 4 (1996), no. 6, 499-523.

The precise version of the surface that I had in mind at the time of the Madison meeting had a fatal flaw: the boundary of each of its hexagonal faces is a chain of one-quarter pitches of circular helices, alternately right-handed and left-handed (cf. Fig. E2.7). Even now, in 2011, it is not known how to derive an analytic solution for a minimal surface bounded by a circuit ('Schwarz chain') composed of such arcs. Although an assembly of these hexagons looks like a single infinitely-connected minimal surface, it is not one. To explain why I had chosen these helical curves for the surface patch boundary, I offer the following more or less chronological summary of the tangled sequence of events that culminated in the construction of the model of L.


In February 1968 I found my first promising lead in the hunt for the gyroid — a pair of related surfaces I named M4 and M6. One might call them wrinkled versions of the gyroid. The original models of these surfaces are shown in Figs. E1.16a and E1.16b. A variety of stereoscopic images of more recent models of M4 and M6 are shown in Figs. E1.19 and E1.20.


Fig. 1.16a                                                          Fig. 1.16b
M4                                                                     M6
view: [111] direction

As the result of a lucky guess about how to remove the 'wrinkles' in M6, I produced the pseudo-gyroid, which is shown in Fig. E1.17. The details of how I got from the pseudo-gyroid to the gyroid are described below following Fig. E1.2p.


Fig.E1.17
The pseudo-gyroid

By September 1968 I had concluded that the gyroid is the only embedded surface among the countably many surfaces associate to P and D, but my 'proof' was based on (a) physically bending assemblies of plastic replicas of surface modules and on (b) computer graphics animation of that bending (cf. Fig. E1.21b). In 1996, Karsten Grosse-Brauckmann and Meinhard Wohlgemuth published a rigorous proof that the gyroid is embedded (free of self-intersections) and contains neither straight lines nor reflection symmetries, in The gyroid is embedded and has constant mean curvature companions, Calc. Var. 4, 1996, 499-523.

It seems likely that before 1968, no one had ever bothered to look at any of the countably many intermediate surfaces associate to Schwarz's P and D surfaces to determine whether any of them were embedded. I confess that before 1968 it had never occurred to me to look there (even though it should have!).

The gyroid has received much attention from physicists, chemists, and biologists since the early nineties, because it has been found to be — at least approximately — a kind of geometrical template for a great variety of self-assembled bicontinuous structures, both natural and synthetic. I first announced its existence at an AMS meeting in August, 1968 (cf. abstract in Fig. E2.10), in the mistaken belief that the pseudo-gyroid is a bona fide minimal surface. (It is remarkably close to one!) The actual gyroid G is described on pp. 48-54 of Infinite Periodic Minimal Surfaces Without Self-Intersections, NASA TN D-5541 (May 1970), and also here in Fig. E1.23.



Weierstrass parametrization of G

Fig. E1.2m
The rectangular coordinates of G,
defined by Schwarz's solution (Weierstrass-Enneper parametrization)
for the entire family of surfaces associate to P and D
If θG in the term eiθG is replaced by zero, the coordinates are those of D.
If θG is replaced by π/2, the coordinates are those of P — the adjoint of D.

(The value given above for θG agrees up to eight significant figures
with the value derived from a more recent analysis
based on a Schwarz-Christoffel mapping,
in Adam Weyhaupt's 2006 PhD thesis (pp. 115-116).



Fig. E1.2n
Bonnet bending of one skew hexagonal face of D (blue)
into its image in G (violet) and P (red)
During the bending, the trajectory of each point on the surface is an ellipse.






Fig. E1.2o
Sequence of Bonnet bending stages from D to G to P (stereo)
One lattice fundamental domain (plus one additional hexagonal face)
  •     0º          D
  • ~38.015º   G
  •   90º          P

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

The above images are eleven frames selected from a movie I made in 1969
using a FORTRAN program written by the computer scientist Charles Strauss.
The cinematographer, Bob Davis of the MIT Lincoln Lab,
used a Bell and Howell 35mm movie camera
modified to accept input data from a PDP-11 computer.
Ken Paciulan and Jay Epstein peformed the data input.
The enthusiastic help of all of these great guys is gratefully acknowledged.

To animate this time-lapse bending sequence, click here
Press on the Page Down/Page Up keys to see the animation.


Stereo views of translation fundamental domain of P, D, and G


Fig. E1.2p
The D surface


Fig. E1.2q
The G surface


Fig. E1.2r
The P surface

For high-resolution pdf versions of these three images, look
here (D),
here (G),
and
here (P).


These pdf images will probably load slowly, because they are large (7 to 10 Mb). For maximum image clarity, zoom in to make the width of the image almost equal to the screen width. The associate surfaces D and P are called adjoint, because the angle of associativity (cf. Fig. E1.23) by which they are related via Bonnet's bending transformation is π ⁄ 2.

Like all of the other intermediate surfaces associate to the helicoid and the catenoid, G contains neither straight lines nor plane lines of curvature.

The round tunnels centered on [100], [010], and [001] axes, which are arranged in square checkerboard arrays, are bounded by approximately helical curves of opposite handedness in the two intertwined labyrinths of the surface. The outermost curved edges of the eight congruent hexagonal faces in Fig. E1.3a correspond to the edges of the regular map with holes {6,6|3}. If you examine these edges closely, you will see that they are not quite plane. Each of them lies half inside and half outside the enclosing cube.

Let us call a geodesic curve on a triply-periodic minimal surface a trivial geodesic if it is either a straight line or a mirror-symmetric plane line of curvature, and a non-trivial geodesic otherwise. Since there are neither straight lines nor mirror-symmetric plane lines of curvature in the gyroid, all of its geodesic curves are non-trivial.

If you apply rubber bands to physical models of the three surfaces P, G, and D (as I have done), you will easily discover a variety of periodic geodesic curves, both closed and unbounded. I will eventually show here images of several non-trivial geodesics, both closed and unbounded, on these three surfaces.

For a discussion of geodesics on multiply-connected surfaces, see Steven Strogatz's 2010 NYTimes essay on geodesics, with links to Konrad Polthier's videos on this topic.



The four images in Figs. E1.15d to E1.15g were made by Ken Brakke, period killer extraordinaire.

The embedded surfaces I-WP (cf. also Fig. 2.5) and F-RD (cf. also Fig. 2.6) are the adjoints of two of the four self-intersecting Schoenflies surfaces (cf. Fig. E2.1). In 1975, I used the laser-soap film technique mentioned just below Figs. E1.15a and b to kill periods in order to obtain the approximate shapes of the curved edges of the octagonal surface patch of F-RD(r). Several years ago, Ken used his Surface Evolver program to obtain the vastly improved modeling of F-RD(r) shown below. Soon afterward I asked Ken to use Surface Evolver to obtain the hexagonal patch of I-WP(r). In order to emphasize how the tubular structure of each cubic surface cell on the right is related to its companion at its left by a one-eighth-turn (45º) about a vertical axis, Ken included '(r)' in its name.


Fig. E1.15d                   Fig. E1.15e
    F-RD                           F-RD(r)


Fig. E1.15f                   Fig. E1.15g
    I-WP                           I-WP(r)


A spectacular large model of G is shown in the
YouTube video,
Gyroid playground climbing structure at the San Francisco Exploratorium
and also in Figs. E1.18a-d (below).

This structure was designed and built by a team that included
Thomas Rockwell, Paul Stepahin, Eric Dimond, and John Kinstler.
Kinstler describes here how the plastic modules of the
Exploratorium gyroid were designed and fabricated.

Here is a photo of a more recent laminated plywood gyroid sculpture at the Exploratorium.
The photographer was Jef Poskanzer, who has informed me that
the young man inside is his nephew Henry.



Fig. E1.18a
Exploratorium gyroid
photo courtesy of Amy Snyder

High-resolution photos of the Exploratorium gyroid
(also courtesy of Amy Snyder) are at:
E1.18a
E1.18b
E1.18c
E1.18d



Figs. E1.19a-e show M6, which was the immediate precursor of the gyroid (cf. abstract in Fig. E2.10). The vertices coincide with the vertices of the Coxeter-Petrie map {6,4|4} on the gyroid, but the edges of the hexagonal faces are line segments, not quasi-helical arcs as in the gyroid. Figs. E1.20a-e show M4, the immediate precursor of M6.



Fig. E1.19a
65 hexagons of M6
view: [111] direction, silhouetted by bright summer sky backlighting



Fig. E1.19b
65 hexagons of M6 (stereo)
view: [100] direction



Fig. E1.19c
65 hexagons of M6
view: [100] direction, silhouetted by bright summer sky backlighting



Fig. E1.19d
65 hexagons of M6
view: [111] direction






Fig. E1.19e
65 hexagons of M6
view: [100] direction




Fig. E1.20a
30 quadrangles of M4 (stereo)
view: [111] direction



Fig. E1.20b
30 quadrangles of M4 (stereo)
view: [110] direction



Fig. E1.20c
30 quadrangles of M4 (stereo)
view: [110] direction



Fig. E1.20d
The edges of the model in Fig. E1.20c,
which define a portion of the 'deficient' graph BCC6
view: close to [100] direction


Fig. E1.20e
30 quadrangles of M4
view: [111] direction, backlit by summer sky



Fig. E1.20f
30 quadrangles of M4
view: [100] direction, backlit by summer sky



Cubic unit cell of G



Fig. E1.22
Cross-eyed stereogram of a gauzy cubic unit cell of G
G is associate to P and D, i.e., its rectangular coordinates
are a linear combination of those of P and D (cf. Fig. E1.23).
This figure shows twice as much of G as is contained in one translation fundamental domain.
The lattice for G is body-centered cubic (b.c.c.).

Among the countable infinity of surfaces associate to P and D, G is the only embedded surface.
A translation fundamental domain has genus 3.



Cubic unit cell of G


Fig. E1.24
A view from a different corner (the lower right rear corner) of the unit cell of G shown in Fig. E1.22

The skeletal graphs of the two enantiomorphic labyrinths of G are enantiomorphic Laves graphs of girth ten (and degree 3).

Since the lattice for G is b.c.c.,
a truncated octahedron is a reasonable choice for a translation fundamental domain.
This cube unit cell has volume equal to that of two translation fundamental domains.


The sequence of three stereo images immediately below illustrates the application of Ossian Bonnet's 1853 bending transformation, without stretching or tearing, to the 'morphing' of Schwarz's D surface into Schwarz's P surface. The orientation of the tangent plane at each point of the surface remains fixed throughout the bending. The G surface shows up a little more than one-third of the way (cf. Fig. E1.23) through the transformation. A simply-connected translation fundamental domain of genus 3 is shown in each figure. Each such domain is composed of eight congruent regular skew hexagonal faces, which are defined by Coxeter's regular map with holes {6,4|4} (cf. Fig. E1.4).

The lattices for the three [oriented] surfaces are face-centered cubic (f.c.c) for D, body-centered cubic (b.c.c) for G, and simple cubic s.c.) for P.


Stereo view of G


Fig. E1.28
Cross-eyed stereogram of a larger piece of G




A hexagonal face of G

Fig. E1.29
Cross-eyed stereogram of a hexagonal face of G that corresponds to
a face of the regular map with holes {6,4|4} {cf. Fig. E1.5).
The six vertices of the face coincide with vertices of a truncated octahedron.
The edges of the face appear to be plane curves, but they are not.
One half of each edge lies inside this truncated octahedron, and one half outside.
view: [100] direction


Two hexagonal faces of G


Fig. E1.30
Cross-eyed stereogram of two hexagonal faces of G


David Hoffman and Jim Hoffman (not relatives!) have animated the associate surface transformation P-> G -> D for a single translation fundamental domain.



Gerd Schröder-Turk has also animated the P -> G -> D Bonnet transformation . In addition, he has created a spectacular 'FlyThrough' animation of a large chunk of G. His movie shows the enantiomorphic skeletal graphs of the two labyrinths of G. Don't be surprised if Gerd's movie takes a while to download this file. It's not small (~91 Mb!).


The "level-set" surface G*, which very closely resembles G, is defined by the equation

cos x sin y + cos y sin z + cos z sin x = 0.

David Hoffman and Jim Hoffman have illustrated the striking resemblance between G and G*.



Below are images of G* made with Mathematica.

The level surface G*


(a)


(b)                      (c)                      (d)                     (e)


                   (f)                        (g)                      (h)                   

Fig. E1.31
Various views of the level surface G*
  • (a) stereo view
  • (b) skew hexagonal face
  • (c) assembly of eight of the skew hexagonal faces, which defines a translation fundamental domain bounded by a cube
  • (d) the assembly shown in (c) viewed from the (1,1,1) direction
  • (e) the assembly shown in (c) viewed from the (− 1,− 1,− 1) direction
  • (f) view of the assembly in (c) from the (1,0,0) direction, showing the square array of quasi-helical tunnels. Adjacent tunnels spiral alternately CW and CCW.
  • (g) view of the assembly in (c) from the (1,1,1) direction, showing the kagomé array of quasi-helical tunnels. Adjacent tunnels spiral alternately CW and CCW.
  • (h) oblique view of the assembly shown in (g)

The quasi-helical curves ('flattened helices') in G* that are centered on c4 axes coincide with right circular cylindrical helices of radius π/4 at just four points in each period, at exactly quarter-period intervals. Helices fail at all other points to satisfy the level surface equation for G*. Halfway between each pair of consecutive points of coincidence, the radius of each flattened helix has a local minimum ≈ π/4 − 0.015941383804981744. Hence this reduced radius ≈ 2.02972 % less than π/4.


