_{
The triplyperiodic CoxeterPetrie regular skew
polyhedron {6,63}, which — like (6.4)2 — has
the same topology and symmetry as Schwarz's
diamond surface D. Here it is shown with just
one of its two skeletal graphs.
}
_{
Abstract 65830, submitted in summer 1968
to the American Mathematical Society for the
Madison meeting in August
}
_{
A summary of my enumeration of the
quasiregular tessellations of the {6,4} family
(abstract published by the American Mathematical Society in 1969)
}
_{
(6.4)2
front view
high resolution image (64mesh quadgrid)
high resolution image (1024mesh quadgrid)
}
_{
(6.4)2
top view
high resolution image (64mesh quadgrid)
high resolution image (1024mesh quadgrid)
}
(6.4)^{2}
— alternatively written
(6.4.6.4)_{G} —
is a triplyperiodic
quasiregular warped polyhedron composed of regular
convex
hexagons and regular
skew
quadrilaterals that are spanned by
minimal surfaces. (The minimal surfaces are approximated here
by hyperbolic paraboloids.)
(6.4.6.4)_{G} has the same topological
structure and symmetry as the gyroid. We adopt the convention
that a triplyperiodic polyhedron is called warped if at least one
of its face species is a skew polygon.
The 4 in (6.4)^{2} is written in
bold face italics to indicate that its quadrilateral faces are skew.
The face angle of the skew quadrilaterals in (6.4)^{2} is
cos^{1}(2/7)
≅ 73.3985°. The tilt angle
(the angle between an edge and the
orthogonal projection of the edge onto the equatorial plane) is
cos^{1}√(5/7) ≅ 32.3115°.
A quasiregular warped polyhedron that is a sort of inverse to
(6.4)^{2} is (6.4)^{2}, in which the regular
hexagons are skew, with
90° face angles, and the regular quadrilaterals are squares. In
the continuous family (6.4)^{2},
the only examples in which one
of the two face species is plane and the other is skew
are
(6.4)^{2}
and (6.4)^{2}.
In the future I will post images of some examples.
P
D
G
{6,44}_{P}
{6,44}_{D}
{6,44}_{G}
s.c. lattice
f.c.c. lattice
b.c.c. lattice
These three TPMS are tiled here in a pattern
called {6,44}. The
prototile for each surface
is a particular variant of hex_{90}, a
regular skew
6gon
with 90º face angles. In this pattern, there are
4 faces
incident at each vertex and 'holes' with
4fold
symmetry.
Alternatively, these surfaces can be tiled in the
patterns {4,64} and {6,63}. The prototiles for
{4,64} are variants of quad_{60}, a regular skew
4gon with 60º face angles. In this tiling, there
are 6 faces incident at each vertex and holes
with 4fold symmetry.
The prototiles for {6,63} are variants of hex_{60},
a regular skew 6gon with 60º face angles. In
this tiling, there are 6 faces incident at each
vertex and holes with symmetry of order 3.
It is conventional to call
skew polygons
regular if they are both equilateral
and equiangular, irrespective of
whether their edges are
straight or curved.
The names {6,44}, {4,64}, and {6,63} are
Coxeter's modified
Schläfli symbols for the
three
infinite regular skew polyhedra. Each
of these three polyhedra is homeomorphic
to — and has the same symmetry as —
one of the two Schwarz surfaces: {6,44}
and {4,64} are homeomorphic to P,
while {6,63} is homeomorphic to D.
P, D, and G are related by
Bonnet bending,
without stretching or tearing. A prototile of
one surface can be continuously transformed
by bending into the corresponding prototile
of either of the other two surfaces:
hex_{90}(D) ↔
hex_{90}(G) ↔
hex_{90}(P).
quad_{60}(D) ↔
quad_{60}(G) ↔
quad_{60}(P).
hex_{60}(D) ↔
hex_{60}(G) ↔
hex_{60}(P).
In fact an entire lattice fundamental region, not
just a prototile, can be transformed in this way.
Below are stills from a 1969 movie showing the
bending of hex_{90}(D) ↔
hex_{90}(G) ↔
hex_{90}(P).
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 triplyperiodic only
when the Bonnet angle θ satisfies the equation
θ_{p,q} = ctn^{1}[(p/q)(K'/K)],
where p and q are any two coprime positive integers.
This obviously necessary restriction hadn't occurred
to me before Blaine Lawson mentioned it — more
than once! — in September 1968. (Thanks, Blaine.)
To animate the timelapse bending sequence above,
click
here, and press on the
Page Down/Page Up keys.
The portion of the
1969 'Part 4'
movie that illustrates this bending
starts at 3^{min}18^{sec} after the beginning.
The combinatorial structure of the three tiling patterns
{6,44}, {4,64}, and {6,63} is illustrated by
Poincaré's hyperbolic disk model of
uniform tilings in the hyperbolic plane.
(image by Eric W. Weisstein,
Wolfram MathWorld)
{4,64}
{6,44}
{6,63}
(images from Wikipedia)
Coxeter and his friend Petrie long ago discovered the three
triply
periodic regular skew polyhedra called {p,qr}.
Their faces are
regular plane polygons.
p is the number of edges of each face,
q is the number of faces at each vertex,
and
r is the number of edges of each hole.
{4,64}
{6,44}
{6,63}
{4,64}, including the skeletal graph of labyrinth A
{4,64}, including the skeletal graphs of both labyrinth A and labyrinth B
{6,44}, including the skeletal graph of labyrinth A
{6,44}, including the skeletal graph of labyrinth B
{6,63}, including the skeletal graph of labyrinth A
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 hex_{90} ({6,44} tiling)
or by
(ii) regular skew quadrilaterals quad_{60} ({4,64} tiling)
or by
(iii) regular skew hexagons hex_{60} ({6,63} tiling).
Below are views of G tiled by hex_{90}
in the CoxeterPetrie {6,44} map.
Each hex_{90} has 90º face angles.
Every hex_{90} face is
related to each of six faces with
which it shares an edge by a halfturn about an axis
of type (110) perpendicular to G at the midpoint of
the shared edge.
(1)
(2)
(3)
(4)
(5)
(1)
[100] orthogonal projection
(2)
(100) viewpoint
(3)
[100] orthogonal projection
(4)
[111] orthogonal projection
(5)
~(110) viewpoint
Below are views of G tiled by the regular
skew hexagon hex_{60}, the prototile of
the CoxeterPetrie {6,63} map.
hex_{60} is so named because it has 60º face angles.
It is related to six other hex_{60} faces, with each of
which it shares an edge, by a halfturn about an
axis of type (110) perpendicular to the surface at
the midpoint of the shared edge.
Stereo image of a cubic unit cell of G
tiled by eight replicas of hex_{60}
This unit cell is comprised of
two lattice fundamental regions.
(100) viewpoint
Highresolution version
Stereo image of the cubic unit cell of G
illustrated just above
(111) viewpoint
Highresolution version
Stereo image of the
hexagonal face hex_{60} of G
(111) viewpoint
Highresolution version
Stereo image of the
hexagonal face hex_{60} of G
and an associated cuboctahedron
The midpoints of the six edges of
hex_{60} coincide with vertices
of the cuboctahedron.
(111) viewpoint
Highresolution version
Stereo image of the
hexagonal face hex_{60} of G
(110) viewpoint
Highresolution version
Stereo image of the
hexagonal face hex_{60} of G
(415) viewpoint
Highresolution version
Gyroid sculpture
by Chaim GoodmanStrauss and Eugene Sargent
Click
here for the authors' account of
how they implemented their design.
Here's their razzledazzle 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 gyroidrelated 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)
IWP
FRD
(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)
SBA1x50
Batwing5x2
(models by Alan Mackay)
The four founding fathers of triplyperiodic minimal surfaces
Georg Friedrich Bernhard Riemann
Karl Hermann Amandus Schwarz
(18261866)
(18431921)
Alfred Enneper
Karl Theodor Wilhelm Weierstrass
(18301885)
(18151897)
(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
'EulerLagrange 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 EulerLagrange equation were the
helicoid
and
catenoid,
both of which were discovered by
Meusnier in 1776.
Meusnier also proved that the mean curvature of every solution surface is equal to zero.
Since some closed curves 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 areaminimizing, i.e.,
the surface patch
inside every sufficiently small closed curve enclosing
any point of the surface has less area than any other surface bounded by that closed curve.
Pioneering investigations of
triplyperiodic minimal surfaces (TPMS)
were performed by
Schwarz,
Riemann,
Weierstrass,
Enneper, and
Neovius
in the middle of the 19^{th} century.
By the early 1960s, however, TPMS had almost faded from view in the
mathematical literature.
Since about 1970, there has been a revival of interest in TPMS
as mathematical research on minimal surfaces of every kind has expanded.
Now it is no longer just mathematicians who study TPMS.
Materials scientists are also interested in them, because they have concluded that some
of the few known examples of low
genus
— especially those on a
cubic lattice —
are useful as templates for the shapes of a variety of socalled
selfassembled structures
that are studied by physical, chemical, and biological scientists.
Below are links to a tiny sample of the
relevant mathematical and materials science studies, but the sample is
neither comprehensive nor uptodate.
I have attempted here to summarize my own study of TPMS, which began
quite unexpectedly in the spring of 1966.
I have included an account
— warts and all — of some of the events that led to my involvement
in this study, during which I frequently wandered down bypaths that
were well off the main route.
(I have long believed that such bypaths sometimes offer a more rewarding
view than the main route.
John Horton Conway
has explained that he finds it fruitful to juggle several ostensibly
unrelated problems at the same time,
because one problem may turn out to be the key to the solution of another.
My discovery of a precursor of the gyroid minimal surface
in 1968 was for me a validation of Conway's truism, as explained below.)
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 newfound passion for TPMS,
even though I lacked the credentials most employers would have considered
a minimum requirement for such an undertaking.
Because of his scientific reputation, he had sufficient clout to shield me
from attacks both by local skeptics
(of whom there were more than a few) and also by officials in NASA
headquarters who wondered what
on earth soap films might have to do with NASA's mission.
Lester C. Van Atta
19051994
Associate Director, NASA Electronics Research Center
But it was all too good to last!
ERC was abruptly shuttered in July 1970 in what many of us concluded
was probably an act of political malice
directed by President Nixon against Senator Ted Kennedy of Massachusetts.
I cannot avoid being somewhat sceptical of the purportedly objective history
of the closing of ERC by the author of this
contemporaneous account,
in which no specific role is ascribed to Nixon.
Until the announcement on December 29, 1969 by the Administrator of
NASA that ERC would close on June 30, 1970,
I felt quite free to decide what to investigate, with few strings attached.
In retrospect, I believe that I would almost certainly have been unable to
concentrate productively on
my research at ERC if I had been aware of the turbulent political winds that
were blowing about our heads.
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 triplyperiodic 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 triplyperiodic
graphs and convex polyhedra since the mid1950s,
but before 1966 I knew nothing about minimal surfaces, aside from a
nodding acquaintance with the helicoid and the catenoid.