Flat-faced polyhedral models of minimal surfaces

Voronoi G

In 1969, with the help of Charles Strauss and Bob Davis, I made a 35mm movie of the collapse of several examples of triply-periodic graphs. It includes images of the continuously transformed Voronoi cells that surround each graph vertex. Several scenarios from this movie may be seen in my 1972 video "Shapes of Soap Films Part 4". In 2013, I noticed to my surprise that the convex polyhedron I call V-G17 appears at approximately 10 min:32 sec after the beginning of the video, as part of the scenario entitled

'Voronoi cells of a vertex of the collapsing graph (degree 6) of the edges of the Coxeter map {4,6|4} on the gyroid surface'.

V-G17 is the enantiomorphic Voronoi cell (cf. Fig. II-2a, p. 89 in NASA TN D-5541) whose two mirror-symmetric versions surround neighboring vertices of a pair of properly juxtaposed dual Laves graphs. "Voronoi G" (cf. Figs. E1.34c to E1.34g), which is a triply-periodic polyhedron with plane faces, may be derived from an infinite packing of these two enantiomorphic polyhedra by removing the three decagon faces from every cell. Voronoi G roughly approximates the gyroid surface.

I recently returned to the study of the transformation of the Voronoi polyhedra associated with graph collapse. If a given vertex V of a graph of degree Z is regarded as fixed, the Z vertices that are endpoints of the Z edges incident at V all rotate in circular arcs centered on V. In its collapsed state, the infinite graph degenerates into the edge complex of a single convex polyhedron whose edges are parallel to the edges of the graph before collapse.

As stated above, V-G17 is the Voronoi cell of a vertex of the union of the Laves graph of degree 3 and its dual graph. But the 1972 video demonstrates that V-G17 also happens to be identical to the Voronoi cell of a vertex of the partially collapsed graph of degree 6 with the combinatorial structure of the 'deficient' graph BCC6, which is the graph with the edges and vertices of M4 (cf. Fig. E1.20e).

Fig. E1.32 shows a set of screen-captured video images selected from a continuous sequence of geometrical transforms of the truncated octahedron Voronoi cell. Polyhedra of three combinatorial types occur in this sequence. V-G17, which appears at the midpoint of the sequence, is the example labeled '10:32'. Each polyhedron that precedes V-G17 by a given number of frames is related to one that follows it by the same number of frames by inversion in its center followed by rotation through a quarter-turn. The approximate value of the video time code for each image is listed directly below the image, and a Schlegel diagram for each polyhedron is shown below its time code. The values of the time code indices demonstrate that the transformation is accelerating at a significant rate.

NOTE:

I expect eventually to post here a detailed analysis of the collapse transformation of triply-periodic graphs. I will include graphic images of selected examples of graphs and their associated Voronoi polyhedra, plus examples of nets of some of these polyhedra, together with vertex coordinate data.

I have observed that the absence of reflection symmetry is an essential property of a triply-periodic graph if the collapse transformation is to reach its final state (the set of edges of a convex polyhedron) with no edges colliding until the final instant. The Laves graph is the progenitor of all of these collison-free graphs. Its symmetry allows the twisting that prevents edges from colliding.

10:14           10:21           10:27           10:29           10:32           10:35           10:37           10:40           10:42

Fig. E1.32
Video images of Voronoi cells of the
vertices of partially collapsed graphs


Fig. E1.33
Schlegel diagrams of the Voronoi cells of Fig. E1.32
(These diagrams show the combinatorial structure of the polyhedra,
but they are not metrically faithful projections.)



        

Fig. E1.34a                                                                                            Fig. E1.34b
Voronoi cells of the vertices of the enantiomorphic Laves graphs
that are skeletal graphs of the gyroid (stereoscopic pairs)


Fig. E1.34c
Voronoi G, viewed in the [100] direction (stereoscopic pair)

(For a higher-resolution [pdf] version of the above image, look here.)


Fig. E1.34d
Orthogonal projection of Voronoi G onto [100] plane (cf. Fig. E1.34c)



Fig. E1.34e
Voronoi G, viewed in the [110] direction (stereoscopic pair)
(For a higher-resolution version of the above image, look here.)
This image demonstrates that the gyroid is symmetrical by a half-turn about a [110] axis.



Fig. E1.34f
Orthogonal projection of Voronoi G onto [110] plane



Fig. E1.34g
Orthogonal projection of Voronoi G onto [111] plane
(For a higher-resolution [pdf] version of the above image, look here.)

Compare the shapes of the tunnel holes with the
corresponding holes in G shown in Fig. E1.19.



Fig. E1.34h
One chamber of Voronoi G
All but six of its thirty vertices lie on the sphere of radius r1 shown here.
The six outliers — shown as red dots — lie on
a slightly larger sphere of radius r2 (not shown).
The ratio r2/r1 ≅ 1.029.



Uniform polyhedra based on G


Figs. E1.35a,b
The uniform gyroid (6.32.4.3), an infinite uniform polyhedron

The faces of (6.32.4.3) are all regular plane polygons, and
its symmetry group is transitive on both faces and vertices.
(For a pdf version of the image in Fig. E1.35a, look here.)

The combinatorial structure and symmetry of (6.32.4.3) are
defined by the
Poincaré hyperbolic disk model of uniform tilings in the hyperbolic plane.


Fig. E1.35c
The Poincaré hyperbolic disk model of 6.32.4.3)
(Wikipedia image)


Let me define two objects to be homologous if they have the same symmetry and the same topology. Since April 1966, I had been mulling over the homologous relations between the two Schwarz minimal surfaces P and D and the Coxeter-Petrie regular skew polyhedra {6,4|4}, {4,6|4}, and {6,6|3}, whose faces are regular plane polygons,


My first model of the gyroid


Fig. E1.4i
A Voronoi surrogate of P — the Coxeter-Petrie
triply-periodic regular skew polyhedron {6,4|4},
which is homologous to P.

And here is one D surrogate:


Fig. E1.2c
A Voronoi surrogate of D — the Coxeter-Petrie
triply-periodic regular skew polyhedron {6,6|3},
which is homologous to D.

After I create an image of {4,6|4}, I will post it here.
(cf. Wikipedia's animated images of the three
Coxeter-Petrie regular skew polyhedra
.)

Both {6,4|4} and its dual {4,6|4} are homologous to P, and the self-dual {6,6|3} is homologous to D. In 1968 I derived an example of a triply-periodic polyhedron with plane faces that is homologous to G — Voronoi G (cf. Figs. E1.34c-g). It was known from the work of Coxeter and Petrie that there exists no regular triply-periodic skew polyhedron with plane faces that is homologous to G. I wondered, however, whether there exists a uniform triply-periodic polyhedron with plane faces that is homologous to G. The abstract in Fig. E1.35c is a condensed summary of the results of my search for quasi-regular tilings of P, G, and D.

Although my notebook and most of the associated files and physical models related to these tilings were destroyed when an intruder ransacked offices at Southern Illinois University in 1982, I plan — time permitting — to reconstruct this work, and I will post here some examples of the quasi-regular tilings ('15 IQRWP') cited in the abstract in Fig. E1.35c.

After I showed my abstract (cf. Fig. E1.35c) to Norman Johnson in 1969, he extended it by deriving the combinatorial structure (although not the geometry) of every quasi-regular or uniform tiling in the {4,6} family. When I investigated the geometry of his examples, I discovered that (6.32.4.3) is the only uniform tiling of the gyroid by plane polygons in this family. The (6.32.4.3) tiling was later discovered independently by John Horton Conway, who named it mu-snub cube. ('Mu' is an abbreviation for multiple here.)

Norman Johnson is well known for his 1966 enumeration of the 92 Johnson solids, which Zalgaller later proved is exhaustive. He is currently writing a book on uniform polytopes in R3 and R4.

(For decades, I have tried — without success — to persuade Norman
to publish an account of his enumeration.)

Norman's list of the tessellations of the {4,6} family,
proved by him to be complete, is shown in Fig. E1.35b.




Fig. E1.35b
Norman Johnson's 1969 enumeration of
the uniform tessellations of the {4,6} family
(unpublished)



Fig. E1.35c
A summary of my enumeration of the
quasi-regular tessellations of the {6,4} family
(abstract published by the American Mathematical Society in 1969)

Fig. E1.35d (below) shows AMS Abstract 658-30 (1968),
cited in the first line of the Abstract in Fig. E1.35c (above).



Fig. E1.35d
Abstract 658-30, submitted in summer 1968
to the American Mathematical Society for the
Madison meeting in August



Uniform polyhedron models of P


Fig. E1.36
(6.43), an infinite uniform polyhedron
model constructed by my Cal Arts student Bob Fuller in 1971



Fig. E1.37
(6.43), an infinite uniform polyhedron
model constructed by Bob Fuller in 1971


Fig. E1.38a
(6.43), an infinite uniform polyhedron
model constructed by Bob Fuller in 1971


Fig. E1.38b
Stereo photo of Bob Fuller's model of (6.43)



Isaac Van Houten's bronze sculpture of G


Fig. E1.39a
Isaac Van Houten's 2006 bronze sculpture of the gyroid G


Fig. E1.39b
Template for casting Isaac's sculpture of G


Fig. E1.40
The uniform gyroid (cross-eyed stereo view)




Bathsheba Grossman's rapid prototype sculpture of G


Fig. E1.41
Two views of one of Bathsheba Grossman's printed models of G


In 1934, the German mathematician Berthold Stessmann published an article in Mathematische Zeitschrift 38, 1934 (414-442) entitled "Periodische Minimalflächen".

A book by Stessmann, also entitled " Periodische Minimalflächen ", was published by J. Springer in 1934.

* * * *
I have been unable to learn what became of Stessmann in the WWII period. (Perhaps there are some leads in here.) If you discover any information about him, please email me.
* * * *
On the first page of Stessmann's article there are drawings of the six skew quadrilaterals that Arthur Moritz Schoenflies proved in 1890 are the only skew quadrilaterals, spanned by minimal surfaces, that generate TPMS by half-turn rotations about their edges, i.e., by repeated applications of Schwarz's reflection principle. These six Schoenflies quadrilaterals are reproduced here in Fig. E2.1.

The six Schoenflies quadrilaterals


Fig. E2.1
The six Schoenflies quadrilaterals

Of the six Schoenflies quadrilaterals, only I and III are patch boundaries for embedded surfaces — Schwarz's D and P, respectively (cf. Figs. E1.2a and E1.3a). By applying Schwarz's reflection principle, it is easy to demonstrate that the other four quadrilaterals — II, IV, V, and VI — define surfaces with transverse self-intersections.

But what about the adjoints (cf. text below Fig. E1.23) of II, IV, V, and VI?
It is a fundamental property of any two adjoint minimal surfaces S1 and S2 that if a boundary edge in S1 (say) is a straight line segment E1, then its image in S2 is a segment of a plane line of curvature C1, lying in a plane perpendicular to E1. Let us call this property 'Property A'.

The adjoint of II is an embedded surface of genus 9 and is called Neovius's surface. It was first analyzed in 1883 by Edvard Rudolf Neovius — an 1869 doctoral student of Schwarz's — in his dissertation, Zweier Speciellen Periodischen Minimalflächen auf welchen unendlich viel gerade linien und unendlich viele ebene geodätischen linien liegen. I sometimes identify Neovius's surface by the alternative name C9(P) — or just C(P) — because it has exactly the same embedded straight lines as P and has genus 9. (The 'C' in C9(P) stands for 'complement'.)

C9(P) is illustrated in Figs. E2.2a,b,c, and its self-intersecting adjoint surface C9(P)† is illustrated in Fig. E2.2d.

Edvard Neovius (1851-1917), who became Professor of Mathematics at the University of Helsinki, was a cousin of Ernst Lindelöf and uncle of Rolf Nevanlinna. (For further information, see this short article about the history of the Finnish Mathematical Society.)


The embedded Neovius surface C9(P) and its self-intersecting adjoint surface C9(P)†



(a)                                                                              (b)
Figs. E2.2a and b
One lattice fundamental region of
the embedded Neovius surface C9(P) (genus 9)
The images are copied from Tafeln II and III of Neovius's 1883 dissertation.

On June 27, 2012, one day after my wife Reiko and I visited the magnificent National Library of Finland (cf. images above), Mr. Jari Tolvanen, a reference librarian there, kindly informed me by email that Neovius's doctoral dissertation, a copy of which is in the library's collection, is also available online here.


(c)                                                                              (d)
Figs. E2.2c and d
(c) An assembly of eight lattice fundamental regions of the embedded Neovius surface C9(P)
and
(d) an assembly of seven lattice fundamental regions of its self-intersecting adjoint surface C9(P)†

These two images are copied from Tafeln IV and III of Neovius's 1883 dissertation.
A complete copy of the dissertation is shown here.



Fig. E2.3a
My first model of C9(P)
The 'see-through' tunnels are aligned in [110] directions.


Fig. E2.3b
A later model of C9(P)






Fig. E2.4a
Announcement in a February, 1969 abstract published by the American Mathematical Society
describing C(D), a TPMS that is complementary to Schwarz's D surface

"Abstract 658-30" mentioned in the third line
and shown here as Fig. E2.10, refers to the gyroid,
which I originally named "L" (for Laves) in 1968.

The possibility that there exists an embedded counterpart of C(P), which it would be reasonable to call 'C(D)', occurred to me as I was crossing the street while returning to my NASA office from the MIT library. I was staring at the illustrations in the photocopy of Neovius's 1883 PhD dissertation made for me a few minutes earlier by the MIT science librarian. When I saw the drawing shown above in Fig. E2.2b, I was startled to see that the set of straight lines in the surface is identical to the set of straight lines in Schwarz's P surface. I imagine that Neovius and Schwarz (his teacher) also must have noticed this matching of lines!

I suddenly decided that what is sauce for the goose may also be sauce for the gander. I reasoned that since both P and D can be regarded as [infinite] regular polyhedra, it is plausible that if one of them (P) has an embedded companion surface that contains the same set of straight lines, then the other (D) probably does too. I concluded that if I could confirm the existence of such a surface for D, I would name Neovius's surface 'C(P)' and the new surface 'C(D)' — where 'C( )' means 'complement of ( )'.

As soon as I reached my office a few minutes later, I needed only to glance at my straw model of the straight lines in D to recognize instantly that the embedded surface C(D) exists. A few months later, I realized that higher order complements of both P and D probably exist too (cf. discussion of 'Notched adjoints' following Fig. E2.82).

The MIT librarian told me that no one had borrowed Neovius's thesis during the fifty years since the library had acquired it. It was printed in large folio format, and as a result she had to cut all the pages into quarters (which took her a good while!) before she could make a photocopy for me. Perhaps that's a good indication of how interested mathematicians were in periodic minimal surfaces in those bygone days.

(My account of how the MIT library copy of Neovius's thesis came to see the light of day for the first time in fifty years would hardly be complete if I failed to mention that I initially asked the librarian to obtain a copy for me via inter-library loan, because I naively assumed that it was much too obscure to be included in the MIT collection. I even mentioned that I was aware that she might have to send to Finland for it, and I was prepared to wait. She did not deign to reply but simply marched off in the direction of the stacks, pausing only to look back once over her shoulder, wearing an expression of polite disdain as if to say, "What kind of institution do you think we are?" I can't imagine that she actually knew that she would find the thesis on the shelves, but she located it in much less than a minute!)



C(D)


Fig. E2.4b
C19(D) (genus 19)
Right: One translation fundamental domain
Left: One-fourth of a translation fundamental domain

C19(D) is called the [first-order] complement of D,
because it has exactly the same embedded straight lines as D.
Ken Brakke's images of a sequence of
higher-order complements of C19(D)
of genus 35, 51, 67, ..., which belong to two families, A and B,
are shown



Fig. E2.4c
C19(D)
Two translation fundamental domains



Fig. E2.4d
C19(D)
One-fourth of a translation fundamental domain,
cut from two different parts of the surface


Fig. E2.4e
What's left in 2011 of the brittle 1970 model of C19(D) shown in Fig. E2.4b
view: [111] direction



Fig. E2.4f
C(D)
view: [-1-1-1] direction



Fig. E2.4g
C(D)
view: [110] direction


What about the adjoints of Schoenflies IV, V, and VI?

When (in 1968) I first examined these three quadrilaterals with Property A (cf. discussion following Fig. E2.1) in mind, I was startled to discover that although the two triply-periodic surfaces derived from Schoenflies IV and from its adjoint are obviously both self-intersecting (both of them have a 120º corner and therefore a branch-point), it seemed to me highly likely that the adjoints of both Schoenflies V and Schoenflies VI were patches for embedded surfaces! I developed persuasive experimental evidence in support of this conjecture by combining modeling with soap films and the Bonnet bending of approximate replicas, made from vacuum-formed plastic sheet material, of these soap films. I chose the name F-RD for the adjoint of V (which has genus 6) and the name I-WP for the adjoint of VI (which has genus 4). (My naming conventions are explained in Infinite Periodic Minimal Surfaces Without Self-Intersections).

Andrew Fogden was the first to confirm that F-RD is embedded, in a 1992 article entitled "A systematic method for parametrizing periodic minimal surfaces: the F-RD surface", Journal de Physique 2 (1992) 233-239. Fogden succeeded in obtaining the Weierstrass polynomial for F-RD, a task that was begun in 1934, but never completed, by Berthold Stessmann (cf. Stessmann B., Periodische Minimalflächen, Mathematische Zeitschrift 33 (1934) 417-442).

In a 1992 collaboration between the versatile mathematician Djurdje Cvijovic and the Cambridge University chemist Jacek Klinowski, the authenticity of I-WP was rigorously established and its several unusual properties were derived (Cvijovic, D. and Klinowski, J.: The computation of the triply periodic l-WP minimal surface, Chemical Physics Letters 226 (1994) 93-99).

F-RD and I-WP were the first identified examples of TPMS in which the two labyrinths of the surface are non-congruent, but in recent years, many additional examples of such surfaces have been discovered..

In every known example of an embedded TPMS that contains no straight lines, the two labyrinths are found to be non-congruent. Elke Koch and Werner Fischer have classified such surfaces as non-balanced and surfaces that contain straight lines — like P and D — as balanced. Because I-WP and F-RD contain no straight lines, they are called non-balanced. They are illustrated in Figs. E2.5 and E2.6.

I-WP



Fig. E2.5a
I-WP (genus 4)
oblique view



Fig. E2.5b
I-WP
Modeled by
oblique view



Fig. E2.5c
I-WP
view: [111] direction (approximately)



Fig. E2.5d
I-WP
view: [100] direction
Portions of two non-trivial geodesics (cf. discussion below Fig. E1.24) are shown.
The one is at the left is closed, and the triply-periodic one at the bottom is unbounded.



Fig. E2.5e
I-WP
view: [100] direction



Fig. E2.5f
I-WP
view: [100] direction



Fig. E2.5g
I-WP
oblique view



F-RD


Fig. E2.6a
F-RD (genus 6)
view: [111] direction

(See my 1997 F-RD poster essay)

I am indebted to John Brennan and Robert Fuller, two of my
outstanding students at California Institute of the Arts in 1971,
for volunteering to complete this large model of F-RD. In 1983,
it was destroyed by a marauder at Southern Illinois University/
Carbondale, together with most of my other models. (I never
learned the identity of this person, but I was assured that he was
not one of my students!)



Fig. E2.6b
F-RD
view: [111] direction



Fig. E2.6c
F-RD
view: [110] direction



Fig. E2.6d
F-RD
view: [100] direction



Fig. E2.6e
F-RD translation fundamental domain
view: [111] direction



Fig. E2.6f
F-RD
One-half of a translation fundamental domain


Fig. E2.6g
F-RD
One translation fundamental domain (stereo view)


Fig. E2.6h
F-RD
One-and-a-half translation fundamental domains



MY MEANDERING ODYSSEY


A more or less chronological account
of some of my physics research
before I became interested
in minimal surfaces

I realize that except in books on the history of science or mathematics, it is not customary to describe the development of mathematical or scientific results in strictly chronological fashion, and only an unusually dedicated reader will have the stamina required to reach the end of this story. So much of what I did depended on chance events that it may sometimes seem to the reader like a sort of random walk. It was more than a decade after my journey began that I first had an inkling of where it might lead. For those readers who make it all the way to the end, I can only say, "Mazel Tov!"


1953-1957


Fig. E2.10
The author in 1954 sectioning a single crystal of silver at the jeweler's lathe
used for radioactive tracer studies of atomic diffusion in metals and alloys.
This apparatus — and most of the other equipment we used —was designed
and/or assembled by Prof. David Lazarus, assisted by Carl T. Tomizuka, my
distinguished predecessor at the University of Illinois in Urbana/Champaign.

At the University of Illinois in the 1950s, research in condensed matter physics was heavily weighted toward the study of point defects in metals, semiconductors, and alkali halides. For my PhD research in David Lazarus's group, I made radioactive tracer measurements of atomic diffusion coefficients as a function of temperature and alloy composition in single crystals of Ag-Cd and Ag-In, using experimental techniques developed by Dave, his post-docs, and the students (principally Carl Tomizuka) who preceded me. Although I enjoyed this work at first, my progress was slow, and as I looked in awe at the accomplishments of some of my classmates, I gradually began to question whether I was temperamentally suited for a career as an experimental physicist. Besides, it seemed to me that my thesis topic did not have much scientific significance, and I saw little prospect of making any fundamental advance in the field of diffusion.

I was fascinated, however, by the mathematics of random walks on lattices — a famous example of Brownian motion. This fascination eventually led me to my first significant discovery in physics — that by measuring the isotope effect for self-diffusion in an elemental crystal, one could distinguish between 'substitutional' diffusion and 'interstitial' diffusion. This had not previously been possible. How this came about is described below.

One day in the spring of 1957, I read a 1952 paper by John Bardeen and Conyers Hering entitled, "Diffusion in Alloys and the Kirkendall Effect". Appendix A of that paper is an analysis by Hering of the difference between the diffusion coefficient of a vacancy (vacant atomic site) and that of an atom. It had long been widely accepted that the mechanism for self-diffusion in noble metals, for example, is the exchange of an atom with an isolated vacancy. The concentration of vacancies was known to relatively dilute, even at the elevated temperatures required for observing self-diffusion. As a consequence, after an atom has exchanged positions with a particular vacancy, it is somewhat more than randomly likely that the next jump of that atom will be an exchange with the same vacancy. When that happens, the two consecutive jumps of the atom will have either partially or wholly canceled each other, and the atom is described by Hering as undergoing a correlated random walk. An atom in an interstitial position (e.g. a lithium atom in a germanium crystal), on the other hand, is believed to hop from one interstitial site to another with no correlation between the directions of consecutive jumps. Its diffusion is characterized as a strictly random walk.

Hering proved that if atoms diffuse by the vacancy mechanism ('substitutional' diffusion) and the vacancies are relatively dilute, then the diffusion coefficient for an atom is smaller than the diffusion coefficient for a vacancy by a fractional amount that depends on the coordination number (number of nearest neighbors of an atom) of the host crystal. The smaller the coordination number, the larger this fraction. For the uncorrelated random walk of a vacancy, which jumps a distance a with frequency Γ, the diffusion coefficient for the vacancy is given by

Dvacancy = a2 Γ / 6.    (E2.11)

Hering showed that for the correlated random walk of an atom in a homogeneous crystal in which the atom-vacancy jump vector has at least two-fold rotational symmetry, the diffusion coefficient for the atom is given by

Datom = f  a2 Γ / 6,    (E2.12)

where the correlation factor f  is given by

f = (1 + < cos θ >Av ) / (1 − < cos θ >Av );    (E2.13)

< cos θ>Av is the average value of the cosine of the angle between two consecutive jumps of an atom.

Since the diffusion of an atom in an interstitial position does not involve an exchange with a vacancy, the directions of its consecutive jumps are uncorrelated.

I had a hunch that in crystals of cubic symmetry, Bardeen-Hering correlation would reduce both the self-diffusion coefficient and the isotope effect for self-diffusion by the same fractional amount. This turned out to be the case. Let Dα and Dβ be the self-diffusion coefficients of isotopes α and β of mass mα and mβ and correlation factors fα and fβ, respectively. I defined the

isotope effect = ((Dβ / Dα) − 1) / ((mα / mβ)1/2 − 1).    (E2.14)

In the absence of correlation effects, the isotope effect would be equal to unity. I conjectured that correlation effects would cause it to be equal instead to fβ. If this conjecture were correct, one could distinguish between interstitial self-diffusion and self-diffusion by the vacancy mechanism simply by measuring the isotope effect for self-diffusion. I first estimated the influence of Bardeen-Hering correlation on the isotope effect by using an approximate model of correlation published by Alan LeClaire and Alan Lidiard in Phil. Mag., 1, 518 (1956). This rough estimate appeared to confirm my hunch that in crystals with the required symmetry, the isotope effect is equal to the correlation factor.

1958-1964

In 1958, in order to refine my calculation of the influence of correlation on the isotope effect, I designed a Fortran program for extending it to a higher order of approximation. In this program, I modeled the infinite crystal by a sequence of four cubically symmetrical sub-crystals of successively larger volumes, each centered at the initial site of the diffusing atom.



Fig. E2.14
The four cubically symmetrical sub-crystals in my
Fortran program for the random walk of a vacancy

The top row lists the number of atomic sites in each sub-crystal.
The vacancy is located at the center of each sub-crystal.
The four sub-crystals contain
nearest neighbors,
2nd nearest neighbors,
3rd nearest neighbors,
and
4th nearest neighbors
of the vacancy, respectively.

A vacancy, starting from a site adjacent to the diffusing atom, was allowed to execute an infinite random walk, during which it had a finite probability of escaping through the boundary of the sub-crystal. Program runs on an IBM 704 computer for each of the five successively larger subcrystals confirmed that correlation does indeed reduce both the isotope effect and the self-diffusion coefficient by exactly the same fractional amount (to within at least eight significant figures), in agreement with the calculation I had made earlier using the Lidiard-LeClaire model. Soon afterward K. Tharmalingam and A. B. Lidiard published an algebraic proof that my results were exact, in an article entitled 'Isotope Effect in Vacancy Diffusion' (Phil. Mag., 4, Issue 44, 1959, pp. 899-906).


Junjiro Kanamori

In 1960, I wrote the the Bardeen-Hering correlation factor as a combinatorial expression. Then I visited the University of Chicago, where the theoretical physicist Junjiro Kanamori made an extremely useful suggestion that enabled Robert W. Lowen, Jr. and me to evaluate the correlation factors for seven structures: four of them 2-dimensional and three of them 3-dimensional. We reduced the correlation factors for the three 3-dimensional cases to triple elliptic integrals and published our results in the Bulletin of the American Physical Society, April 1960, 4, No. 5, p. 280 (cf. Figs. E2.15a,b). I don't recall whether we ever completed our calculations for the face-centered cubic structure, for which Hering had obtained a value of 0.78.