In April of that year,
Konrad Wachsmann,
chairman of the architecture department at the University of Southern California,
suggested that I visit the North Hollywood architect/designer
Peter Pearce,
who had a grant from the Graham Foundation for a oneyear study of polyhedra,
crystal structure, and related topics.
Although Peter did not claim to be an expert on the mathematics of minimal
surfaces, he had developed a novel
application of minimal surfaces to the design of periodic structures that
led me to make a radical change in the direction of my research. Below I
summarize how this happened.
Peter Jon Pearce
Architectural designer
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 crosseyed 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 (triplyperiodic graphs)
and polyhedral packings.
The intended purpose of my heuristic scheme was to represent every atomic
site in a crystal strucure by a polyhedron with
(i) the
same number f of faces as the number Z of edges incident at
the corresponding node of the graph, and
(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 ballandstick model of the lattice,
and (ii) the cube also has the same symmetry as the node with respect
to the surrounding lattice.
A piece of the 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 facecentered 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 bodycentered 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 secondnearestneighbor 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 14faced 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 spacefilling
eightfaced 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 vertexenclosing polyhedra nodal polyhedra, irrespective of whether
they turn out to be saddle polyhedra or convex polyhedra.
For about ten years, I had been calling the triplyperiodic
graph whose edges correspond to the edges of a packing of expanded regular octahedra the WP graph,
because it mimics the pattern of string tied around a wrapped cubic box (see image below).
Curiously, the ERO was about to introduce me to TPMS!
The expanded regular octahedron ERO,
which is the nodal polyhedron of the 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 socalled 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 triplyperiodic graph.
In June 1966, I undertook some soap film experiments in order to explore these questions.
To my surprise I found that the boundary curve C_{0}
(shown at left below), which has the shape of one of the several
Hamilton cycles
on the
cuboctahedron,
spans at least two disktype soap films of different shape.
One of these two surfaces, S_{1}, is a surface of least area and is called stable. The other,
S_{0}, is not a surface of least area. It is called unstable, because it can be formed
as a soap film on C_{0} only if one or more wires or threads are added to C_{0}
along appropriate curves embedded in S_{0} — i.e., curves that partition S_{0}
into an assembly of smaller surface patches each of whose boundaries spans a unique stable minimal surface.
S_{1} can drape the boundary frame C_{0}
in either of two positions. Let's call it S_{1a} if it's in one of these positions
and S_{1b} if it's in the other.
S_{1a} and S_{1b} are
related by a halfturn about the axis A_{1}A_{2}.
(See Ken Brakke's computed images of S_{1a} and S_{1b} below.)
If a wire frame in the shape of C_{0} is withdrawn from a solution of soap and water,
it will span a soap film in the shape of one or the other (but not both) of these surfaces.
S_{0}, which drapes the wire frame C_{0} in only one position,
forms as a soap film if threads or wires are incorporated in C_{0}
along one or both of the lines A_{1}A_{2} or B_{1}B_{2}.
C_{0}, a curve that spans at least
Cuboctahedron
two soap films of different shape
The three [orthogonal] c2 axes of the boundary curve C_{0}.
The two axes A_{1}A_{2} and B_{1}B_{2}
each intersect C_{0};
the vertical axis V_{1}V_{2} does not.
The soap film S_{1a}, one of two
congruent stable surfaces spanned by C_{0},
is an areaminimizing ('least area') minimal surface.
S_{1a}
S_{0}
S_{1b}
(Stable)
(Unstable, unless string
(Stable)
or wire is added along
either A_{1}A_{2} or B_{1}B_{2}
or
both
A_{1}A_{2} and B_{1}B_{2})
Incorporating internal threads or wires along (a) either A_{1}A_{2}
or B_{1}B_{2} or (b) both
A_{1}A_{2} and B_{1}B_{2}
in the wire frame C_{0}, which has twelve edges, partitions C_{0}
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 simplyconnected
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 5gons and 7gons have total curvatures of only
2^{1}⁄_{6} π and
3^{1}⁄_{3} π, respectively,
it follows from Nitsche's theorem that each of them spans only one minimal surface of disk type.
This implies that incorporating a wire or thread along either or both of the axes
A_{1}A_{2} and B_{1}B_{2}
will convert the bare frame C_{0} into a frame that spans the surface S_{0}.
Note that the soap film in each of the photos below is a piece of S_{0}.
extra wire added along A_{1}A_{2}
extra wire added along B_{1}B_{2}
extra wires added along A_{1}A_{2} and B_{1}B_{2}
In October 1967, three months after I joined NASA/ERC, I was a selfinvited
guest at Hans Nitsche's home in Minneapolis.
Although Hans showed considerable interest in my wire frame of C_{0} and in
my plastic model of S_{1}, he never mentioned that the groundbreaking paper
in which he introduced and proved the 4π theorem was about to be published!
Since I found Hans to be both kind and modest, I later concluded that perhaps he thought
I was so ignorant of the mathematics of minimal surfaces
that he would only confuse and embarrass me if he discussed such a subtle problem.
I extend my warm 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 C_{0}.
Triplyperiodic 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 selfdiffusion in f.c.c crystals, the
isotope effect
and the
BardeenHering 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 selfdiffusion.
I modeled interstitial diffusion pathways (strictly random walk)
by the edges of one triplyperiodic graph and pathways for diffusion by the vacancy mechanism in the same crystal
(correlated random walk) by the edges of a second triplyperiodic graph intertwined with the first graph.
I defined these two graphs as duals, and I attempted to discover whether it is possible to define which symmetry
and combinatorial properties are required of a triplyperiodic graph in order for it to have a unique dual,
by analogy with the
dual of a planar graph
or the
dual of a convex polyhedron.
Using essentially ad hoc methods to identify
dual pairs of triplyperiodic graphs, I found that while the dual of the diamond
graph is also a diamond graph, the dual of the 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 vacuumforming 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 vacuumforming machine.)
A piece of Schwarz's D surface
Next I replaced the 90º skew hexagon by
one with 60º corners, and a second such labyrinthine surface appeared!
A portion of Schwarz's P surface
tiled by 60º skew hexagons
A transparent model of P
I had unwittingly stumbled onto the two classical examples of
adjoint
(or conjugate) TPMS, which were discovered and analyzed in 1866 by H.A. Schwarz
(and also — independently — by Riemann and Weierstrass).
It took a telephone call to the minimal surface authority Hans Nitsche in Minnesota for me to identify these surfaces.
I decided to name them
D (for diamond) and P (for primitif), after the crystal structures with matching topology and symmetry.
I recognized that the chambers in the two complementary labyrinths of P define the sites of the cesium
and chlorine ions, respectively, in the ionic crystal CsCl.
Only after consulting a handbook of crystal structures
did I learn that the atoms of sodium and thallium in the binary solid solution
NaTl occupy sites that correspond to the symmetrical 'chambers' in the respective labyrinths of D.
I began to study in earnest both differential geometry and the complex analysis used in investigations
of minimal surfaces.
With the benefit of hindsight, I later recognized that if — at some time during the year after
I stumbled onto D and P in April 1966 — I had taken the time to
read Schwarz's Collected Works more carefully, I might possibly have noticed
the following theorem on p. 174:
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_hex_{90} and
P_hex_{90}
that tile the map {6,44} in these surfaces are
inscribed in a cube of the same size, that linear combination becomes the
arithmetic mean of
the coordinates of corresponding points of D and P!
It is perhaps surprising that Schwarz doesn't seem to have taken the trouble to
sum the coordinates of at least a few pairs of corresponding points of D and P.
It seems very likely that if he had done so, he would have discovered the gyroid.
Dual graphs and skeletal graphs
The concept of a dual relation for pairs of triplyperiodic graphs had suddenly acquired new significance for me.
I began to think of such graphs as potential skeletal graphs of the two labyrinths of an
embedded TPMS. The geometry of such paired graphs would dictate the geometry of the TPMS.
A literature search in the UCLA library indicated that besides D and P,
only three other examples of embedded TPMS — H, CLP, and Neovius's surface — had been known since 1883.
But I found it hard to imagine that there were not others!
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 triplyperiodic graphs
I had identified — including the oppositely congruent
Laves graphs
— might also be skeletal graphs of the two labyrinths of an embedded TPMS.
The Laves graphs were a troublesome case, because the absence of reflection symmetries made it impossible
for me to imagine how such a surface could be generated.
Sometimes the makeshift rule I had refined by exploiting the relation
between triplyperiodic graphs and saddle polyhedra yielded a dual pair of triplyperiodic graphs
that were neither directly nor oppositely congruent — for example,
the 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 eveningsandweekends hobby.
(Further details of this story are described below.)
I believed that Peter Pearce had made an inspired choice
when he chose minimal surfaces
for the faces of his diamond and 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 triplyperiodic graph.
Although the recipe worked without a hitch for every graph I tested,
I felt distinctly uneasy, because I suspected that there must exist cases for which
it would be ineffective.
Although I modified the recipe several times during the next several months,
I was never able to give it a solid theoretical foundation.
Beginning in June 1966, as a sparetime hobby I set out
to discover a 'counterexample' —
a graph for which the recipe fails to produce either
interstitial or nodal polyhedra.
I continued to test a variety of graphs, gradually
accumulating a diversified collection
of vacuumformed interstitial and nodal polyhedra.
I had the additional goal of finding a way to construct a
hypothetical TPMS I originally named L (for Laves).
Here, however, I will refer to it as G (for gyroid),
even though I didn't invent that name until
almost two years later.
It was obvious that G couldn't have any reflection symmetries,
since the union of its two enantiomorphic skeletal graphs has no such symmetries.
I also recognized that there could be no straight lines embedded in G,
since by a theorem of H. A. Schwarz,
a straight line embedded in a minimal surface is an axis of
2fold rotational symmetry.
This implies that a halfturn rotation of G about such an axis
would interchange the two labyrinths of G — and therefore also
interchange the two skeletal graphs of G.
But that is impossible, since the two skeletal graphs are enantiomorphic.
I couldn't imagine how to define
the boundary curves of an elementary surface patch whose edges
are neither straight line segments nor curved geodesics
(mirrorsymmetric plane lines of curvature).
For each of the five examples of TPMS known before 1968,
there exists a skew polygon surface patch with straight edges.
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 'cylinderized' labyrinths is surprisingly small.
I was startled to observe how snugly the cylinders of the intertwined labyrinths
nestle against each other.
When I compared the contours of my toy models of P and D
with the smoother contours of my vacuumformed models of P and D,
it seemed entirely plausible that the junctions between
cylinders in the toy model of G could be flared
and filleted so that the envelopes of the two labyrinths
would coalesce into one
single surface — a TPMS — just as they do for P and D.
I understood, of course, that vague intuitive arguments like
these 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 'BCC_{6}'
(see stereo image below) that provided the longsought 'counterexample'
to my empirical recipe for deriving the interstitial and nodal polyhedra of
a triplyperiodic graph.
(b) Although this graph failed spectacularly to yield a
finite interstitial polyhedron, it pointed toward something
much more interesting — an infinite triplyperiodic
saddle polyhedron
that I call M_{4}. The symmetry and combinatorial structure
of M_{4} strongly
strongly suggested to me that the hypothetical G minimal
surface might exist after all.
I immediately lost almost all interest in saddle polyhedra and
began to concentrate
instead on confirming the existence and embeddedness of the G
surface and on searching for other new examples of TPMS.
A portion of the deficient symmetric graph BCC_{6} of degree six
I define a deficient symmetric graph on a given set of
vertices as a symmetric
graph of degree less than the maximum possible for that set. (With
two additional edges incident at each vertex, BCC_{6} would
be transformed into BCC_{8},
the bcc graph.)
viewpoint: close to [100] direction
30 quadrangles of M_{4} (stereo)
view: [111] direction
30 quadrangles of M_{4} (stereo)
view: [110] direction
30 quadrangles of M_{4} (stereo)
view: [110] direction
30 quadrangles of M_{4}
view: [111] direction, backlit by summer sky
30 quadrangles of M_{4}
view: [100] direction, backlit by summer sky
Abstract 65830 submitted in summer 1968
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 shortlived, however.
On December 30, 1969, the director of NASA visited Cambridge to
announce to a gathering of all employees that ERC would be permanently closed in exactly six months.
We were of course startled — as well as disheartened — by this unexpected news. The six yearold ERC
was by far the youngest of the eighteen NASA centers.
It was the only federal research center
with electronics research for its mission,
a legacy inherited from the Kennedy presidency (though
it was President Lyndon Johnson who presided over its development).
ERC was famously topheavy (or perhaps I should say bottomheavy) with a bloated support
infrastructure of low and midlevel administrators, clerks, etc.,
many of whom were from the Boston area, hired in the early days before President Kennedy was assassinated.
Soon after it opened, the hiring of scientists and engineers slowed down abruptly, and it appeared
that the original plan to develop a wellrounded
scientific and technical staff had been abandoned.
It was our impression that Johnson preferred to support NASA activities elsewhere, especially in Texas.
However, according to
the Wikipedia entry for ERC:
"Although it was the only Center NASA ever closed, ERC actually grew while NASA eliminated major programs and cut staff.
Between 1967 and 1970, NASA cut permanent civil service workers at all Centers with one exception,
the ERC, whose personnel grew annually."
Whatever the case, I arrived at ERC in July 1967 in a state of blissful ignorance.
Only after I began work did I begin to learn from my colleagues
about discrepancies between ERC's officially stated 'mission' and what seemed to be its actual potential for significant accomplishment.
My own position there was relatively comfortable, however, with
one exception: the absence of inhouse colleagues who shared my scientific and mathematical interests.
I would have benefited from having someone close by for chitchat
about — and even collaboration in — those areas of research in physics and mathematics in which I had a special interest.
There were spectacular compensations for this deficiency, however.
For one, I was acquainted with a few extremely bright young mathematicians in the greater Boston area who showed a friendly interest in my work,
and I benefited greatly from my few conversations with them.
If only I had shown more initiative, I could have benefited even more from knowing them than I actually did.
They included
Thomas Banchoff (differential geometry),
Norman Johnson (convex polytopes),
Nelson Max (computer graphics), and
Charles Strauss (computer graphics).
In 1966, a year before I joined NASA/ERC, Norman Johnson introduced me to the analysis by
Coxeter
and Moser of
the infinite regular maps {6,44}, {4,64}, and {6,63} (cf.
the book by these authors that is cited below, following Fig. E1.1k).
These three regular maps describe the combinatorial structure of the flatfaced
CoxeterPetrie
infinite regular skew polyhedra.
But they also describe the combinatorial structure of H. A. Schwarz's
P and D surfaces, the
canonical 19th century examples of triplyperiodic minimal surfaces,
as well as that of
their only embedded associate surface, the
gyroid G
(which I nearly discovered in February 1968,
when by chance I found a doppelganger that is spookily similar).
(Parenthetical note: In 1969 Donald Coxeter was my guest in
Cambridge, Mass., where he presented a lecture at MIT. To my surprise,
he told me that he had never heard of the Schwarz surfaces!)
H.S.M. "Donald" Coxeter
Thomas Banchoff at Berkeley in 1973
photo by George Bergman
Charles Strauss (seated) and
Thomas Banchoff
at Brown University in 1979
Norman Johnson
Nelson Max
MIT was directly across the street from ERC, and Harvard was only a 20minute 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 modelmaker
who lived in a nearby suburb. We hit it off immediately.
After examining my plastic minimal surface models,
Hal easily convinced me that he could make more accurate and more durable vacuumforming tools than I could.
Dr. Van Atta was acquainted with Hal's father, an engineer who was president of High Voltage Engineering Corp.,
the manufacturer of Van de Graaff generators. I persuaded Dr. Van Atta
to hire Hal in a flexible parttime arrangement so that he could fabricate vacuumforming tools for me.
Hal wan't interested in a fulltime job, since he had other clients, and in any event I expected to have only enough
projects to keep him busy intermittently. From then until the end of my stay at ERC about thirty months later,
Hal was my invaluable collaborator.
Dr. Van Atta also arranged to hire — one or two at a time — parttime workstudy students from area universities
(Boston University, Northeastern University, Harvard, and MIT)
to help with FORTRAN programming and the assembly of new minimal surface models.
These young superstars were Kenneth Paciulan, Richard Kondrat, Randall Lundberg, Jay Epstein, and Dennis ____(?).
I am grateful to them all.
Dr. Van Atta also hired James Wixson, an experienced applied mathematician and computer programmer.
Jim helped me with a variety of chores. One of his several accomplishments was the invention and programming of a computer algorithm
for generating every possible skew quadrangle that serves as a module for a compound periodic minimal surface
on a cubic lattice — an assembly of
finite surface patch modules whose four straight edges include at least one edge
along a [111] direction, coincident with an axis of 3fold rotational symmetry.
Decades later, these solutions have become of some interest as models for structures investigated by
physicists and chemists who are soft matter specialists.
I recall now with some embarrassment that during my first week at ERC, I visited the Harvard mathematics department
and stopped by the offices of one after another member of the faculty to ask naive questions about the
rather prosaic problem of how to go about enumerating those examples of triplyperiodic graphs that are symmetric.
Professors
Zariski
and
Ahlfors
were both polite, but it was clear that my questions held little interest for them, and the interviews were mercifully short.
Andrew M. Gleason
Andy Gleason
was another matter, however. He cordially invited me into his office,
where we spent the next ninety minutes or so discussing my problem.
First he asked me why I was interested in this question.
When I explained my still rather halfbaked ideas about the connections to triplyperiodic minimal surfaces,
he showed considerable interest. Although he didn't provide me with definitive solutions for any of my problems,
he did ask me a number of stimulating and provocative questions.
I never met him again. It was only a few years ago that I learned of the great range of his highly original accomplishments
in both 'pure' and 'applied' mathematics.
He was a very kind person, and I shall never forget him.
If I had known then that (a) Andy Gleason and I both graduated from high schools in Westchester County, N.Y. (he in Yonkers
and I in Mount Vernon),
(b) he graduated from Yale in 1942, the year I entered Yale, and (c) we were both in Naval Intelligence during WWII
(he helping to crack the Japanese code and I
passively studying the Japanese language),
I would undoubtedly have attempted some small talk about these coincidences,
but that would hardly have advanced our discussion of mathematics!
As a federal civil service employee, I had unfettered access to the WATS government longdistance telephone line.
I made good use of it now and then, including having several fruitful conversations about
the stability of minimal surfaces — beginning in 1968 — with
Fred Almgren
at Princeton.
I first met him facetoface in September 1969, when we arranged to have sidebyside seats
on a flight to the USSR. For a week Fred was my roommate at the Hotel Iberia in Tbilisi, Georgia, while we were
attending a conference on minimal surfaces. Afterward Fred went on to St. Petersburg for an extended sabbatical visit.
Frederick J. Almgren, Jr.
In April 1968, shortly after I discovered experimentally a remarkably close approximation to what I subsequently called the gyroid
(but before I had any proof that such a minimal surface exists),
I telephoned
Robert Osserman
at Stanford to ask for his help with a proof. I sent him a plastic model of the surface, and soon aferwards he asked
his PhD student
Blaine Lawson to investigate the problem.
What followed is described below, just after Fig. E2.68c.9.
From then on, I occasionally used the WATS telephone line to discuss some of my conjectures about minimal surfaces with Blaine,
whom I found to be extremely knowledgeable about every conceivable aspect of minimal surface theory.
Robert Osserman
H. Blaine Lawson, Jr.
I was free — within reason — to attend meetings of the American Mathematical Society, of which I was a member.
In contributed 15minute talks at one or two of those AMS meetings, I described my work and showed some of my minimal surface models.
Once or twice someone in the audience would express interest in the mathematics,
but more often it seemed that they were curious mostly about how I had constructed the models!
After one of those AMS meetings in New York City, I visited the Courant Institute,
where I had the enormous good luck to meet
Stefan Hildebrandt,
already one of the upandcoming leaders
in the mathematics of minimal surfaces. During the next few years,
Stefan more than once saved me from making a serious blunder as I groped my way toward a fuller understanding
of minimal surfaces.
Stefan Hildebrandt at Berkeley (1979)
photo by George M. Bergman
©George M. Bergman
Source: Mathematisches Forschungsinstitut Oberwolfach gGmbH
Stephen Hyde
told me in 2011 that according to Stefan, the reason it fell to his lot to interview me during my 1968 visit to the Courant Institute
was that he was at that time one of the youngest members of the research staff. It was the custom for junior members
to be assigned the chore of hosting the cranks and crackpots who invited themselves to the Institute. Since I was selfinvited, for all anyone could tell I
— with my bizarre colored models of surfaces in tow — might turn out to be one of those unwelcome visitors.
I was gratified to learn from Stephen that Stefan concluded — after listening to my spiel and examining my surfaces — that I was
probably neither crank nor crackpot! He took some photos of me and my models on the roof of the Courant Institute.
Here is one of me holding my plastic model of the gyroid.
A.H.S. and the gyroid at the Courant Institute, 1968
Photo by Stefan Hildebrandt
At ERC I buried myself in my research with little thought about the future.
Dr. Van Atta provided even more support for my work than I ever asked for.
On that day at the end of 1969 when we were informed about the impending shutdown, Richard Nixon had been president for almost a year.
According to a contemporary news account, a prominent science journalist
overheard some interesting remarks in the White House by the physicist
Lee DuBridge, the former
CalTech president who was Nixon's scientific advisor. DuBridge was alleged to have said that
the president's decision to close ERC was prompted by his wish to damage the presidential aspirations of
the senior senator from Massachusetts, Teddy Kennedy.