Fig. E2.15a
Computed values of Bardeen-Hering correlation factors (APS abstract)

  STRUCTURE    Z              − < cos θ >Av          f

 linear chain    2   1          0
 honeycomb layer    3   1/2     .333333
 square layer    4   1 − 2/π     .466942
 triangle layer    6   5/6 − √ 3/π     .566057
 diamond    4   1/3     .500000
 simple cubic    6   .209841     .653120
 body-centered cubic    8   1 − Γ4(1/4)/8π3 − 8π/Γ4(1/4)     .727194
Fig. E2.15b
Correlation factors for seven crystal structures


After 1959, I tried — with limited success — to invent a systematic duality rule ('partitioning algorithm') for associating infinite periodic graphs in pairs that represent the sets of all possible geometrical pathways for diffusing atoms in (a) 'vacancy-exchange' diffusion and (b) 'interstitial' diffusion. It didn't occur to me to imagine any kind of surface separating such infinite graphs until 1964, when I first learned about the Coxeter-Petrie [infinite] regular skew polyhedra, (Only in 1966 did I realize that these three infinite polyhedra were nothing more than 'flattened and folded' incarnations of Schwarz's P and D surfaces. When I met Donald Coxeter for the first time at a 1966 geometry conference in Santa Barbara, he told me that he had never heard of the Schwarz surfaces. His face expressed something close to shock when I showed him my plastic models of P and D. He examined them closely for a couple of minutes or more, before saying a word.)

I catalogued a variety of examples of crystal structures that could be neatly partitioned into two disjoint substructures, and I computed the shapes of the Voronoi cells for many examples of unary, binary, and ternary crystal structures. From time to time I made wooden models of many of these polyhedra and used some of them as nodes for ball-and-stick network models of crystal structures. An especially useful resource for me in those days was 'Third Dimension in Chemistry', by Alexander F. ('Jumbo') Wells. It was there that I learned (in 1958) of the existence of the Laves graph (cf. John Tanaka's oral history interview of Wells).

A graph is called symmetric if all of its vertices are symmetrically equivalent and all its edges are symmetrically equivalent. Another way of saying this is: A symmetric graph is one that is both edge-transitive and vertex-transitive. A regular graph, on the other hand, is one in which every vertex has the same degree. Hence every symmetric graph is regular, but not every regular graph is symmetric.

Of the three symmetric graphs on cubic lattices known to me, then or now, the Laves graph is the one of smallest degree — three. (The other two such graphs are the simple cubic graph and the diamond graph.) I imagined in fantasy an elemental crystal whose atomic sites correspond to the vertices of a single Laves graph, with self-diffusion occurring by means of atom-vacancy exchanges. As an additional part of the fantasy, I imagined measuring the isotope effect for self-diffusion in this hypothetical crystal. I believe that the Bardeen-Hering correlation factor (cf. Eq. E1.3) has never been computed for the Laves graph, but it is likely to have a value of less than one-half, since the coordination number of the Laves graph is smaller than that of the diamond graph. (The data in Fig. E2.15b suggest the possibility that it is equal to 1/3. I may get around to calculating it some day, just for fun!) Consequently the isotope effect, which is predicted to be equal to the correlation factor, would also be less than one-half. But I realized that the fantasy was far-fetched, because the interstices in this hypothetical crystal would be so large that self-diffusion would not necessarily occur by a simple vacancy exchange mechanism.

On the other hand, suppose there exists a strongly ordered binary intermetallic compound in which the atoms of the two elements sit on the respective sites of two dual Laves graphs. Such a structure would be analogous to Zn-S (zinc sulfide), or In-Sb (indium antimonide), but with coordination number (degree) of each subgraph equal to three, not four. In a hard sphere model of such a structure, the interstitial cavities would be of modest size if the sphere radii for the two species were not grossly different. Radioactive tracer measurements for each species of the isotope effect for self-diffusion would provide evidence for or against the hypothesis that self-diffusion occurs by the vacancy mechanism.

Below is a stereo pair of recent photos of an ancient model of the two inter-penetrating Laves graphs.


Two dual Laves graphs

Every node (wooden triangle) in each graph is joined by a
wooden dowel not only to its three nearest neighbors on the
graph, but also to its two nearest neighbors on the dual graph.


UPDATE:


Fig. E2.16
Toshikazu Sunada


In a highly original article in the Notices of the American Mathematical Society in 2007, Crystals that Nature Might Miss Creating, the mathematican Toshikazu_Sunada — who was unaware of the history of the Laves graph — independently predicted its existence, making use of results of his research on random walk on crystal lattices. In his remarkable analysis, Sunada's 'K4 crystal' (i.e., the Laves graph) emerges as the unique mathematical twin of the diamond crystal. He proves that diamond and K4 are the only three-dimensional crystals with the property he calls strong isotropy, and also that the honeycomb (cf. graphene) is the only two-dimensional crystal with this property. (I confess that I understand only the easy parts of Sunada's article!)


1964 − April 1966

In July 1964, after spending a few stimulating months consulting for a new division of Beckman Instruments on the design of apparatus for measurements of the Mössbauer effect, I joined the Physics Research Laboratory of Space Technology Laboratories (STL) in Los Angeles. Within a few months, STL was acquired by TRW and changed its name to TRW Systems. I continued — from time to time — to ponder the question of how to develop a 'partitioning algorithm' for inter-penetrating pairs of triply-periodic graphs.

April 1966 −April 1967

One day in April, 1966, in a hallway of the TRW Physics Research Laboratory, I noticed an engineer who was drawing polygons on a large plastic sphere. When I [politely] asked him what he was doing, he replied with some impatience that he was trying to model a fly's-eye lens by arranging a few hundred hexagons on a sphere but was having some difficulties. (In 'Ernst Haeckel (1843 − 1919) is still a problem', Eclectica (2009), Alan Mackay describes a similar error made — and subsequently corrected — by Ernst Haeckel.)

Trying not to sound patronizing, I suggested to the engineer that he might consider including some pentagons, and I explained why the patterm he was searching for didn't exist. I told him the famous story about Euler and the bridges of Königsberg, and I explained that Euler had derived a simple equation that accounts for every possible combination of polygons that tile the sphere. Because it was obvious that he was somewhat less than pleased by my butting in, I decided not to pursue the matter further. However, I did casually mention the incident to my supervisor.

A few days later, I was invited by the research vice-president of TRW Systems to spend one or two days every month as a kind of informal consultant to a group of company engineers who were designing a manned space station. (I knew very little about structural engineering, but in 1965 — four years before the first lunar landing — I had submitted an invention disclosure to TRW describing a modular building system designed for use on the moon. It employed hollow columnar space-frames, based on the geometry of space-filling tetragonal disphenoids. Each column, which was clad in aluminum and stored in a flat collapsed configuration, was designed to be self-deployed after delivery to the moon. Columns could be filled with lunar sand, so that a shelter constructed from an assembly of columns would provide effective shielding from dangerous radiation.)

In order to catch up on gossip about the current state of the art in modular building systems, I paid a visit to the distinguished architect Konrad Wachsmann, chairman of the architecture department at nearby University of Southern California. Wachsmann in turn referred me to the North Hollywood architect/designer Peter Pearce, who was studying polyhedra, crystal structures, periodic three-dimensional networks, and the design of a modeling kit for both polyhedra and networks. Peter had received a grant from the Graham Foundation to study natural and man-made periodic structures. He showed me many ball-and-stck models of crystal structures he had constructed with the help of his assistant, Bob Brooks. Illustrations of these models appeared twelve years later in Peter's book, 'Structure in Nature is a Strategy for Design', MIT Press (1978). Peter told me that he had been inspired especially by R. Buckminster Fuller, Alexander F. Wells, D'Arcy Thompson, and Charles Eames, his former employer.

Two of Peter's models each contained a specimen of what he called saddle polyhedra and made a profound impression on me. Peter had seen a museum exhibit designed by Charles and Ray Eames in collaboration with the mathematician Ray Redheffer, in which a motor-driven quadrangular wire frame emerged repeatedly from a beaker of soap solution with a physical approximation to a minimal surface spanning its boundary. Peter recognized that by spanning appropriate circuits of edges in triply-periodic graphs with plastic polygons that approximated minimal surfaces, he could fill the interstitial cavities in those graphs with saddle polyhedra.

Peter's concept of saddle polyhedron struck me instantly as the critical ingredient required to complete the duality rule ('partitioning algorithm') I had been mulling over in my struggle to develop a systematic relation between substitutional and interstitial sites in crystal structures. Although I never expected to find a rule applicable to every possible triply-periodic graph, I did hope to find one that would work at least for every symmetric graph — a graph which is both edge-transitive and vertex transitive, i.e., a graph in which all vertices are symmetrically equivalent and all edges are symmetrically equivalent. As explained in pp. 76-85 of Infinite Periodic Minimal Surfaces Without Self-Intersections), however, I discovered that although it is not necessary for the graph to be symmetric, it is apparently necessary to add the stipulation that for symmetric graphs,

  • each vertex of the graph g is joined by an edge to every one of the Z nearest neighbor vertices (the graph g is described as being of maximum degree with respect to the vertices.

  • each vertex lies at the centroid of the positions of the Z nearest neighbor vertices (the graph g is described as being locally centered).

(The two saddle polyhedra shown below in Figs. E2.50-E2.53 and in Figs. E2.55a,b,c demonstrate that these conditions are too restrictive.)

One (cf. Fig. E2.20) of Peter's two saddle polyhedra filled an interstitial cavity of the diamond graph, a symmetric graph of degree four on the vertices of the diamond crystal structure, while the other (cf. Fig. E2.25) filled an interstitial cavity of the body-centered cubic (b.c.c.) graph, a symmetric graph of degree eight on the vertices of the b.c.c. lattice. Each of these saddle polyhedra is called the interstitial polyhedron of the graph g and has the following properties:

  • (a) the interstitial polyhedron and the graph g have the same point group symmetry with respect to the center of the cavity;
  • (b) the number of faces of the interstitial polyhedron is equal to the degree Z (number of edges incident at each vertex) of a second [dual] graph ginterstitial, in which there is a vertex v at each cavity center and Z edges — incident at v — that protrude through the faces of the interstitial polyhedron. Each of these Z edges is incident also at a vertex v of one of the Z adjacent interstitial polyhedra.

Because the diamond graph happens to be self-dual, if every vertex of the graph is enclosed by a replica of the interstitial polyhedron, the assembly of such polyhedra — just like the assembly of interstitial polyhedra that occupy the interstitial cavities — define a packing of R3. In this role, these saddle polyhedra are called nodal polyhedra. The nodal polyhedron of the b.c.c. graph is shown in Fig. E2.27.

For some infinite symmetric graphs — depending on the proximity of vertices in coordination shells beyond the first — the number F of faces of the space-filling Voronoi polyhedron that encloses each vertex is greater than Z. The simplest example of a pair of symmetric graphs that illustrate the duality expressed by properties (a) and (b) is a pair of simple cubic (s.c.) graphs (cf. Fig. E2.58). My goal was to incorporate the concept of saddle polyhedron in a procedure that defines this duality in a systematic way.

Among the many triply-periodic graphs that exhibit both properties (a) and (b) defined above are the seven symmetric graphs listed in the table below. The f.c.c. graph and the FCC6(I) graph are the only examples among these seven for which the dual graphs are not also symmetric. For the f.c.c. and s.c. graphs, both the nodal and interstitial polyhedra happen to be convex.



  GRAPH    Z    F     RELEVANT FIGS.

 Laves    3   17     E2.41, E2.42
 WP    4   12     E2.36
 diamond    4   16     E2.19c
 s.c.    6    6     E2.58 − E2.60
 FCC6(I)    6   12     E2.46 − E2.49
 b.c.c.    8   10     E2.24b, E2.36
 f.c.c.    12   12     E2.56d − E2.56f
Fig. E2.17
Examples of symmetric graphs on cubic lattices
Z = degree of the graph (coordination number).
F = number of faces of the Voronoi polyhedron
associated with each vertex.



   Diamond graph
  dual graph   diamond graph
  nodal polyhedron   expanded regular tetrahedron ERT
  interstitial polyhedron   expanded regular tetrahedron ERT
Fig. E2.18


Fig. E2.19a                                 Fig.2.19b
Schwarz's D surface
(images courtesy of Ken Brakke)
The diamond graph is the skeletal graph of both labyrinths.



Fig. E2.19c
A portion of the diamond graph (Z=4)
The edges of the expanded regular tetrahedron ERT (cf. Fig. E2.20),
interstitial polyhedron of the diamond graph,
are shown in blue.


Fig. E2.20
The expanded regular tetrahedron ERT,
interstitial polyhedron of the diamond graph
It is also the nodal polyhedron of the diamond graph.
(The diamond graph is self-dual.)

ERT is the saddle polyhedron Peter Pearce constructed in
an interstitial cavity of the diamond graph (cf. Fig. E2.19c).
Each face is a regular skew hexagon
with face angle θ = cos-1(− 1/3) =~109.47°.



Fig. E2.21
The regular tetrahedron and the edges (black lines) of ERT



Fig. E2.22
The 16-face Voronoi cell for the vertices of the diamond graph
For a sharper [pdf] version of this image, look here.


  b.c.c. graph
  dual graph   WP graph
  nodal polyhedron   expanded regular octahedron ERO
  interstitial polyhedron   tetragonal tetrahedron TT
Fig. E2.23


Fig.2.24a
A cubic unit cell of I-WP
(image courtesy of Ken Brakke)
The skeletal graphs are
the b.c.c. graph − aka the I graph −and the WP graph.