(Kennedy was widely regarded at the time as Nixon's most formidable potential rival.)
NASA had been funneling about $60 million annually into Massachusetts, and a significant fraction of those funds
supported ERC, with substantial collateral benefits to the state economy.
During the late winter and early spring of 1970,
ERC director James Elms made frantic efforts to find another federal agency to
occupy the new $40 million building into which we had moved a week or so before the announcement of the shutdown.
By late spring, it was decided that a handful of members of the technical staff— mostly engineers and a few applied mathematicians —
would be retained to work for a newly minted federal agency that would be
called the Transportation Systems Center, as part of the U. S. Department of Transportation.
The rest of us were told, "Good luck!" (Thanks to the good offices of Peter Pearce,
I had already been invited to teach at the abouttobeformed California Institute of the Arts,
in Valencia, California, so my distress over the demise of ERC was somewhat less acute than that of many of my colleagues.)
In my last six months at ERC, I tried to record as much as possible
of what I had learned about TPMS in a NASA technical note entitled
'Infinite Periodic Minimal Surfaces Without SelfIntersections'.
Meanwhile, I had been commissioned to design and construct an 11ft.diameter model of
the gyroid minimal surface for the Museum of Modern Art in New York City,
where an Art and Mathematics exhibition was scheduled to open in mid1970.
Here's how the commission came about:
Arthur Drexler,
Director of the Department of Architecture and Design at MOMA, having heard about the gyroid from one of my colleagues,
visited Cambridge in the late summer of 1969 to
inspect my collection of minimal surface models. He immediately chose the gyroid as the
surface he would like to see me sculpt for the exhibition.
Dr.Van Atta then telephoned NASA headquarters and almost overnight
obtained a grant of $25,000 to support the project. My friend Keto Soosaar, an expert structural engineer at MIT,
introduced me to his colleague Jeannie Freiburghouse, an experienced Fortran programmer,
and I immediately hired her to use the Weierstrass integrals to compute the coordinates of 8000 points on a hexagonal
patch of the gyroid that corresponds to a face of the CoxeterPetrie regular map {6,44}.
I arranged for ERC to award a contract to the Gurnard Engineering
Corporation of Beverly, Massachusetts to manufacture two CNCmilled aluminum
dies for vacuumforming two kinds of thin zincalloy modules —
one in the shape of the hexagon of {6,44}
and the other in the shape of the quadrilateral of {4,64}, its dual.
These two kinds of module were to be joined by epoxy,
in a facetoface, overlapping arrangement, resulting in a
design in which one side of the gyroid surface is tiled by hexagons
while the other side is tiled by quadrilaterals.
Such a design avoids the need for unsightly connectors along module edges.
(I recognized that it would be more efficient to use a single module shape based
on the hexagon of the selfdual
map {6,63}, which is twice as large as that of {6,44}, but I didn't pursue this idea, because I knew from experience that
the larger negative draft angle of such a big curvaceous module
would probably prevent successful vacuumforming, by introducing
ugly ridgelike 'wrinkles' in each module.)
In the spring of 1970, NASA funding for the project was abruptly
subjected to a special kind of 'midcourse correction': it was cancelled.
On investigation, I was informed that
at a retirement party — presumably well lubricated —
for the senior ERC comptroller responsible for my MoMA account,
someone had 'accidentally hit the wrong key on his computer',
with the result that the money still left in the account
was sent back to Washington
(i.e., NASA headquarters).
The new heir to the comptroller's
office told me that there was no way to recover this money.
Having had earlier experience with bureaucracies, I recognized that the project was finished.
(But see Figs. E1.18ad below.)
Dr. Van Atta resigned from ERC in the autumn of 1969 to become research vicepresident
of the University of Massachusetts/Amherst. Months later my colleagues and I guessed that
he might have received early warning signals about the impending demise of ERC.
His successor, Lou Roberts, an able electrical engineer and administrator,
generously arranged for the last remaining technical typist in our division to be assigned the single task of
typing my technical note, but the deadline was so tight that much of what I wrote
was litle more than a first draft, since I had no opportunity
for either proper editing or for review by another person.
(Personal computers had not yet been invented. If computer work stations that
allowed for some kind of wordprocessing existed in those days, I never heard of them.) Immediately after I submitted my manuscript to NASA,
I handed in a list of typos and other errors for final corrections. Although I was promised that they would be dealt with, they were not.
The one hundred complimentary copies I was promised turned out to be three copies.
We all had the feeling that we were now ancient history, and nobody much cared.
(Perhaps that is the way it always is with institutions that are in their death throes.)
In August 1969, before I had any suspicion that my sojourn at NASA would soon end, I received an invitation —
thanks to the kind intervention of the mathematicians
Robert Osserman
and
Lipman Bers —
to describe my research in a postdeadline presentation at
a September conference in Tbilisi, Georgia, USSR on Optimal Control Theory,
Partial Differential Equations, and Minimal Surfaces. The conference chairman was
Revaz Gamkrelidze
of the Steklov Institute.
When I flew to the USSR consulate in Washington to apply for a visa,
the apparatchik in charge at first turned me down, using the excuse that there wasn't enough time.
Just at that moment, the distinguished UCLA plasma physicist
Burton Fried
happened to enter the office. Recognizing me (we had chatted at an APS January meeting a few years earlier), he instantly
addressed me by my first name.
The apparatchik, who somehow realized that Burt was an important personage, was clearly startled at this show of familiarity.
He turned away from me and quickly processed Burt's visa (for an upcoming
conference on plasma physics in Russia).
Meanwhile, I had retired to a couch a few feet in front of the counter, determined not to give up my own quest for a visa.
Once Burt's application was processed, he stopped by the couch for
a brief chat and then departed.
For the next several minutes, the apparatchik
pretended to ignore me while he was shuffling papers at the counter.
Finally he looked up and asked, "Why are you still here?"
I replied that I expected him to change his mind about my visa application,
since he had managed to grant Fried's request in spite of the fact that Fried's schedule was even tighter than mine.
Perhaps he was impressed by my skill in what he may have perceived as Marxist dialectics. In any event,
he appeared to have a sudden change of heart and grumbled, "Perhaps I can do something for you after all."
(This was my first — but not my last — observation of
obsequious behavior by a petty Soviet bureaucrat.)
When I landed at Vnukovo International Airport in Moscow about ten days later,
I was welcomed by Revaz Gamkrelidze, but I still had to get my bulky collection of
plastic models of TPMS through customs.
When a stolid Ukrainian customs agent showed signs of balking at the sheer number
of boxes I had brought with me
(I suppose he suspected they contained contraband),
Gamkrelidze put on an impressive show of commanding authority.
He announced in a magisterial voice (in Russian) that the boxes contained "mee'neemal soor'facez".
The agent echoed in a bewildered voice, "Mee'neemal soor'facez?"
Gamkrelidze replied with great emphasis that the conference would be impossible without them,
and that was that.
After the conference began, I recognized with dismay
that it would have been more sensible to bring fewer models. A few of the very dignified Russian and Western European mathematicians
at the conference appeared to be somewhat offended by the sheer quantity of models I had brought.
In any event, by the end of the fourday conference, several of the larger models had magically disappeared from
the locked auditorium storage room in which they were kept overnight after each day's session,
thereby lightening my load on the trip home.
The conference was nominally hosted by
Lev Pontryagin,
the giant of mathematics at the Steklov Institute, and
Ilia N. Vekua,
the amiable Georgian mathematician who was then Rector of Tbilisi State University. Gamkrelidze had been Pontryagin's doctoral student.
Lev Pontryagin
Ilia N. Vekua
Although I was introduced to Pontryagin, I felt far too intimidated to attempt
conversation with him. With Vekua, who was a convivial sort of man, it was another matter.
During a banquet at his home one evening, he told me that Southern California
was one of his favorite places in the world.
After I told him that I had lived in San Diego and L.A. for ten years,
he spent the next hour showing me his color slides of the California landscape
and telling me stories about his visits to California.
Here is the program of the 1969 Tbilisi conference:
page 1
page 2
page 3
page 4
page 5
page 6
page 7
(Because I was invited at the last minute, I was not listed in the program.)
Once I have digitized my stereoscopic Kodachrome slides of Tbilisi and the surrounding countryside, I will post
images here.
After the conference, Revaz Gamkrelidze generously arranged a special visit by four or five of us to a small local research institute, where his brother
Tamaz, who is a distinguished orientalist,
showed us a breathtakingly beautiful treasure that had recently been unearthed in Georgia.
It was a tiny sculpture of a chariot and horses, composed entirely of thin gold wires (perhaps less than 1 mm. in diameter).
I do not remember exactly how old it was estimated to be, but I vaguely recall hearing that it was about 4000 years old.
(If one of my readers has information about this object, please share it with me.)
Gamkrelidze invited those of us who were planning to be in Moscow during the week after the conference to attend
a party at his Moscow apartment. Since I had a Moscow appointment scheduled at just the right time with V.A. Koptsik,
the Lomonosov University specialist in
Shubnikov
—
Belov
color symmetry theory,
I was able to attend the party. Koptsik graciously arranged for the 78 yearold
Nikolai Belov,
who had long since retired, to make a special trip to the university so that I could meet him.
Belov was one of the truly memorable people I met during my two weeks in the USSR.
I regret that I did not have the opportunity to spend more time with him.
In his booming voice and heavily accented English, while loudly thumping his chest, he told me that he was "not a Communist, but a Russian!"
(I thought that he was being somewhat indiscreet, but perhaps he was confident that he was too distinguished for Brezhnev to bother him.)
I shot a stunning pair of stereoscopic photographs of Belov that I am still trying to locate in my cluttered files,
because I would like to post them here. Since I am not a particularly skillful photographer, it is all the more remarkable
that his portrait looks almost as if it had been taken by
Yousuf Karsh.
I first met Prof. Koptsik in 1968 at a geometry conference
at the Ledgemont Laboratory of Kennecott Copper Co. in Massachusetts.
It was organized and hosted by a metallurgical physicist, the late
Arthur Loeb,
whose specialty was crystallography.
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. SchröderTurk have effectively summarized
the state of our understanding in 2012 of the role of triplyperiodic 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, 529538.
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 SchröderTurk, on which these conference papers are based. In my own presentation, entitled
Reflections concerning triplyperiodic minimal surfaces,
I described how I came to be involved in the investigation of minimal surfaces, beginning in 1966.
(There is considerable overlap between some parts of this account and the material on this website.)
I recommend the following authoritative online introductions
to the sprawling subject of minimal surfaces, including triplyperiodic ones:
Hermann Karcher and Konrad Polthier's
"Touching Soap Films, An Introduction to Minimal Surfaces"
Elke Koch and Werner Fischer's
3periodic surfaces without selfintersections
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 online 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 interpenetrating 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 'crosseyed' 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 (bodycentered 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 triplyperiodic minimal surface (TPMS) that is embedded, i.e., free of selfintersections,
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 triplyperiodic
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 triplyperiodic
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 triplyperiodic
quasiregular skew polyhedron (6.4)^{2} (cf. Fig. 1.1b),
which is derived from the CoxeterPetrie regular skew polyhedron {6,44}
(cf. Fig. E1.35c). Quasiregular polyhedra are edgetransitive, but not facetransitive.
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 quarterturn 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 CoxeterPetrie triplyperiodic
regular skew
polyhedron {6,63} (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 {6,63}
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 {6,63}

Dual pairs of diamond graphs are shown below in both 'medium thick' and 'thick' versions. In the thick
version (cf. Figs. E1.3eh), 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 ≥ 2^{1/2}/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 triplyperiodic quasiregular 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. E1.4eh), 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 CoxeterPetrie triplyperiodic regular skew polyhedron {6,44},
which has the same topology and symmetry as
Schwarz's primitive surface P
Fig. E1.4j
{6,44} with embedded [skinny] dual graphs
Fig. E1.4k
{6,44} with embedded [fat] dual graphs

{6,44} is a is an infinite regular polyhedron.
There is only one variety of chamber — a truncated octahedron whose
six square faces have been removed.
Hence it is bounded by eight regular hexagons.
At the center of every chamber there is a vertex of one of the dual skeletal graphs
illustrated in Figs. E1.4ah.


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. E1.5il), 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 ≥ 3^{1/2}/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 'counterclockwise' Laves graph,
skeletal graph of the other labyrinth of the G surface
Fig. E1.10
The enantiomorphic skeletal graphs of the two disjoint labyrinths of the G surface
Fig. E1.11
Orthogonal projection on [100] plane of the enantiomorphic Laves graphs
Fig. E1.12
Another view of the 'counterclockwise' Laves graph
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 TOTD
(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 TOTD existed, it would be a nonbalanced TPMS—
i.e., its two labyrinths would be noncongruent.
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 mirrorsymmetric plane lines of curvature.
In 1974, I tested for the existence of TOTD by using a lasergoniometer method I had devised in 1968
at NASA/ERC. This extremely tedious method requires the construction of a set of several straightedged boundary frames
of various proportions. The laser is used to measure the orientation
of the normal to the surface of a [longlasting] polyoxyethylene soap film S'
bounded by each of these frames at many points
that are as close as possible to the edges of the frame.
Each S' is a candidate for the surface adjoint to S. The adjoint curves computed for
the edges of S demonstrated that it is
impossible to 'kill the periods' and therefore that TOTD does not exist.
In 2001, Ken Brakke used his Surface Evolver program to confirm this conclusion with
enormously greater speed and accuracy than is possible with the soap filmlaser technique.
Fig. E1.13c
The dual pair of skeletal graphs for another
hypothetical but nonexistent embedded TPMS
The green vertices define the sites of the Cu atoms,
and the blue vertices define the sites of the Mg atoms
in the binary alloy Cu_{2}Mg, which has the structure called
Cubic Laves phase C15.

I call the Cu graph FCC_{6}(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 twosheeted
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 infinitelymultiplyconnected,
but it is nevertheless easy to characterize the topological
complexity of every example of such a surface by computing the
genus p of a single lattice fundamental domain.
Except where it is specifically stated to the contrary,
it will be assumed in all that follows that TPMS refers to an
embedded surface,
i.e., one that is free of transverse selfintersections.
Since the smallest posssible value for the genus is three,
Schwarz's P and D surfaces are members of a
very small select group of topologically simplest examples of TPMS.
Below is a recipe for computing the genus of a TPMS.
It is based on one of Gauss 's most astounding discoveries,
the GaussBonnet theorem, which links the topology
and the geometry of a surface.
One can very crudely express the essence of the GaussBonnet
theorem in this context by saying that
the larger the value of the integrated Gaussian curvature for
one lattice fundamental region of the surface,
the steeper the saddlelike surface contours, and —
therefore — the larger the number of tubular 'handles' in the surface
as it 'bends around' this way and that.
On p. 233 of the 13^{th} edition of 'Mathematical Recreations and Essays' by W.W. Rouse Ball and H.S.M. Coxeter,
the authors use Euler's
formula
F − E + V = 2,
which relates the number F of faces, the number E of edges, and the number V of vertices of a convex polyhedron
to prove that adding a handle to an orientable surface reduces the EulerPoincaré characteristic
Χ = 2 − 2p by 2 and therefore increases the genus
p by 1. The proof simply updates the values of F, E, and V
after two different ngons of a map on the surface are joined by a 'bent prism' (which is a convenient device
for representing a handle). F is increased by n − 2, E is increased by n, and V remains unchanged.
Since
X = F − E + V,
Χ is reduced by 2.
Fig. E2.4
Recipe for calculating the genus
of one lattice fundamental domain of a TPMS,
applied to Schwarz's P surface
Another way to calculate the genus is to
substitute for d_{G} from Eq. 5 in the equation p =1− d_{G}.
d_{G} 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, d_{G}= − 2.
The sign of d_{G} for surfaces of negative Gaussian curvature,
like minimal surfaces, is negative because the sense
of a geodesic edgecircuit 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 EulerPoincaré characteristic Χ=2 − 2p,
look here.
For information about the GaussBonnet theorem,
look here.

E3. The DGP 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 Dcatenoid'
wireframe 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 Dcatenoid'.
Fig. E3.1g
Four translation fundamental domains of Schwarz's D surface
Each face is one of the hex_{90}
faces shown below in Fig. E3.1h.

The lattice for D is facecentered 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 nearestneighbor links in the diamond crystal structure.
Fig. E3.1h
The hexagonal face hex_{90} of D is
defined by the Coxeter map {6,44}.
Its face angles are 90º, and its area is
half the area of the hexagonal face
hex_{60} defined by the Coxeter map {6,63},
shown below in Fig. 3.1i.
Fig. E3.1i
The hexagonal face hex_{60} of D is
defined by the Coxeter map {6,63}.
Its face angles are 60º, and its area is
twice the area of the hexagonal face
hex_{90} defined by the Coxeter map {6,44},
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 {6,63}
larger image
The inscribed regular skew hexagon with straight edges
is a face of {6,44},
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, SpringerVerlag, New York, 1965
and in
Infinite Periodic Minimal Surfaces Without SelfIntersections,
NASA TN D5541, 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 mirrorsymmetric
plane lines of curvature (plane geodesics) in the other.
Fig. E3.2a
Stereoscopic image of
the linear asymptotics (blue)
and plane geodesic curves (green)
in the 'square catenoid' of P (cf. Fig. 1.2d, e, f)
Fig. E3.2b
Stereoscopic image of the linear asymptotics
embedded in the 'crossed triangles Pcatenoid'
wireframe of Schwarz's primitive surface P
(cf. Figs. 1.2c, d, e)
Here the triangles are only half as far apart as
the 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 'crossedtriangles 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 halfturn 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 Pcatenoid' 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 junglegymlike soap film.
Any finite portion of such a soap film can be made stable if threads are stretched along a sufficient number of the embedded straight lines ('linear asymptotics').
As a sort of metaphor for the 'pipejoint' module of P in Figs. E3.2d, e, imagine that you are
inside a spherical soap bubble at the center of a cube.
Now deform the bubble by blowing against its interior surface in the six directions
x, −x, y, −y, z, −z simultaneously,
forming six cylindrical tubules attached symmetrically to the inside of the cube
faces at their centers.
P partitions R^{3} into two congruent interpenetrating labyrinths.
The skeletal graph (NASA TN D5541, pp. 3839) of each labyrinth
is the graph of degree 6 whose edges are those of a packing of congruent cubes.
In Figs. E1.4ah below are images of tubular simple cubic graphs shown in both 'medium thick' and 'thick' versions. In the
thick version, the diameter d of the cylindrical tubes is the largest possible, consistent
with the requirement that the dual pair of tubular graphs not intersect.
Intersection occurs when the ratio d/e ≥ 1/2, where e is the edge length of the [thin] skeletal graph.
The three CoxeterPetrie 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.
{6,63} is homeomorphic to D, while
{6,44} and {4,64} are homeomorphic to P.
Regular skew polyhedron {6,63}
(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 {6,44}
(homeomorphic to P)
Fig. E3.2i
Translation fundamental domain
The lattice of the oriented surface is s.c.
Fig. E3.2j
Assembly of twentyseven
translation fundamental domains
Regular skew polyhedron {4,64}
(homeomorphic to P)
Fig. E3.2k
translation fundamental domain
The lattice of the oriented surface is s.c.
Fig. E3.2l
Assembly of twentyseven
translation fundamental domains
For a comprehensive discussion
of the CoxeterPetrie 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 CoxeterPetrie {6,44} map defines the
arrangement of the hex_{90} 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 hex_{90} face is
related to each of six faces with
which it shares an edge by a halfturn about an axis
of type (110) perpendicular to G at the midpoint of
the shared edge.
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 hex_{90} face has been added here at the
top of the piece of G shown in Figs. E3.3a,b.
This orthogonal projection of G onto the (100) plane
shows that the
projected outline S of the spiralling geodesic that bounds each
(100)type tunnel in G is approximately circular.
For a higherresolution 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 highresolution view, look
here.
For a stereoscopic perspective view, look
here.
Fig. E3.3e
The hexagonal tile hex_{90} of G
(front view)
The regular skew hexagon hex_{90}
is a face of the
CoxeterPetrie {6,44} map. Its face angles
are 90º, and its area is onehalf the area
of the hexagonal face hex_{60} defined
by the CoxeterPetrie {6,63} map.
(The hexagon hex_{60} is shown in Figs. 3.3m, n.)
Fig. E3.3f
The hexagonal face hex_{90} of G
(back view)
The front and back surfaces are not the same!
Fig. E3.3g
The hexagonal face hex_{90} of G
(side view)
Fig. E3.3h
This image suggests that hex_{90} of G can be inscribed
in a truncated octahedron, but that is impossible.
Although the vertices of hex_{90}
coincide with six
vertices of the truncated octahedron, its edges are
not plane curves. Each edge approximates the shape
of a quarterpitch of a helix. One half of each edge
lies inside the truncated octahedron, and the other
half lies outside. Alternate edges are curves of
opposite handedness.
G contains one replica of hex_{90}
in three of every four
truncated octahedra in a packing of truncated octahedra.
Fig. E3.3i
The quadrangular tile quad_{60} of G,
a regular skew polygon
(front view)
Fig. E3.3j
The quadrangular tile quad_{60} of G
(back view)
Fig. E3.3k
The quadrangular tile quad_{60} of G
(side view)
The tile quad_{60} is a face of the CoxeterPetrie
{4,64}
map, which is the dual of {6,44}. Its face angles are
60º, and its area is
equal to twothirds that of hex_{90}.
Every quad_{60} face is
related to each of four faces with
which it shares an edge by a halfturn about an axis of
type (110) perpendicular to G at the midpoint of the
shared edge.