Fig. E2.24b
The b.c.c. graph (Z=8)
The edges of the tetragonal tetrahedron TT (cf. Fig. E2.25)
are shown in blue.


Fig. E2.24c
The b.c.c. graph (green vertices) and its dual,
the WP graph (orange vertices)


Fig. E2.25
The tetragonal tetrahedron TT,
interstitial polyhedron of the b.c.c. graph

TT is the saddle polyhedron Peter Pearce constructed in
an interstitial cavity of the b.c.c. graph (cf. Fig. E2.24).
Each face is a regular skew quadrangle
with face angle θ = cos-1(1/3) =~70.13°.



Fig. E2.26
The eight edges (purple) of TT
For a sharper [pdf] version of this image, look here.



Fig. E2.27
The expanded regular octahedron ERO,
nodal polyhedron of the b.c.c. graph


Fig. E2.28
It appears that ERO is identical to the
asymptotic limit surface suggested by this image
from Ken Brakke's Surface Evolver sequence of surfaces
of successively higher genus in the Neovius C(P) family. (I
first observed this curious result in 1975 and described it in a
letter to the physicist Tullio Regge in response to some questions
from him about minimal surface soap film experiments. I had met
Tullio a few weeks earlier at a Providence conference hosted by
Tom Banchoff. Tullio was a formidable expert in differential
geometry. He had studied the work of the pioneering
19th century Italian masters of the subject (Bianchi
et al) when he was a young student.)



Fig. E2.29
The twenty-four edges (black) of ERO
For a sharper [pdf] version of this image, look here.





Fig. E2.30
The 14-face Voronoi cell for the vertices of the b.c.c graph
For a sharper [pdf] version of this image, look here.




Fig. E2.31               Fig. E2.32


Fig. E2.33               Fig. E2.34

The surfaces in Figs. E2.31 and E2.33 are portions of F—RD and I—WP, respectively.
Note that each of the surfaces in Figs. E2.32 and E2.34 can be regarded, approximately, as
the image under rotation by 45 degrees about a vertical axis of the surface at its left. I obtained
experimental confirmation of the existence of the surface in Fig. E2.32 in 1975, using a laser
to measure the surface normal orientation for a set of hypothetical adjoint soap films
near their boundaries. This method can be described as an extremely tedious (and
far from precise) way to kill periods. In 2001, Ken Brakke accomplished the
same task with enormously greater precision using his Surface Evolver
and also confirmed the existence of the surface in Fig. E2.34.




  WP graph
  dual graph   b.c.c. graph − aka the I graph
  nodal polyhedron   tetragonal tetrahedron TT
  interstitial polyhedron   expanded regular octahedron ERO
Fig. E2.35


Fig. E2.36
The WP graph (Z=4)
The edges of ERO (cf. Figs. E2.27, E2.37) ,
interstitial polyhedron of this graph, are shown in blue.



Fig. E2.37
The expanded regular octahedron ERO,
interstitial polyhedron of the WP graph

In 1966, while assembling the faces of this model,
I discovered that if adjacent hexagons are related
by rotation instead of reflection,
the result is an infinite smooth surface — Schwarz's D surface.
(That was my introduction to triply-periodic minimal surfaces.
A few minutes later, I replaced the 90° hexagons by 60° hexagons
and obtained Schwarz's P surface.)



Fig. E2.38
The tetragonal tetrahedron TT,
nodal polyhedron of the WP graph (cf. Fig. E2.36)



Fig. E2.39
The expanded octahedron EO,
the Voronoi cell for the vertices of WP
For a sharper [pdf] version of this image, look here.


  Laves graph
  dual graph   enantiomorphic Laves graph
  nodal polyhedron   trigonal trihedron T
  interstitial polyhedron   trigonal trihedron T' (enantiomorph of T)
Fig. E2.40


Fig. E2.41
A portion of the Laves graph (Z=3)
The edges of the trigonal trihedron TT ,
interstitial polyhedron of the Laves graph,
are shown in blue
(cf. model in Figs. E2.43 and E2.44).


Fig. E2.42
A view of the edges of TT
from a direction different from that in Fig. E2.41



Fig. E2.43
The trigonal trihedron TT,
nodal polyhedron for the Laves graph (cf. Figs. E2.41 and E2.42)
is a skew decagon with 120° face angles.



Fig. E2.44
Another view of TT



      FCC6(I) graph
  dual graph   a non-symmetric graph of degree 10
  nodal polyhedron   pinwheel polyhedron PP
  interstitial polyhedron   doubly expanded tetrahedron DET
Fig. E2.45


Fig. E2.46
Four vertices' worth of the FCC6(I) graph (Z=6),
a locally-centered deficient symmetric graph (LCDSG)

The vertices of the FCC6(I) graph are those of the f.c.c. lattice.
At each vertex, six of the twelve edges
of the standard f.c.c. graph are omitted.

FCC6(I) is described on pp. 47-48 of
Infinite Periodic Minimal Surfaces Without Self-Intersections.



Fig. E2.47
Four vertices' worth of the FCC6(I) graph
inscribed on the surface of the polyhedron VPdiamond
(Voronoi polyhedron for the diamond crystal structure)



Fig. E2.48
Five VPdiamonds' worth of the FCC6(I) graph



Fig. E2.49
Five VPdiamonds' worth of the FCC6(I) graph



Fig. E2.50
Doubly expanded tetrahedron ('DET'),
the interstitial polyhedron of the FCC6(I) graph
Six faces are regular skew quadrangles,
and four faces are regular skew hexagons.



Fig. E2.51
Two views of DET
(stereo)


Fig. E2.52
The Doubly Expanded Tetrahedron is so named because
its twenty-four edges are produced by reflecting each edge of every face
of a regular tetrahedron in each of the two other edges of that face.
(Every edge of the tetrahedron is reflected four times,
since it is incident at two faces.)



Fig. E2.53
When an infinite set of DETs is assembled by gluing hexagonal faces together in pairs,
the quadrangular faces remain exposed and define Schwarz's D surface.



Fig. E2.54
When an infinite set of DETs is assembled by gluing quadrangular faces together in pairs,
the hexagonal faces remain exposed and define Schwarz's P surface.





Fig. E2.55a
Pinwheel polyhedron PP,
the nodal polyhedron of the FCC6(I) graph


Fig. E2.55b
Pinwheel polyhedron PP,
rendered in Mathematica



Fig. E2.55c
The red edges are the edges of
the nodal polyhedron PP of the FCC6(I) graph.
It has the same volume as the
Voronoi polyhedron (rhombic dodecahedron).
















Fig. E2.55d
Partial packings of PP, rendered in Mathematica
(The quadrangular surface patch module is rendered here as a doubly-ruled surface.
Although it resembles the actual minimal surface, it is only an approximation.)


Below are four views of a portion of the compound [self-intersecting] surface
composed of replicas of just one of the two enantiomorphous quadrangular surface
patches that make up the pinwheel polyhedron PP (above). Three patches are incident at
edges of type [111], but along edges of type [100], two patches are smoothly related by a half turn.

I learned of the concept of compound surfaces of this type from the mathematician
Dennis Johnson, who explained that intersections of three patches along directions
of type [111] are analogous to the 120º intersections in froths of soap bubbles.



Fig. E2.55e
Viewed in [100] direction
Click here for higher resolution image



Fig. E2.55f
Viewed in [110] direction
Click here for higher resolution image



Fig. E2.55g
Viewed in [111] direction
Click here for higher resolution image



Fig. E2.55h
Orthogonal projection on [111] plane
Click here for higher resolution image



      FCC graph
  dual graph   fluorite graph
  nodal polyhedron   rhombic dodecahedron
  interstitial polyhedra   regular tetrahedron, octahedron
Fig. E2.56a



Fig. E2.56b                                              Fig. E2.56c
Two views of F-RD
Image at left courtesy of Ken Brakke


Fig. E2.56d
The nodal polyhedron of the FCC graph
is the Voronoi polyhedron (rhombic dodecahedron).


Fig. E2.57
The fluorite graph is the dual of the FCC graph.
Its nodal polyhedra are
the regular tetrahedron and
the regular octahedron.


Fig. E2.58
The FCC graph and the fluorite graph


      s.c. graph
  dual graph   s.c. graph
  nodal polyhedron   cube
  interstitial polyhedron   cube
Fig. E2.59


Fig. E2.60
The s.c. graph,
skeletal graph of one labyrinth of Schwarz's P surface



Fig. E2.61
The s.c. graph, which is also the
skeletal graph of the other labyrinth of Schwarz's P surface



Fig. E2.62
The congruent skeletal graphs of the two disjoint labyrinths of Schwarz's P surface
(stereo pair)

A few days after I met Peter Pearce, I observed with astonishment that for certain shapes of saddle polygons spanned by a minimal surface, e.g., the 90° regular skew hexagon (a module for Schwarz's D surface) or the 60° regular skew hexagon (a module for Schwarz's P surface), if two specimens of the saddle polygon are related by a half-turn about a common edge, instead of by mirror reflection in a plane containing that edge (which is the arrangement in most, although not all, of the saddle polyhedra I had explored by then), not only does the junction between the two polygons appear to be perfectly smooth, but an endless sequence of these half-turns produces a single smooth, embedded infinite labyrinthine surface with the global topology and symmetry of a Coxeter-Petrie regular skew polyhedron!

I had accidentally stumbled onto two examples of the application of Schwarz's reflection principle, his P and D surfaces − two objects that I had never heard of. I was unable to locate a reference to either of these surfaces in my books on geometry or differential geometry. Because I realized that some contemporary mathematicians must be familiar with these two surfaces, I paid a visit to the UCLA math department, where I showed my plastic models to the two faculty specialists in differential geometry. But neither of them recognized the two surfaces!

Next: a visit to the UCLA science library, where I learned that the late Johannes C. C. Nitsche, a prolific mathematician on the faculty of the University of Minnesota, was a noted authority on minimal surfaces (cf. his summary of the field, "A Course in Minimal Surfaces"). When I telephoned him and described what I had been doing, he kindly explained that I had probably made models of the two TPMS for which H. A. Schwarz (and also Riemann and Weierstrass, as I was to learn later) had developed solutions in 1866. He referred me to Vol. 1 of Schwarz's Collected Works. From a quick perusal of this tome, I learned that Schwarz had also discovered two other examples of TPMS — H and CLP, both also of genus 3 — and I made plastic models of them too.

I soon noticed that on p. 271 of Hilbert and Cohn-Vossen's 'Geometry and the Imagination'; the authors write that

In this way, Neovius4 succeeded in constructing a minimal surface that extends over the entire space without singularity or self-intersection and has the same symmetry as the diamond lattice (italics added).
4E. R. Neovus, Bestimmung zweier speziellen periodischen Minimälflachen, Akad. Abhandlung, Helsingfors, 1883


I foolishly assumed that the authors had simply become confused here and were actually referring to Schwarz's D surface. Eighteen months later, I learned that Neovius had treated an entirely different surface of genus 9.

I invented a naive scheme for identifying and labeling these surfaces, each of which I regarded as lying between the two triply-periodic graphs of a dual pair. I named these pairs of graphs 'skeletal graphs', because I thought of them as the skeletons of their respective hollow labyrinths. I found it helpful to regard the skeletal graph edges as thin hollow tubes that could be enlarged by inflating them until the whole graph was transformed into the TPMS. Then if the tubes were overinflated, the graph would eventually shrink down into the dual graph! I imagined that for at least a portion of this inflation cycle, the surface of the graph would define a triply-periodic surface of non-zero constant mean curvature.

As I began my admittedly superficial study of the mathematical underpinnings of these surfaces, beginning with the two Schwarz reflection principles, I couldn't help wondering what other examples of embedded 'TPMS' might exist. In particular, I wondered whether there was a TPMS whose skeletal graphs were the enantiomorphous pair of Laves graphs I had learned about seven years earlier in 'Third Dimension in Chemistry', by Alexander F. Wells. A pair of enantiomorphic Laves graphs that are related by inversion has b.c.c. translation symmetry. It struck me as curious that of the three different cubic lattice symmetries — simple cubic (s.c.), face-centered cubic (f.c.c.), and body-centered cubic (b.c.c.) — b.c.c. was missing from the inventory of cubic lattice symmetries for known examples of TPMS of ultimately simple topology (genus three).

A second reason for my focus on the Laves graph was that it was apparently the only other example of a triply-periodic graph — besides the simple cubic and diamond graphs, which are the skeletal graphs of P and D, respectively — in which congruent regular polygons are incident at each edge. In the Laves graph, there are precisely two regular polygons incident at each edge. They happen to be infinite helical polygons, centered on lines parallel to two of the three coordinate axes. (I had been strongly influenced by Coxeter's 'Regular Polytopes', and I believed one should take regularity very seriously!) The Laves graph is not a reflexive regular polyhedron, however, and its lack of reflection symmetries made it impossible for me to imagine just how it could serve as the skeletal graph of the labyrinth of an embedded TPMS.

But the most compelling reason for my conviction that there must exist an embedded TPMS whose skeletal graphs are enantiomorphic Laves graphs was that the simple cubic graph, the diamond graph, and the Laves graph were the only examples I could identify of symmetric triply-periodic graphs of cubic symmetry that are self-dual. Even though I knew of no theoretical justification for claiming that such graphs — regarded as skeletal graphs of embedded TPMS — play a unique role in defining embedded TPMS of cubic symmetry, I nevertheless believed that they must play such a role! I was aware of the fact that the concept of skeletal graph was itself somewhat ill-defined. It seemed to me to be a very 'natural' construct when applied to the then known examples of embedded TPMS, but I had no idea how to prove that for every possible example of an embedded TPMS there is a unique pair of skeletal graphs.