Each of these midpoints is also the midpoint of an
edge of a dual hex_{90}
face (cf. text below Fig. E3.3a).
(Fig. E3.3l illustrates a wellknown 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 semiregular skew 12gon composed
of six replicas of quad_{60} of G
Its face angle sequence is
..., 60º, 120º, 60º, 120º,...
Unlike the three regular skew polygons
quad_{60}, hex_{90},
and hex_{60},
this 12gon does not tile G.
For a highresolution view, look
here.
Fig. E3.3n
The hexagonal face hex_{60} of G is defined
by the selfdual Coxeter map {6,63}.
Its face angles are 60º, and its area is
twice the area of the hexagonal face
hex_{90} shown above in Figs. 3.3e,f,g.
Fig. E3.3o
The hexagonal face hex_{60} of G
(side view)
Every hex_{60} tile is
related to each of six faces with
which it shares an edge by a halfturn about an axis
of type (110) perpendicular to G at the midpoint of the
shared edge.
Fig. E3.3p
Assembly of approximately octahedral shape
tiled by the hexagonal faces hex_{90} of G
view: (100) direction
Fig. E3.3q
Another view of the model of G shown in Fig. E3.3p
view: (111) direction
How I got started (19661970)
In the spring of 1966, I accidentally 'discovered' the Schwarz surfaces
P and D and then observed their close connection to the
CoxeterPetrie regular skew polyhedra. In order to represent the symmetry and
combinatorial structure of both the surfaces and their flatfaced relatives, the CoxeterPetrie polyhedra,
I employed the metaphorical device of dual skeletal graphs, which I'll call
g_{1} and g_{2}. These are triplyperiodic graphs regarded
as lying centered in the interiors of the two intertwined labyrinths
of these structures. In the discussion that follows, it is assumed
(although not stated!) that
g_{1} and g_{2} are either directly or oppositely congruent.
If this restriction is dropped, the assumption (stated below) that the two labyrinths
each contain exactly half of space in their interiors must also be dropped.
Now imagine that every edge of g_{1}
is replaced by an infinitely thin hollow tube with walls composed of some soapfilmlike material,
and that the space inside the entire connected network of these tubes defines a single
hollow — but shrunken — labyrinth t_{1}. (Assume that t_{1} has no selfintersections.
Remember: this is a metaphorical concept, not
a rigorous mathematical construction.)
Here is how I described the relation between g_{1} and g_{2},
the two skeletal graphs of a TPMS in 1970, on p. 79 of
Infinite Periodic Minimal Surfaces Without SelfIntersections:
"Assume that the skeletal graph is given for one labyrinth of a given intersectionfree TPMS.
Let each edge of the skeletal graph be replaced by a thin open tube, and let these tubes be smoothly
joined (without [self]intersections) around each vertex
so that the whole tubular graph forms a single infinitely multiplyconnected surface, which contains the
skeletal graph in its interior. Such a tubular graph is globally homeomorphic to the corresponding minimal surface.
If the tubular graph is sufficiently "inflated", it becomes deformed into a dual tubular graph which contains in its interior
the skeletal graph of the other labyrinth of the surface. The "outside" of the first tubular graph is the "inside" of the second tubular graph.
The two tubular graphs of a given TPMS are required to have the same space group as the TPMS, and to correspond, respectively, to two
tubular graphs which are globally homeomorphic to the TPMS."
Now imagine inflating t_{1} so that at its summit,
(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 g_{1} and g_{2}, and
(iv) it has no selfintersections.
At the inflation summit, the surface t_{1} is an embedded TPMS.
Until it reaches the summit, t_{1} exhibits the symmetry of g_{1}.
At the summit, it has the symmetry of the union of the [intertwined] g_{1} and g_{2}.
As the inflation proceeds beyond the summit, the surface exhibits the symmetry of g_{2}.
It eventually shrinks down to the thin tubular graph t_{2}, which has the symmetry of g_{2}.
Curiously, the outside of t_{1} is transformed into the inside of t_{2}.
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
18942000
In 1967, I had the rare privilege of becoming acquainted with Dirk Struik.
We both enjoyed hiking along the nature trails in Concord near my home.
When Prof. Struik was more than 100 years old, I attended his lecture
on the history of mathematics at an AMS meeting in Cincinnati.
He was in tiptop form.
who had shown him a draft copy of my
Infinite Periodic Minimal Surfaces Without SelfIntersections.
Dennis invited me to his office to explain
that the transformation of the tubular graphs g_{1} and g_{2} is an example of the classical
AlexanderPontryagin duality
(which I had never heard of before).
For the Schwarz surface P, the skeletal graphs
g_{1} and g_{2} are identical to the 6valent simple cubic graph
defined by the edges of an ordinary packing of cubes.
For the Schwarz surface D, both g_{1} and g_{2} are copies of the 4valent diamond graph,
whose edges correspond to the nearest neighbor links in the diamond crystal structure.
Both of these graphs are symmetric, i.e.,
there is a group of symmetries that is transitive on all the edges and all the vertices.
I knew of only one other example of a symmetric triplyperiodic graph on a cubic lattice — the Laves graph.
In the spring of 1966, I was seized by the notion that there must exist a TPMS with
[enantiomorphic] Laves graphs for its skeletal graphs.
I called it the Laves surface L.
Unlike the skeletal graphs of P and D, however, the configuration of
two dual Laves graphs has no reflection symmetries, and
its axes of rotational symmetry lie in directions that I determined could not possibly correspond to lines embedded in the surface
I was seeking.
As a consequence, I had no idea how to generate a surface patch bounded by either straight line segments or plane geodesics
a ("Schwarz chain").
By the time I moved to NASA in July 1967, I had made a reasonably thorough study of Schwarz's writings on periodic minimal surfaces,
and I understood the Bonnet associate surface transformation that defines the relation between P and D
(and also the relation between the catenoid and the helicoid).
The brightly colored plastic models of P and D I had constructed were almost literally screaming out to me
that I should explore the territory between these two surfaces (where one surface is bent continuously into the shape of the other),
but I did not hear their screams!
In February 1968, I stumbled accidentally on a very close approximation of the gyroid. I'll call it the pseudogyroid.
The models illustrated in Figs. E1.2k, E1.2l, and E1.17 show the final steps in the procedure that led to this pseudogyroid.
The resemblance between this virtual doppelgänger and the true gyroid is so close
that with the naked eye it is impossible to tell them apart.
I still had no idea yet that the gyroid is just a surface associate to P and D that happens to be free of selfintersections.
In those days there were not yet any known examples of embedded TPMS derived by examining intermediate stages of the 'morphing'
transformation that bends one minimal surface into its adjoint surface
via the associate surface transformation,
and I didn't have the imagination to think of that possibility.
In 1990, it occurred to
Sven Lidin and Stefan Larsson
to look for an embedded surface among the surfaces associate to
Schwarz's [embedded] surface H
and its selfintersecting adjoint surface,
and they found exactly one — 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 vertextransitive and edgetransitive).
In H and CLP, the two other examples of Schwarz's TPMS, which — like P and D — are of genus 3,
the skeletal graphs are merely regular and not symmetric.
My intuition suggested that a symmetric graph is so homogeneous that it is very likely to be the skeletal graph of
a labyrinth of some embedded TPMS. (I eventually discovered that although some of the few known examples of symmetric
triplyperiodic graphs are skeletal graphs of labyrinths of such surfaces, by no means all of them are.)
During the spring and summer of 1968, I concentrated on the writing of
a socalled preliminary report (an internal NASA document, not intended for general circulation),
entitled "ExpansionCollapse Transformations on Infinite Periodic Graphs",
NASA/Electronics Research Center Technical Note PM98 (September 1969),
draft versions of two patent applications, and
computer graphics animations of collapsing graphs.
The considerably less timeconsuming one of the two patent drafts was eventually entitled,
"Honeycomb Panels Formed of Minimal Surface Periodic Tubule Layers".
I had discovered no useful ideas about how to prove that the pseudogyroid (cf. Fig. E1.17)
was the basis for a bona fide minimal surface.
Blaine Lawson told me in early August that he too had made no progress toward figuring out how to
prove that a skew hexagonal face of the pseudogyroid, with its strictly helical edges,
could somehow be analytically continued to generate an embedded periodic minimal surface.
But ever since my first phone conversation with Blaine in late spring, I had found it extremely helpful to
discuss with him a variety of questions concerning minimal surfaces other than the gyroid. I used him
as a sounding board on some of my still tentative ideas about how to derive new examples of embedded TPMS
by
(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 spacefiller 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 pseudogyroid model shown in Fig. E1.17 to illustrate my 15minute talk (cf. Fig. E2.10).
I was still calling the surface the 'Laves surface' in those days.
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 spaceframes.
I had investigated the transformation for graphs associated with
the P, G, and D surfaces (cf. Figs. E2.68b.0, E2.68b.1, and E2.68b.2, for example).
I described what I called 'just a coincidence' (or words to that effect): that
the trajectory of every graph vertex is an ellipse not only in the associate surface transformation of Bonnet
but also in the totally unrelated graph collapse transformation.
I emphasized that there is no fundamental connection between these two transformations.
I described how I had found that of the triplyperiodic graphs that are associated with
P and D, either as embedded graphs or skeletal graphs, those that have reflection symmetries
are not candidates for expandable spaceframes
because of pairwise collisions of edges (called webs or struts in the spaceframe) that occur early in the collapse.
In contrast to this behavior, for all of the twisted graphs
derived from the pseudogyroid, including the Laves graph, no such collisons of edges occur. The only collisions are
the ones that would occur in actual physical spaceframes, in which struts collide somewhat before the 'complete collapse' stage
because of their finite thickness.
I had not previously even mentioned graph collapse to Blaine, and it's hardly surprising that he
didn't seem to understand the details of what I said to him.
It was obvious that I hadn't explained the elliptical trajectories coincidence very well, because Blaine's response was something like:
"Are you saying that the gyroid is associate to Schwarz's P and D surfaces?"
I hadn't said that at all, but it hardly mattered, because
at that instant, everything suddenly fell into place. The fog had finally lifted!
Thanks to Blaine's question, I finally understood that
the gyroid is just a surface associate to P and D that
happens to be embedded (free of selfintersections). It is the only such surface,
as I was soon able to confirm by means of simple 'morphing' sketches similar to the computer drawings in Fig. E1.21.
Because I had spent the summer analysing the details of graph collapse
transformations on P, D, and G,
I was aware that the surface orientation at the vertices of
the hexagonal faces of the CoxeterPetrie map {6,44} on the pseudogyroid
is identical to the surface orientation
at the corresponding vertices of P and D.
That was a powerful hint pointing to the
Bonnet transformation that had been 'staring me in the face' every day
since March, when I assembled my first plastic model of the gyroid.
I was hugely embarrassed, realizing how obvious it should have been to me
that the gyroid is associate to P and D!
After all, I was familiar with the properties of the Bonnet transformation.
I had long since traced out the geometrical relation between the
equatorial circle in the catenoid and the central axis of the helicoid,
which I had found illustrated in
Dirk Struik's marvellous
Lectures on Classical Differential Geometry.
I had also sketched the corresponding curves in P and D
countless times.
Those relations should have been the clue.
I had also spent days studying not only H. A. Schwarz's Collected Works, but also
Erwin Kreyszig's
Differential Geometry,
Luther Pfahler Eisenhart's
A Treatise on the Differential Geometry of Curves and Surfaces,
and
Barrett O'Neill's
Elementary Differential Geometry.
Although I knew from experience that ideas that should be obvious
are sometimes anything but obvious, I nevertheless felt stupid when I
realized that I had posed the wrong question to Bob Osserman back in March.
when I asked him whether there might be a way to derive the Weierstrass
parametrization for a Schwarz chain composed of six helical arcs.
I had mistakenly assumed that the edges of the hexagonal faces of
the {6,44} map on the gyroid were perfect helices.
Immediately after Jim Wixson joined NASA/ERC in January 1968,
I asked him to write a FORTRAN program for
calculating — from Schwarz's equations — the coordinates of
a set of closely spaced points on
the equatorial geodesic of the 'square catenoid'
in P. A simple soapfilm demonstration suggests that
although this curve appears to be approximately circular, it cannot
be a circle. Consider its shape in the limit of very small
separation of the boundary squares
of the 'square catenoid'. In that limit it can
be described roughly as a square with slightly rounded corners.
As the separation of the boundary squares is increased,
the curve looks more and more like a circle, but I found it impossible
to imagine that it becomes exactly circular when the separation
becomes equal to its value in the P surface.
When I plotted the points computed by Jim, I found that the equatorial geodesic
departs from perfectly circular shape by slightly less than 0.5%.
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 quasihelical image S_{G} in G
of the equatorial geodesic S_{P} in P.
I found that this projection of S_{G}
(cf. Fig. E3.2b) also departs from perfectly circular shape by
slightly less than 0.5%.
Fig. E3.2a
Stereoscopic view of the linear asymptotics (blue) and plane geodesic curves (green)
in the 'square catenoid' of P (cf. Fig. 1.2d, e, f)
Fig. E3.2b
Orthogonal projection on the [100] plane
of the quasihelical geodesic S_{G} in G
(cf. Fig. E3.3c)
(I didn't learn about
Björling's Strip Theorem
until several years after I left NASA/ERC. This theorem proves very simply
that the equatorial geodesic S_{P} in Schwarz's P surface
cannot be circular.)
I felt only slightly less stupid when I discovered that
Blaine's response to my harangue about
the elliptical trajectories of the vertices of collapsing graphs
was not actually the result of his understanding that the
gyroid was associate to P and D.
He had been justifiably confused by my rambling description of those irrelevant elliptical trajectories.
When I explained to him the evidence for the associate surface relationship,
he agreed that it was a reasonable idea.
I immediately proposed that we publish together an announcement about the gyroid.
He courteously refused, explaining that his crucial question to me was prompted by a misunderstanding of what I was saying.
But I insisted that if he had not asked me that question in precisely those words, it would have been impossible to say how long
it might have taken me to understand what was going on.
He then reluctantly agreed to collaborate on a paper about the gyroid.
Two days later, Dr. Van Atta returned to ERC from an outoftown trip.
He had been following my struggles with the pseudogyroid for months.
As soon as I told him my exciting news,
including my plan to copublish with Blaine Lawson,
he scolded me in no uncertain terms! He insisted that I phone Blaine and explain that I had made a serious error, and that I must publish alone.
(It was the only time Dr. Van Atta ever displayed anger or impatience toward me.)
Blaine was courteous when I relayed my new message to him, but I realized that my vacillation must have offended him.
Gradually I succeeded in feeling very slightly less stupid than I had at first, after reflecting on the fact that
neither Schwarz, Riemanm, Weierstrass,
nor any of their successors seem to have suspected the existence of an embedded surface associate to P and D,
in spite of the fact that they were all experts on the Bonnet transformation.
On the other hand, I realized that it had been pure dumb luck for me to stumble onto
M_{4} and M_{6}, the precursors of the gyroid.
I was able to derive the angle of associativity (cf. Fig. E1.23) easily,
because I had already made a detailed study of the geometrical calculations Schwarz carried out
in his analysis of the P and D surfaces.
In January 1968, because I was curious about the precise shape and arc length of the
quasicircular edges of a {6,44} hexagon of P
(cf. Fig. E1.2c), I sketched the outline of a computer program for getting answers to these questions.
My colleague Jim Wixson coded the program in FORTRAN and ran it on ERC's PDP11 minicomputer.
The output of Jim's program, combined with Schwarz's analysis,
provided the required clues to the value of the angle of
associativity of G (cf. Fig. E1.23).
These results demonstrated that the departure from perfect
circularity of the quasicircular holes in a pipejoint unit cell of P
is in the range of approximately ± 0.5% of the hole's mean radius,
implying a comparably small departure
from perfect helicity of their image curves in the gyroid.
Not only did the quasihelical curves in the pseudogyroid (cf. Fig. E1.17) turn out to be
very close approximations to the corresponding curves in the true gyroid,
but these curves are also extremely close to — but not quite the same as —
the corresponding curves in the 'levelset' gyroid (cf. Fig. E1.31.)
In late 1968, I decided that I must somehow force a nodal polyhedron for BCC_{6} into being,
and by trialanderror I produced the
spacefilling saddle polyhedron shown In Figs. E2.70a, b, and c.
It is described in
Infinite Periodic Minimal Surfaces Without SelfIntersections.
Fig. E2.70a
BCC_{6} Pinwheel polyhedron:
the 6faced nodal polyhedron
of
the deficient symmetric graph BCC_{6} of degree 6
(stereo pair)
The vertices of the graph are the complete set of vertices of the b.c.c. lattice.
BCC_{6} is described on p. 82 of
Infinite Periodic Minimal Surfaces Without SelfIntersections.
Fig. E2.70b
An oblique view of the BCC_{6} Pinwheel polyhedron
(stereo pair)
Fig. E2.70c
The BCC_{6} graph (orange) and its dual graph (black)
(stereo pair)
The edges of the black graph are the edges
of the BCC_{6} 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 BCC_{6} graph.)
Fig. E2.7
The six components of the brass tool, shown before assembly
and final machining, that I designed for vacuumforming the plastic
hexagons of the 1968 model of the pseudogyroid 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 halfcube, 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
Mirrorsymmetric 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 65830 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 pseudogyroid (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 pseudogyroid to
Bob Osserman,
who passed it along to his PhD student
Blaine Lawson,
Blaine promised to think about the problem in his spare time, even though he was already fully occupied with his dissertation research.
It wasn't until about ten days after the Madison meeting that at last I
understood — thanks to a fruitful phone conversation with Blaine —
that the gyroid is an associate surface in the
Schwarz P−D family.
(The details are described just after Fig. E2.68c.9.)
By means of drawings based on hand calculations, I confirmed that there are no other intersectionfree
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 selfintersecting.
Twentyeight years later, this claim was at last proved rigorously by
Karsten GrosseBrauckmann 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, 499523.
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 onequarter pitches of circular helices,
alternately righthanded and lefthanded (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 infinitelyconnected 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 M_{4} and M_{6}.
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
M_{4} and M_{6} are shown in Figs. E1.19 and E1.20.
Fig. 1.16a
Fig. 1.16b
M_{4}
M_{6}
view: [111] direction
As the result of a lucky guess about how to remove the 'wrinkles' in M6, I produced the pseudogyroid,
which is shown in Fig. E1.17. The details of how I got from the
pseudogyroid to the gyroid are described below following Fig. E1.2p.
Fig.E1.17
The pseudogyroid
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 GrosseBrauckmann and Meinhard Wohlgemuth published a rigorous proof that
the gyroid is embedded (free of selfintersections) and contains neither straight lines nor reflection symmetries, in
The gyroid is embedded and has constant mean curvature companions, Calc. Var. 4, 1996, 499523.
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 selfassembled 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 pseudogyroid is a bona fide minimal surface.
(It is remarkably close to one!)
The actual gyroid G is described on pp. 4854 of
Infinite Periodic Minimal Surfaces Without SelfIntersections,
NASA TN D5541 (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 (WeierstrassEnneper parametrization)
for the entire family of surfaces associate to P and D
If θ_{G} in the term e^{iθG} is replaced by zero,
the coordinates are those of D.
If θ_{G} is replaced by π/2, the coordinates are those of P — the adjoint of D.
(The value given above for θ_{G} agrees up to eight significant figures
with the value derived from a more recent analysis
based on a SchwarzChristoffel mapping,
in
Adam Weyhaupt's 2006 PhD thesis (pp. 115116).
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)
The surfaces at all other angles between 0º and 90º are selfintersecting.
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 PDP11 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 timelapse 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 highresolution 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 {6,63}.
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 triplyperiodic minimal surface a trivial geodesic
if it is either a straight line or a mirrorsymmetric plane line of curvature, and a nontrivial geodesic otherwise.
Since there are neither straight lines nor mirrorsymmetric plane lines of curvature in the gyroid,
all of its geodesic curves are nontrivial.
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 nontrivial geodesics,
both closed and unbounded, on these three surfaces.
For a discussion of geodesics on multiplyconnected 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 IWP (cf. also Fig. 2.5) and FRD (cf. also Fig. 2.6) are the
adjoints of two of the four selfintersecting Schoenflies surfaces (cf. Fig. E2.1).
In 1975, I used the lasersoap 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 FRD(r).
Several years ago, Ken used his Surface Evolver program to obtain the vastly improved modeling of FRD(r) shown below.
Soon afterward I asked Ken to use Surface Evolver to obtain the hexagonal patch of IWP(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 oneeighthturn (45º) about a vertical axis, Ken included '(r)' in its name.
Fig. E1.15d
Fig. E1.15e
FRD
FRD(r)
Fig. E1.15f
Fig. E1.15g
IWP
IWP(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. E1.18ad (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
Highresolution photos of the Exploratorium gyroid
(also courtesy of Amy Snyder) are at:
E1.18a
E1.18b
E1.18c
E1.18d
Figs. E1.19ae show M_{6}, which was the immediate precursor of the gyroid (cf.
abstract in Fig. E2.10). The vertices coincide with the vertices of the
CoxeterPetrie map {6,44}
on the gyroid, but the
edges of the hexagonal faces are line segments, not quasihelical arcs as in the gyroid.
Figs. E1.20ae show M_{4}, the immediate precursor of M_{6}.