All of these considerations at times seemed to me to smack more of theology than of mathematics. (I am reminded that the young Riemann, who was probably the first to solve the equations for what we now call Schwarz's P and D surfaces, as a young man abandoned the study of theology (his pastor father's choice) for a career in mathematics!

I resolved to learn more about TPMS, which I recognized as far more interesting objects than saddle polyhedra, but in the meantime I was determined to continue exploring the relation between triply-periodic graphs and saddle polyhedra. On evenings and weekends throughout the spring and summer of 1966, I used a toy vacuum-forming machine and home-made moulds cast from polyester resin poured against a thin stretched rubber membrane to make dozens of saddle polyhedra of different shapes, all of which I shared with Peter Pearce. He preferred to make his saddle polygons by draw-forming— pushing a tool in the shape of a skew polygon outline against a transparent vinyl sheet that had been softened by heating. I preferred vacuum-forming with solid moulds, but it was clear that Peter's method also worked well. It has the advantage of not requiring the extra labor involved in making a mould, but the disadvantage is that it cannot replicate the shape of a minimal surface as well as a carefully crafted mould can.

During these months of experimenting, I found no counterexample to my improvised duality rule, even for triply-periodic graphs that are not symmetric. In May, I hit on the idea of what I rather lamely called a 'defective' symmetric graph (I decided later that 'deficient' might be a more appropriate name) — a symmetric graph A derived from a second symmetric graph B by omitting some of the edges but none of the vertices of B. In A, not every pair of nearest neighbor vertices is joined by an edge. I required that every deficient symmetric graph be locally-centered, i.e., that every vertex lie at the centroid of the vertices with which it shares an edge.

For the simple cubic lattice, it's easy to prove — simply by enumerating each of the possible locally-centered subsets of edges that contains at least three edges — that it is impossible to construct a locally-centered deficient symmetric graph (LCDSG) on the vertices. I have no idea why I failed to ask myself in those days whether there exists a LCDSG on the vertices of the body-centered cubic lattice. The only example of a LCDSG that I examined in 1966 was a graph of degree six I call FCC6(I) (cf. Fgs. E2.45 - E2.55b). Its vertices are those of the face-centered cubic lattice. I used the name FCC for the familiar symmetric graph of degree twelve on the vertices of the f.c.c. lattice, in which every pair of nearest neighbor vertices is joined by an edge. I derived the deficient graph FCC6(I) by removing a symmetrical set of six out-of-plane edges from each vertex, leaving behind a flat six-edge cluster that occurs in each of the four possible [111] orientations. Although FCC6(I) proved not to be a counterexample to the duality rule, I was not confident that the rule would always hold even if I were to restrict it to symmetric graphs only.

As it happened, not only did the rule not fail in the case of FCC6(I) — it yielded an unexpectedly interesting pair of saddle polyhedra. I call the interstitial polyhedron — shown in Fig. E2.51 — the Doubly Expanded Tetrahedron ('DET'), and the corresponding nodal polyhedron — shown in Fig. E2.55a — the Pinwheel Polyhedron. The Doubly Expanded Tetrahedron is the first example I had encountered of a space-filling saddle polyhedron in which the faces are of two kinds — hexagons and quadrangles. As illustrated in Figs. E2.53 and E2.54, Schwarz's P and D surfaces can be formed from either the quadrangular or the hexagonal faces of an infinite 'porous packing' of DETs, according to whether neighboring DETs share quadrangular faces (P) or hexagonal faces (D).



April 1967 − July 1970

In the spring of 1967, the physicist Lester C. Van Atta, who was Associate Director of the NASA Electronics Research Center ('ERC') in Cambridge, Massachusetts, came to Los Angeles for a few days to visit his physicist son Bill, a friend and colleague of mine who happened to be a whiz at solving combinatorial puzzles. A year earlier, stimulated by a Martin Gardner column in Scientific American that described Piet Hein's SOMA puzzle, I had become hooked on investigating the possible symmetries of complementary half-cube packings by the eight solid tetrominoes (cf. Fig. E1.13). After Bill lent his father a set of these puzzle pieces, his father told Bill that he 'wanted to meet the guy who had cost [him] a night's sleep'.

At the end of a very long late evening visit to my home, Lester abruptly invited me to join the NASA/ERC research staff in Cambridge, Massachusetts. He explained that I would be required only to 'follow my nose'. I found it difficult to believe that he was making me a serious job offer, and I didn't make any response. A few days later, Van Atta phoned me from Cambridge and told me in no uncertain terms that I had only three or four days left to get the paper work (a few dozen pages of federal employment application forms) in the mail, because it would be impossible to keep the position open longer than that. This time I took him seriously, and in July 1967 I moved to Massachusetts.

I wanted to immerse myself immediately in the study of TPMS. However, I believed that I should first make a better organized attack on my embryonic duality rule ('partitioning algorithm'). I knew it was unlikely that the rule would be applicable to every possible triply-periodic graph, but I had no idea how to characterize those graphs for which it worked and those for which it didn't. Inspired by Polya's rules for problem solving, I continued to emphasize symmetric graphs — those graphs for which there is a symmetry group transitive on both vertices and edges. I wondered whether my duality rule worked for every possible symmetric graph. If I could find a counterexample, I wanted it to be as simple as possible.



I had been hired at NASA/ERC as a mathematician (chief of a special section Van Atta created for me, called the 'Office of Geometrical Applications'), even though I was at best an amateur mathematican. Curiously it was my forays into recreational mathematics that had led Van Atta to hire me. I was never told by him or anyone else what I should work on, but of course it was understood that if I saw possible applications of what I was doing that might be of interest to NASA, I should not fail to pursue those applications.

From the start, I undertook to learn more about the mathematics used in the study of minimal surfaces. I pored over three books on differential geometry and Schwarz's Collected Works. I was especially curious to know whether there were additional examples of TPMS just waiting to be discovered, but when I began to read the published literature in the field (some of it in German, of course), I often felt overwhelmed. I believed that I didn't have time to become sufficiently knowledgeable about the deep foundations of the relevant branches of mathematics — differential geometry and complex analysis — to make any 'breakthrough' advances in the field.

I had developed a special interest since 1958 in the properties of infinite periodic graphs, and this interest had eventually led me to papers and books by Donald Coxeter. My interest in these graphs had sprung from my research on atomic diffusion in crystalline solids and from the mathematics of correlated random walks on discrete lattices. Because I had discovered (in 1957) that the magnitude of the isotope effect for atomic diffusion in crystals could distinguish between interstitial diffusion and substitutional diffusion, I was curious to learn which elements and compounds were likely to be good candidates for measuring the isotope effect for diffusion. I tried — with only slight success, initially — to develop an algorithm ('duality rule') for deriving the infinite periodic graph whose vertices correspond to the principal interstitial sites of a crystalline solid.


WARNING: Continue at your own risk.
Most of the following text will be integrated into earlier discussions.

I was able to confirm my hunch that the duality rule was valid for many pairs of triply periodic graphs — one of the pair being called substitutional and the other interstitial. In every case I tested, both nodal and interstitial polyhedra exhibited the following properties:

(a) the number of faces of the polyhedron is equal to the number of edges of the associated periodic graph;
(b) the symmetry of the polyhedron is identical to the symmetry of the associated vertex of the periodic graph.

It didn't matter which graph was called substitutional and which was called interstitial. The dual relation between them is, after all, symmetrical. (Of course from a physical point of view, it would be absurd to call the large interstitial cavities in silicon — which can be occupied by smaller atoms like lithium, for example — substitutional and the silicon atomic sites interstitial!)

I call the saddle polyhedron in Fig. E2.25 the interstitial polyhedron for the b.c.c. graph of degree 8, because it is bounded by edges [of a periodic graph] that are the bars of a sort of interstitial cage. This same polyhedron is the nodal polyhedron for another triply periodic graph, which I named WP. The nodal polyhedron encloses a vertex of the periodic graph at its center, and the number of faces of the nodal polyhedron is equal to the number of edges incident at that vertex.

The saddle polyhedron in Fig. E2.20 is both nodal polyhedron and interstitial polyhedron for the diamond graph. That means that in a space-filling array of these saddle polyhedra, each polyhedron can either (a) enclose at its center a vertex of the diamond graph or (b) occupy a single interstitial region bounded by the edges and vertices of the diamond graph.

For some periodic graphs, either the nodal polyhedron or interstitial polyhedron (or both) may turn out to be a convex polyhedron with plane faces. For the simple cubic graph of degree six, for example, both polyhedra are cubes.

In the simplest cases, the graph is unary, i.e., all the vertices of the graph are equivalent. But the duality rule works smoothly without requiring any ad hoc adjustments even for many non-unary graphs. (I plan eventually to post a picture or two of the polyhedra for such a graph.)

As explained below, in early 1968 I searched for — and eventually found — a periodic graph for which the duality rule failed, and that failure led to the discovery of the pseudo-gyroid, which is composed of hexagons with perfectly helical boundary curves.

Recall that FCC6(I) is a locally-centered deficient graph of degree six on the vertices of the f.c.c.lattice (cf. Fig. E2.48). In February, 1968, I realized that I had never searched for the obvious b.c.c. counterpart to FCC6(I): a locally-centered deficient graph on the vertices of the b.c.c. lattice. (It's easy to prove that no locally-centered deficient graph on the vertices of the s.c. lattice exists.) Once I started looking, it didn't take me long to discover BCC6, a symmetric graph of degree six on the vertices of the b.c.c. lattice. A portion of this graph is shown in Fig. E1.20d. BCC6 proved to be the long-sought counterexample to my 'duality rule'. Its edges are those of the infinite regular warped polyhedron ('IRWP') that I call M4 (cf. Fig. E1.16a and E1.20a to E1.20f).

In the case of BCC6, the breakdown in the duality rule occurs at the very first step — the construction of the interstitial polyhedron. Instead of the finite interstitial polyhedron the duality rule was intended to generate, an infinite one — M4 — appeared. I shed no tears over this failure of the duality rule, because the two-labyrinth character of M4 suggested that something of potentially greater interest might be in the offing: an example of a previously unknown TPMS. I observed that the skeletal graphs of M4 were enantiomorphic Laves graphs. This was potentially exciting, because it suggested that M4 might somehow be transformed into the minimal surface (the gyroid) whose existence I had speculated about almost two years earlier.

Appendix II of Infinite Periodic Minimal Surfaces Without Self-Intersections explains why I constructed M6 (in February 1968). M6 was the result of an attempt to improve on its predecessor, M4 (cf. Figs. E1.16a and E1.16b), which is composed of skew quadrangles spanned by minimal surfaces. I was looking for a way to 'smooth out the wrinkles' in M4. The hexagons in M6 are the duals of the quadrangles in M4. In M4, the dihedral angle between adjacent faces is 60°. In M6, it is only ~44.4°. I hoped that this small reduction would enable M6 to look at least a little more like a continuous minimal surface than M4 did.

As soon as I had constructed the model of M6 shown in Fig. E1.16b, I noticed that its edges define regular helical polygons with straight edges, centered on lines parallel to the rectangular coordinate axes (cf. Figs. E1.19b, c, e). I speculated that if I replaced the straight edges of M6 by helical ones, the ~44.4° dihedral angle between adjacent faces might shrink to nearly zero. This was a wild and woolly guess, with no theoretical justification whatsoever, but the new physical model I constructed (cf. Fig. E1.17) was encouraging. The shapes of these edges in the true gyroid just happen to differ so slightly from circular helices that it is virtually impossible to detect the error by eye.

The glaringly obvious hint that I had missed from the outset lay in the identical combinatorial structure of each of the three regular tessellations of the three surfaces — P, D, and the pseudo-gyroid surface in Fig. E1.17. All three of these surfaces can be constructed of surface patches that correspond to faces of any of the three Coxeter-Petrie infinite regular skew polyhedra, with tangent planes identically oriented at corresponding vertices. Since I was already familiar with the details of how curves transform and how the tangent plane remains invariant at each point in the Bonnet bending of the catenoid into the helicoid, it should have occurred to me (but didn't!) that the almost perfectly circular lines of curvature in the coordinate planes of Schwarz's P surface would be transformed by Bonnet bending into almost perfectly helical lines of curvature before they finally became the linear asymptotics in Schwarz's D surface parallel to the coordinate axes.

For the previous several months, I had begun to feel pressure to do something 'useful'. Even though Dr. Van Atta himself never once hinted that he was less than satisfied about how I chose to spend my time, there were growing signs that I could not afford to ignore indefinitely NASA's expectations that my work suggest at least the possibility of some 'practical' offshoots. Since I had no idea how to obtain an analytic solution for a minimal surface patch bounded by six helical arcs, I decided to give up trying to prove that the pseudo-gyroid is a minimal surface. Instead I sent a physical model of the pseudo-gyroid to Bob Osserman, who passed it along to his PhD student Blaine Lawson at Stanford. Blaine agreed to think about the problem, but he warned me that his dissertation would be keeping him extremely busy.

I delved more deeply into the analysis of the 'continuous transformation on vertices and edges' mentioned in the abstract of Fig. E2.10. I attempted to identify every possible example of non-self-intersecting 'infinite regular warped polyhedra' (and 'infinite quasi-regular warped polyhedra'), whose faces are regular skew polygons. (cf. Figs. E1.35b and E1.35c.). At the same time, I analyzed the geometry of what I called the 'graph collapse' transformation, which is diagrammed for the 2-dimensional square graph in Fig. E2.68c. To the exclusion of almost everything else, I concentrated for several weeks on the engineering requirements for the application of this transformation to the design of expandable space frames. Finally I wrote a patent application with the help of two NASA patent attorneys who for two weeks flew up to Cambridge every morning from Washington. Here is a synopsis of the expandable space-frame patent, which was issued in 1975, and here is the complete text of the NASA patent, from which an illustration is shown in Fig. E2.68a. I estimated that with realistically designed struts, a value of 80:1 was feasible for the ratio of the expanded to collapsed volume of the space frame.

With the help of Charles Strauss, Randy Lundberg, Bob Davis, Ken Paciulan, and Jay Epstein, I made an animated film of the collapse of the Laves graph and of three other symmetric graphs. The portion of the video '1969 'Part 4' that shows examples of the graph collapse transformation begins at 7min00sec after the beginning of the video.

A few single frames from the film that illustrate the geometry of the collapse transformation applied to the Laves graph are on pp. 86-88 of Infinite Periodic Minimal Surfaces Without Self-Intersections. In the fully collapsed state, the vertices and edges of the [infinite] Laves graph are mapped onto the four vertices and six edges, respectively, of a single regular tetrahedron (cf. the tetrahedron AOBC in Fig. E2.68b.2). If one vertex (vertex O in Figs. E2.68b.0, E2.68b.1, and E2.68b.2) is fixed, the collapse trajectories of all the other vertices are ellipses centered on that vertex. For every vertex V, the major radius of the ellipse is equal to the initial distance of V from O. The minor radius of the ellipse is equal to the edge length of the graph if V is related to vertex a, b, or c by a translation that is a symmetry of the Laves graph — i.e., if V is red, green, or blue. The minor radius is equal to zero if V is related to vertex O by a translation that is a symmetry of the Laves graph — i.e., if V is yellow. Collapse onto tetrahedron AOBC occurs twice in each period of the transformation: at the two moments when either one-quarter or three-quarters of each elliptical trajectory has been traversed.

Each of the vertices a, b, c rotates on a circular trajectory in one of the three orthogonal coordinate planes. Because of the screw isometries of the Laves graph, edges collide only at the two instants of collapse in each period. In an actual physical space-frame, however, struts are of finite thickness, and this causes edge collisions to occur well before collapse. (Precisely how early the collisions occur in each period depends on the thickness of the struts.) The analogous transformations for those regular graphs derived from Coxetrie-Petrie maps that contain reflection isometries are not physically realizable, because edges collide early in the transformation even though they are of zero thickness.


Fig. E2.68a
A hinged joint in the expandable space frame

The collapse of the Laves graph is readily depicted by regarding the graph as initially embedded in the D surface (cf. Fig. E2.69b.0) and then allowing every vertex to be translated along a linear trajectory in a direction normal to the surface. Vertices related by a translational symmetry of the graph are colored the same. If the two sides of the D surface are labeled A and B, with motion in the direction from A to B defined as positive and motion in the direction from B to A defined as negative, then the two vertices incident on each edge move along normals of opposite sense. The computed positions of the vertices at each stage of the collapse are scaled by the requirement that edge lengths remain invariant, thereby causing the graph to shrink continuously, with all of its edges finally collapsing onto the six edges of a single regular tetrahedron — the tetrahedron with vertices O, A, B, C in Fig. E2.68b.2.


Fig. E2.68b.0
The Laves graph embedded in the D surface
The arrows indicate the initial directions of the curvilinear displacements of the vertices.
Green diplacements are called positive, and
red diplacements are called negative.


Fig. E2.68b.1
The yellow vertex at O is now fixed.
The arrows here indicate the initial directions of the curvilinear displacements
in a coordinate system in which the yellow vertex O is at the origin.


Fig. E2.68b.2
The circular trajectories of vertices a, b, and c
and the elliptical trajectory of vertex d, in Fig. E2.68b.1
One-fourth of a complete trajectory period is shown here for these four vertices.


Perhaps the easiest way to illustrate the collapse is to depict
the circular trajectories, in orthogonal coordinate planes,
of the three vertices that are nearest neighbors of any vertex V
if V is regarded as fixed..

The next four images illustrate these trajectories for
V = a yellow, red, green, or blue vertex.


Fig. E2.68b.3
Rotation of R, G, and B vertices around Y vertex
stereo view


Fig. E2.68b.4
Rotation of G, B, and Y vertices around R vertex
stereo view


Fig. E2.68b.5
Rotation of B, Y, and R vertices around G vertex
stereo view


Fig. E2.68b.6
Rotation of Y, R, and G vertices around B vertex
stereo view


The stereo images in Fig. E2.68c illustrate the application of the graph collapse transformation to the square graph. In this 2-dimensional example, all the edges of the graph coalesce into a single vertical edge. The images below are parametrized by the value of θ, the angle of rotation of each edge of the graph out of the horizontal plane.
The vertex at A is regarded as fixed. Collapse onto a single vertical edge occurs at θ=90° and θ=270°.


Fig. E2.68c.0
Square grid graph before the start
of the collapse transformation
θ=0°


Fig. E2.68c.1
θ=18°


Fig. E2.68c.2
θ=52.2°


Fig. E2.68c.3
θ=81°


Fig. E2.68c.4
θ=90°


Fig. E2.68c.5
θ=279°


Fig. E2.68c.6
θ=307.8°


Fig. E2.68c.7
θ=345°


Fig. E2.68c.8
θ=360°


Fig. E2.68c.9


Some additional history

Inserting handles into a TPMS
(See also Figs. E2.83 to E2.85.)

It occurred to me one day in the spring of 1969 that for some examples of moderately low-genus embedded TPMS S1 and S2 for which the respective adjoint surfaces S1† and S2† have elementary patches that are bounded by simply-connected straight-edged polygons P1 and P2, there must exist an embedded hybrid TPMS Sh whose adjoint surface Sh† has an elementary patch with boundary polygon equal to a linear combination of P1 and P2. (I defined a linear combination of two polygons P1 and P2 to be a polygon interpolated between P1 and P2. The idea of constructing a linear combination of two polygons occurred to me after I recalled a description of linear combinations of convex polyhedra that I had read in a Russian book on polyhedra. I no longer recall the name of the book's author, but it may have been Aleksandr Aleksandroff. I have been unable to trace the book.)

Fig. E2.70 shows a 1969 sketch illustrating my scheme for constructing the hybrid of P and C(P).

Fig. E.2.70
1969 proposal for a genus-14 embedded hybrid of P (genus 3) and C(P) (genus 9)
The two scribbled captions 'Schwarz's "D" ' are erroneous.
They should read 'Schwarz's "P" '.
The arrows indicate the directions
of the local surface normals.

Consecutive edges of the quadrangle P1 at the upper right are
12, 34, 45, 51.
Consecutive edges of the quadrangle P2 at the lower right are
12, 23, 34, 41.
The transformation of P1 into P2 can be described as follows:

  1. Edge 12 remains fixed in place.
2. Edges 34 and 45 are translated
   along a linear trajectory in the
[101] direction.
        3. New edge 23 grows at a steady rate.
                              4. Old edge 51 is reduced to nothing at a steady rate.

The relative weights assigned to the adjoint polygons P1 and P2 require
trial-and-error adjustment to make arcs 23 and 15 coplanar.
The cognoscenti call this process 'killing periods'.


After I sketched the notes shown in Fig. E.2.70, I phoned Blaine Lawson, who was already an expert on minimal surfaces. He was then approaching the last stages of his PhD dissertation research at Stanford under Bob Osserman. I asked him if he thought it was plausible that a hybrid derived from these two straight-edged polygons would define an embedded surface. Blaine replied that it was not an unreasonable idea, because the intermediate value theorem guarantees successful 'period killing' — or words to that effect. I'm sure 'period killing' wasn't the exact expression he used. I recall first hearing those words several years later, in a telephone conversation with David Hoffman who — in collaboration with Bill Meeks— derived the spectacular family of Costa-Hoffman-Meeks surfaces. The original surface from that family is shown below, in a stereo image due to Hermann Karcher.


The original Costa-Hoffman-Meeks surface

I mailed (faxed?) Blaine a copy of my Fig. E.2.70 sketch, but then I abandoned the P—C(P) hybrid, because I realized that vacuum-forming a surface patch with a severe undercut — like the one shown at left center in Fig. E.2.70 — would be difficult or impossible. Determined to build a physical model of some hybrid surface, I turned instead to a topologically simpler case — the hybrid of P and I-WP that I call O,C-TO (cf. Figs. E2.79 to E2.81). Its genus is only 10. Then for the next forty-two years, I forgot all about the P—C(P) hybrid!

This process of hybridization is equivalent to attaching a handle to a minimal surface. During the next six years, I applied this method to several examples of TPMS, using an extremely laborious procedure based on the use of a laser to measure the orientation of the tangent plane at points near the boundary of a long-lasting [polyoxyethylene] soap film spanned by a straight-edged polygon P, which is a candidate for the boundary of an elementary patch of the adjoint of the surface with added handles. By 1991, Ken Brakke's Surface Evolver had rendered this tedious method obsolete.


Hal Robinson's original laser spectrometer (1968)
for measuring the surface orientation of a soap film
(I don't have a photo of the similar laser
spectrometer I constructed in 1975.)

Many other people subsequently discovered handle attachment and applied it to a variety of minimal surfaces, not just periodic ones. (I have been publicly scolded by several mathematicians — most often by J.C.C. Nitsche — for failing to publish my work in refereed journals. Mea culpa.)

In May, 2011, I discovered in my files the long-forgotten sketch shown in Fig. E.2.70 and emailed a copy to Ken Brakke. He quickly confirmed (with his Surface Evolver) that the P—C(P) hybrid is embedded. Ken's pictures of this surface, which he dubbed 'N14', are shown in Figs. E2.71, E2.72, and E2.73 and also at his Triply Periodic Minimal Surfaces web site, where it is called "N14".


Fig. E2.71
Ken Brakke's May 2011 Surface Evolver solution for an elementary patch of
the embedded hybrid of P and C(P)
(cf. Fig. E2.70)
(image courtesy of Ken Brakke)


Fig. E2.72                                                    E2.73
A cubic unit cell of the P—C(P) hybrid (genus 14)
The unit cells in the two images are displaced with respect to
each other by one-half of the body diagonal of an enclosing cube.
(cf. Fig. E2.71)
(images courtesy of Ken Brakke)




Fig. E2.74
1969 sketch showing how to generate O,C-TO
The 'bcc labyrinth surface' referred to at the top
of the page is the surface I later renamed I-WP.



Fig. E2.75
O,C-TO (genus 10), a hybrid of I-WP (genus 4) and P (genus 3)
cubic unit cell
view: [100] direction



Fig. E2.76
O,C-TO
unit cell
view: [111] direction



Fig. E2.77
O,C-TO
1.5 unit cells
oblique view



Fig. E2.78 shows Ken's picture of the Manta surface, which is one of many examples of hypothetical minimal surfaces whose existence I conjectured in 1971. Some of them were inspired by experiments with soap films blown inside a kaledioscopic cell. Others were inspired by considering the structure of various highly symmetrical inorganic crystals. Manta is a balanced surface; the P—C(P) hybrid is non-balanced. If you compare Fig. E2.73 and Fig. E2.74, you will see that the P—C(P) hybrid has a simpler topology than Manta. Manta has [100] tunnels, while the P—C(P) hybrid does not.

A few days after Ken sent me his image of the P—C(P) hybrid, which is shown in Fig. E2.73, he sent me images of two slightly more complicated surfaces he told me he had obtained by 'poking holes' in the P—C(P) hybrid. Images of this new pair — N26 and N38 — can be seen at his website, together with several other hybrids. Of course, purists rightly claim that the existence of all of these hypothetical minimal surfaces is somewhat suspect, since it has not been established by rigorous mathematical proof.

A diagram of the unit cell of the cubic phase of the compound BaTiO3 is shown in Fig. E2.75 for comparison with Manta. One can try to match the sizes and positions of the ions in BaTiO3 to symmetrical cavities in the labyrinths of the surface. At my request, Ken produced the three orthogonal projections of Manta shown in Figs. E2.76, E2.77, and E2.78, together with the following numerical data on the radii of spheres that fit snugly against the surface in three classes of symmetrical cavities:

Radii of tangent spheres in 1x1x1 unit cell of manta genus 19 surface:
       corner sphere radius: 0.18560130
       center sphere radius: 0.18560130
       midedge sphere radius: 0.23553163

(Here's a useful Wikipedia article about ionic radii.)




Fig. E2.74
Manta (genus 19)
(image courtesy of Ken Brakke)



Fig. E2.75
Unit cell of cubic phase of barium titanate (BaTiO3)
Ba2+ red
Ti4+ green
O2− blue



Fig. E2.76
Orthogonal projection of Manta unit cell on [100] plane
(image courtesy of Ken Brakke)



Fig. E2.77
Orthogonal projection of Manta unit cell on [110] plane
(image courtesy of Ken Brakke)



Fig. E2.78
Orthogonal projection of Manta unit cell on [111] plane
(image courtesy of Ken Brakke)



'Notched adjoints': grafting handles onto embedded surfaces


Fig. E2.82a
A note by Ernst Eduard Kummer (H.A. Schwarz's father-in-law)
that is included in Schwarz's Gesammelte Werke
The illustration shows eight triangular Flächenstücke of
Schwarz's diamond surface D inside a tetragonal disphenoid.


Fig. E2.82b
Free translation of the note by Kummer in Fig. E2.82a

One weekend in 1968, while I was reading p. 150 of Schwarz's Collected Works (cf. Fig. E2.82a), it occurred to me that one might be able to model a small portion of a triply-periodic minimal surface, like the portion of Schwarz's D surface shown in Fig. E2.82, by means of a soap film in the interior of the appropriate polyhedral cell. (Schwarz and Plateau had a very active correspondence for many years about soap films and minimal surfaces. I'm surprised that they seem not to have performed experiments of this type.) I quickly constructed a transparent model of the tetragonal disphenoid from four vinyl triangles, stretching cotton threads along the two internal symmetry axes, which are clearly visible in Kummer's Fig. E2.82 sketch. I cut a hole in one of the faces of the cell large enough to provide access to the interior with a soda straw. Using a soap solution containing some glycerine, I discovered that the modeling of the minimal surface is quite easy! One of its elegant features is that by blowing through the straw on one face or the other of the soap film, you can toggle back and forth between two stationary states: a triangular patch of D and a quadrangular patch of C19(D) (cf. Figs. E2.4b, c, d). You stretch the soap film by blowing one corner of it right up to a corner of the cell, and then with a light puff of air through the straw, you push that part of the film just beyond the cell corner. At that point it automatically slides down into its other equilibrium position.



Fig. 2.82c
Soap film model of Schwarz's H surface (1968)
(Photograph copied from an article about Harald Robinson, on p. 55
of the Nov. 1969 issue of Innovation, published by The Innovation
Group of Technology Communication, Inc., Saint Louis, Missouri)

My colleague Hal Robinson constructed this triangular-prism-shaped Coxeter cell.
The nylon thread that is stretched horizontally across the interior of the cell lies on
an axis of 2-fold rotational symmetry of the surface and serves to stabilize the soap
film. Without the thread, the film would be in an unstable stationary state and would
quickly slide away from its equilibrium position and collapse.

Many of the vacuum-formed plastic models of TPMS I constructed after Hal began
to work with me were made from modules whose boundary curves were derived by
tracing the curved edges of soap films like this one.

My model of the tetragonal disphenoid was a flimsy one. When I arrived at NASA on the following Monday morning, I phoned Hal Robinson, the sculptor and model-maker who had recently started to work with me part-time and asked him to make a more physically rugged tetragonal disphenoid. Within a couple of days or so, he produced a lucite model with highly accurate proportions, using monofilament nylon instead of cotton threads for the internal symmetry axes. Next I asked Hal to make me a lucite model of another Coxeter cell relevant for cubic TPMS — the quadrirectangular tetrahedron, which is one-quarter of the tetragonal disphenoid. It contains only one internal axis of two-fold rotational syrmmetry, not two. With this cell, you can toggle back and forth between a patch of P and a patch of the Neovius surface C9(P) (cf. Figs. E2.2a, b), again by blowing on the soap film to stretch it over one corner of the cell. Just as with the tetragonal disphenoid, from there the soap film slides into its other stable stationary state automatically.

Not until the spring of 1970 did it occur to me that perhaps Schwarz's P surface and Neovius's surface C9(P) are merely the topologically simplest members of a countably infinite sequence of embedded surfaces of progressively higher genus:

P, C9(P), C15(P), C21(P), ...

I obtained experimental evidence for the existence of C21(P) by producing the four-sided Flächenstück of C21(P) as a soap film in a stationary — but unstable — state inside the quadrirectangular tetrahedron. This was a more difficult soap film experiment than toggling back and forth between P and C9(P), which are in stable equilibrium. Although the C21(P) soap film corresponds to a [mathematical] stationary state, its area is larger than that of nearby lying [non-minimal-surface] soap films, and I had to struggle to maneuver the film into its unstable equilibrium position long enough for a camera to capture it.

In 1992 I made an impromptu video about minimal surfaces in which I attempted to demonstrate the art of blowing these unstable soap films inside Coxeter cells, but I had run out of glycerine that day. After many tries, I succeeded for a fleeting moment in capturing the gracefully curved Flächenstück of C21(P). I plan to post here a snapshot or two from these videos, but the images of the soap films are somewhat obscured by the transparent tape I used to join the faces of the vinyl tetrahedra. I plan to obtain clearer photos of soap films inside glass tetrahedra I have recently made.



Fig. E2.83a
Twelve elementary triangular Flächenstücke
of Neovius's embedded surface C9(P)


Fig. E2.83b
Twelve elementary triangular Flächenstücke
of Neovius's self-intersecting surface C9(P)†

In both C9(P) and C9(P)†, the triangular Flächenstück abc
is analytically continued by reflection in its edges.
The normal vectors (red arrows) at corresponding points of the
two adjoint surfaces C9(P) and C9(P)† have the same directions.


In 1971, with the assistance of my Cal Arts students John Brennan and Bob Fuller, I performed additional soap film experiments aimed at modeling the elementary Flächenstücke for higher-genus variants of the P and D surfaces and a variety of non-cubic TPMS. All of these experiments involved modifying the shapes of stationary-state soap films inside transparent plastic models of Coxeter cells by blowing on them.

Fig. E2.84 shows some of Ken Brakke's Surface Evolver for some of these higher-genus variant surfaces. I was unable to produce the genus-15 soap film, but occasionally I succeeded with the genus-21 case.


genus=9


genus=15


genus=21


genus=27


genus=33

Fig. E2.84
Ken Brakke's Surface Evolver solutions for the first few
high-genus variants of Neovius's C9(P)


'NOTCHED' ADJOINT SURFACES


genus=3




genus=9




genus=15




genus=21




genus=27




genus=33

Fig. E2.85
Stereo images of the sequence of 'notched' variants of
the adjoints of P, C9(P), ... (left)
and Ken Brakke's images of
the corresponding embedded surfaces (right)

The relative lengths of the line segments in the serrated edges
that make up the notched outlines of the adjoint surfaces that are
illustrated here (in stereo) only roughly approximate the actual values,
which Ken Brakke derived with high precision when he used his Surface Evolver
software to kill periods, thereby generating each of the embedded surfaces (shown at the
right of the corresponding adjoint surface outlines).


The genus pk of the kth surface Mk in the family {Mk} is defined as pk = p0 + k gap (k = 0, 1, 2, ...); the values of p0 and the positive integer gap are characteristic of the family. These families include — but are not limited to — high-genus complements of P and D. For the P and D families, for example, p0=3 and gap=6. Hence the surfaces in these two families are of genus 3, 9, 15, 21, ... .

On that day in 1971, John and Bob and I blew a large variety of "finely filigreed" soap films in a variety of Coxeter cells, and we made detailed drawings of our results. Most — but not all — of these soap films included one or two nylon threads stretched along 2-fold symmetry axes of the enclosing polyhedral cell. The curved soap film boundary edges lying in face planes of the cell are 'mirror-symmetric plane lines of curvature'. Every face plane is a plane of reflection symmetry for both the assembly of cells and the soap films in their interiors. The soap films meet the enclosing face planes orthogonally. Each soap film is an approximate model of the stationary state of the adjoint of a minimal surface bounded both by straight line segments and by either one or two curved edges — according to whether the number of rotational symmetry axes through the cell is one or two. Films with k=0 or 1 are in stable quilibrium. If k=2, the film is in a delicate state of unstable equilibrium. I found it impossible to produce films for k>2. Some skill is required to arrest a film for k=2 in the neighborhood of its equilibrium position long enough to confirm the existence of the equilibrium. (The films were composed of a mixture of distilled water, detergent, and glycerine and were thick and viscous enough for both gravity and capillarity effects to impose some limits on the accuracy of the modeling.)

In 1999, I began sending Ken Brakke data from these 1971 experiments as well as some additional data for surfaces whose existence I conjectured during the following three years, for authentication with his Surface Evolver computer program. Many of these authenticated surfaces are illustrated on his In this work, Ken uses Surface Evolver to 'kill periods' — i.e., to derive the unique values for relative edge lengths that allow the surface to be embedded (cf. the brief description of this problem on pp. 45-46 of Infinite Periodic Minimal Surfaces Without Self-Intersections). In Infinite Periodic Minimal Surfaces Without Self-Intersections, I included only two examples of hybrid surfaces — C(H) (genus 7) and O,C-TO (genus 10), because at the time of writing, these were the only examples of such surfaces for which I had already constructed and photographed vacuum-formed plastic models. (In a footnote on p. 46, I mentioned a third example, of genus 5, that I called g-g'. I soon renamed that surface g-W, after I confirmed that its dual skeletal graphs are related to the structures of hexagonal graphite and of wurtzite.) Figs. E3.7, E3.8, and E3.9 show three views of g-W.



E3. Triangle lattice surfaces

All of the minimal surfaces described in this section are named according to the conventions in Infinite Periodic Minimal Surfaces Without Self-Intersections.


Fig. E3.1a
Schwarz's H surface (genus 3)



Fig. E3.1b
A smaller piece of Schwarz's H surface



Fig. E3.2
C(H) (genus 7)
The [first-order] complement of Schwarz's H surface
unit cell
view: c-axis



Fig. E3.3
C(H)
view along c-axis



Fig. E3.4
C(H)
view: c2 axis (intersection of horizontal and vertical mirror planes) in basal plane



Fig. E3.5
C(H)
view: line in basal plane that is below and parallel to a linear asymptotic (2-fold axis embedded in the surface)
Note the infinitely long straight tunnels with pointy oval cross-section



Fig. E3.6
C(H)
view: c-axis, silhouetted by bright summer sky backlighting
Note the infinitely long straight tunnels.
(The trigonal symmetry of the surface would be slightly more apparent
if the image had been rotated 60º in the image plane,
as in Fig. 3.3!)




Fig. E3.7
g-W ("graphite-wurtzite") (genus 5)
oblique view



Fig. E3.8
g-W
oblique view



Fig. E3.9
g-W and C(H)
oblique view




Fig. E3.10
H''-R (genus 5)
view: c-axis


Fig. E3.11
H''-R (genus 5)
view: c-axis, silhouetted by bright summer sky backlighting


Fig. E3.12
H''-R
oblique view



Fig. E3.13
H''-R
view: c2 axis (intersection of horizontal and vertical mirror planes) in basal plane



Fig. E3.14
H''-R
view: line in basal plane that is below and parallel to a linear asymptotic (2-fold axis embedded in the surface)
Note the infinitely long straight tunnels with pointy oval cross-section




Fig. E3.15
H'-T (genus 4)
view (stereo): c-axis




E4. Surfaces on other lattices



Fig. E4.1
S'-S''
genus 4
view: oblique




E5. Background



E6. Bibliography

For online minimal surface videos, discussion, analysis, and images, including — but not restricted to — examples of embedded triply periodic surfaces, see





E7. A few minimal surface people from around the world


Christian Bär (right), and Hermann Karcher



Ken Brakke at Selinsgrove, Pennsylvania



Tomonari Dotera and Junichi Matsuzawa
at Carbondale, Illinois (November, 2013)

higher resolution image



Shoichi Fujimori at Bloomington, Indiana (2008)



Wojciech Góźdź



Bathsheba Grossman at Santa Cruz



Stefan Hildebrandt at Berkeley (1979)
photo by George M. Bergman
©George M. Bergman
Source: Mathematisches Forschungsinstitut Oberwolfach gGmbH



David Hoffman
at the University of Granada Minimal Surface Conference, June 17, 2013
higher resolution image



Stephen Hyde at Canberra



Hermann Karcher (left),
David Hoffman (center), and
Manfredo Perdigão do Carmo (right) at Granada (1991)
photo by Dirk Ferus
©Dirk Ferus
Source: Mathematisches Forschungsinstitut Oberwolfach gGmbH




Katsuei Kenmotsu at Oberwohlfach (2009)
photo by Renate Schmid
Source: Mathematisches Forschungsinstitut Oberwohlfach gGmbH



Rafael López Camino (right) and me
at the University of Granada, June 11, 2013
higher resolution image


Blaine Lawson (left) and Bill Meeks (right) at Rio (1980)
photo by Dirk Ferus
©Dirk Ferus
Source: Mathematisches Forschungsinstitut Oberwolfach gGmbH


Bill Meeks
at the University of Granada Minimal Surface Conference, June 17, 2013
higher resolution image


Johannes C. C. Nitsche (1925-2006)
photo by Ludwig Danzer
©Ludwig Danzer
Source: Mathematisches Forschungsinstitut Oberwolfach gGmbH



Robert Osserman (1926-2011) at Berkeley in 1979
photo by George M. Bergman
©Mathematisches Forschungsinstitut Oberwolfach gGmbH



Raymond Redheffer (1921-2005)



A light moment during a break at the University of Granada
Minimal Surface Conference, June 17, 2013
higher resolution image



Magdalena Rodriguez, Matthias Weber, and Bill Meeks
at the University of Granada Minimal Surface Conference, June 17, 2013
higher resolution image



Antonio Ros Mulero, me, and my wife Reiko Takasawa
at the University of Granada Minimal Surface Conference, June 17, 2013
high resolution image



Harold Rosenberg and Magdalena Rodriguez
at the University of Granada Minimal Surface Conference, June 17, 2013
high resolution image



Gerd Schröder-Turk at Erlangen-Nürnberg



Isaac Van Houten at Carbondale (2008)
photo by the author



Matthias Weber at Oberwohlfach (2009)
photo by Renate Schmid
©Mathematisches Forschungsinstitut Oberwohlfach gGmbH



Adam Weyhaupt (right) at Edwardsville
with two of his students — Darren Garbuz and Caroline Coggeshall
photo by the author




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