Fig. E1.19a
65 hexagons of M_{6}
view: [111] direction, silhouetted by bright summer sky backlighting
Fig. E1.19b
65 hexagons of M_{6} (stereo)
view: [100] direction
Fig. E1.19c
65 hexagons of M_{6}
view: [100] direction, silhouetted by bright summer sky backlighting
Fig. E1.19d
65 hexagons of M_{6}
view: [111] direction
Fig. E1.19e
65 hexagons of M_{6}
view: [100] direction
Fig. E1.20a
30 quadrangles of M_{4} (stereo)
view: [111] direction
Fig. E1.20b
30 quadrangles of M_{4} (stereo)
view: [110] direction
Fig. E1.20c
30 quadrangles of M_{4} (stereo)
view: [110] direction
Fig. E1.20d
The edges of the model in Fig. E1.20c,
which define a portion of the 'deficient' graph BCC_{6}
view: close to [100] direction
Fig. E1.20e
30 quadrangles of M_{4}
view: [111] direction, backlit by summer sky
Fig. E1.20f
30 quadrangles of M_{4}
view: [100] direction, backlit by summer sky
Cubic unit cell of G
Fig. E1.22
Crosseyed 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 bodycentered 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 onethird of the way (cf. Fig. E1.23) through the transformation.
A simplyconnected 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 {6,44} (cf. Fig. E1.4).
The lattices for the three [oriented] surfaces are
facecentered cubic (f.c.c) for D,
bodycentered cubic (b.c.c) for G, and
simple cubic s.c.) for P.

Stereo view of G
Fig. E1.28
Crosseyed stereogram of a larger piece of G
A hexagonal face of G
Fig. E1.29
Crosseyed stereogram of a hexagonal face of G that corresponds to
a face of the regular map with holes {6,44}
{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
Crosseyed 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 SchröderTurk 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 "levelset" 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 quasihelical 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 quasihelical tunnels.
Adjacent tunnels spiral alternately CW and CCW.

(h) oblique view of the assembly shown in (g)
The quasihelical 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 quarterperiod 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.

Flatfaced 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 triplyperiodic 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
VG_{17} 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 {4,64} on the gyroid surface'.
VG_{17} is the enantiomorphic Voronoi cell
(cf.
Fig. II2a, p. 89 in
NASA TN D5541) whose two mirrorsymmetric 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 triplyperiodic 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, VG_{17} 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 VG_{17} 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 BCC_{6}, which is the graph with the
edges and vertices of M_{4}
(cf. Fig. E1.20e).
Fig. E1.32 shows a set of screencaptured 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.
VG_{17}, which appears at the midpoint of the sequence,
is the example labeled '10:32'.
Each polyhedron that precedes VG_{17} 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 quarterturn.
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 triplyperiodic 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 triplyperiodic 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 collisonfree 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 higherresolution [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 higherresolution version of the above image, look
here.)
This image demonstrates that the gyroid is symmetrical by a halfturn 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 higherresolution [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 CoxeterPetrie regular
skew polyhedra {6,44}, {4,64}, and {6,63}, whose faces are
regular plane polygons,
My first model of the gyroid
Fig. E1.4i
A Voronoi surrogate of P — the CoxeterPetrie
triplyperiodic regular skew polyhedron {6,44},
which is homologous to P.
And here is one D surrogate:
Fig. E1.2c
A Voronoi surrogate of D — the CoxeterPetrie
triplyperiodic regular skew polyhedron {6,63},
which is homologous to D.
After I create an image of {4,64}, I will post it here.
(cf.
Wikipedia's
animated images of the three
CoxeterPetrie regular skew polyhedra.)
Both {6,44} and its dual {4,64} are homologous to
P, and the selfdual {6,63} is homologous to D. In 1968 I
derived an example of a triplyperiodic polyhedron with plane faces that
is homologous to G — Voronoi G (cf.
Figs. E1.34cg). It was known from the work of Coxeter and
Petrie that there exists no regular triplyperiodic skew polyhedron
with plane faces that is homologous to G.
I wondered, however, whether there exists a
uniform triplyperiodic 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 quasiregular 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 quasiregular 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 quasiregular 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 musnub 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
quasiregular tessellations of the {6,4} family
(abstract published by the American Mathematical Society in 1969)
Fig. E1.35d (below) shows AMS Abstract 65830 (1968),
cited in the first line of the Abstract in Fig. E1.35c (above).
Fig. E1.35d
Abstract 65830, 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 (crosseyed 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 (414442) 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 halfturn 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 selfintersections.
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 S_{1} and S_{2}
that if a boundary edge in S_{1} (say) is a straight line segment E_{1}, then its image in S_{2}
is a segment of a plane line of curvature C_{1}, lying in a plane perpendicular to E_{1}.
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 C_{9}(P) —
or just C(P) —
because it has exactly the same embedded straight lines as P and has genus 9.
(The 'C' in C_{9}(P) stands for 'complement'.)
C_{9}(P) is illustrated in Figs. E2.2a,b,c, and
its selfintersecting adjoint surface C_{9}(P)† is illustrated in Fig. E2.2d.
Edvard Neovius (18511917), 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 C_{9}(P)
and its selfintersecting adjoint surface C_{9}(P)†
(a)
(b)
Figs. E2.2a and b
One lattice fundamental region of
the embedded Neovius surface C_{9}(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 C_{9}(P)
and
(d) an assembly of seven lattice fundamental regions of its selfintersecting adjoint surface
C_{9}(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 C_{9}(P)
The 'seethrough' tunnels are aligned in [110] directions.
Fig. E2.3b
A later model of C_{9}(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 65830" 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 interlibrary 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
C_{19}(D) (genus 19)
Right: One translation fundamental domain
Left: Onefourth of a translation fundamental domain
C_{19}(D) is called the [firstorder] complement of D,
because it has exactly the same embedded straight lines as D.
Ken Brakke's images of a sequence of
higherorder complements of C_{19}(D)
of genus 35, 51, 67, ..., which belong to two families, A and B,
are shown
Fig. E2.4c
C_{19}(D)
Two translation fundamental domains
Fig. E2.4d
C_{19}(D)
Onefourth 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 C_{19}(D) shown in Fig. E2.4b
view: [111] direction
Fig. E2.4f
C(D)
view: [111] 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 triplyperiodic surfaces derived from Schoenflies IV and from its adjoint
are obviously both selfintersecting (both of them have a 120º corner and
therefore a branchpoint), 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 vacuumformed plastic sheet material, of these soap films.
I chose the name FRD for the adjoint of V (which has genus 6)
and the name IWP for the adjoint of VI (which has genus 4).
(My naming conventions are explained in
Infinite
Periodic Minimal Surfaces Without SelfIntersections).
Andrew Fogden was the first to confirm that FRD is embedded,
in a 1992 article entitled
"A systematic method for parametrizing periodic minimal surfaces: the
FRD surface", Journal de Physique 2 (1992) 233239. Fogden succeeded in obtaining
the Weierstrass polynomial for FRD, a task that was begun in 1934, but never completed,
by Berthold Stessmann (cf. Stessmann B., Periodische Minimalflächen,
Mathematische Zeitschrift 33 (1934) 417442).
In a 1992 collaboration between the versatile mathematician Djurdje Cvijovic and the
Cambridge University chemist Jacek Klinowski, the authenticity of IWP
was rigorously established and its several unusual properties were derived (Cvijovic, D. and Klinowski, J.: The computation of the triply periodic lWP
minimal surface, Chemical Physics Letters 226 (1994) 9399).
FRD and IWP were the first identified examples of TPMS in which the two
labyrinths of the surface are noncongruent, 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 noncongruent.
Elke Koch and Werner Fischer
have classified such surfaces as nonbalanced and surfaces that contain
straight lines — like P and D — as balanced.
Because IWP and FRD contain no straight lines, they are called nonbalanced.
They are illustrated in Figs. E2.5 and E2.6.
IWP
Fig. E2.5a
IWP (genus 4)
oblique view
Fig. E2.5b
IWP
Modeled by
oblique view
Fig. E2.5c
IWP
view: [111] direction (approximately)
Fig. E2.5d
IWP
view: [100] direction
Portions of two nontrivial geodesics (cf. discussion below Fig. E1.24) are shown.
The one is at the left is closed, and the triplyperiodic one at the bottom is unbounded.
Fig. E2.5e
IWP
view: [100] direction
Fig. E2.5f
IWP
view: [100] direction
Fig. E2.5g
IWP
oblique view
FRD
Fig. E2.6a
FRD (genus 6)
view: [111] direction
(See my 1997 FRD
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 FRD. 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
FRD
view: [111] direction
Fig. E2.6c
FRD
view: [110] direction
Fig. E2.6d
FRD
view: [100] direction
Fig. E2.6e
FRD translation fundamental domain
view: [111] direction
Fig. E2.6f
FRD
Onehalf of a translation fundamental domain
Fig. E2.6g
FRD
One translation fundamental domain (stereo view)
Fig. E2.6h
FRD
Oneandahalf 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!"
19531957
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 AgCd and AgIn, using experimental techniques developed
by Dave, his postdocs, 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 selfdiffusion 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 selfdiffusion 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 selfdiffusion.
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
D_{vacancy} = a^{2} Γ / 6. (E2.11)
Hering showed that for the correlated random walk of an atom in a homogeneous crystal
in which the atomvacancy jump vector has at least twofold rotational symmetry,
the diffusion coefficient for the atom is given by
D_{atom} = f a^{2} 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,
BardeenHering correlation would reduce both the selfdiffusion
coefficient and the isotope effect for selfdiffusion
by the same fractional amount.
This turned out to be the case.
Let D_{α} and D_{β} be the selfdiffusion coefficients of isotopes α and
β of mass m_{α} and m_{β} and correlation factors f_{α} and f_{β},
respectively. I defined the
isotope effect =
((D_{β} / D_{α}) − 1)
/ ((m_{α} / m_{β})^{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 f_{β}.
If this conjecture were correct, one could distinguish between interstitial selfdiffusion and selfdiffusion by the vacancy mechanism
simply by measuring the isotope effect for selfdiffusion.
I first estimated the influence of BardeenHering 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.
19581964
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 subcrystals of successively larger volumes, each
centered at the initial site of the diffusing atom.
Fig. E2.14
The four cubically symmetrical subcrystals in my
Fortran program for the random walk of a vacancy
The top row lists the number of atomic sites in each subcrystal.
The vacancy is located at the center of each subcrystal.
The four subcrystals contain
nearest neighbors,
2^{nd} nearest neighbors,
3^{rd} nearest neighbors,
and
4^{th} 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 subcrystal.
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 selfdiffusion 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 LidiardLeClaire 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. 899906).
Junjiro Kanamori
In 1960, I wrote the the BardeenHering 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 2dimensional and three of them 3dimensional.
We reduced the correlation factors for the three 3dimensional 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 facecentered cubic structure,
for which Hering had obtained a value of 0.78.
Fig. E2.15a
Computed values of BardeenHering 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

bodycentered cubic

8

1 − Γ^{4}(1/4)/8π^{3} − 8π/Γ^{4}(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) 'vacancyexchange' 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
CoxeterPetrie [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 ballandstick 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 edgetransitive and vertextransitive.
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 selfdiffusion occurring by means of atomvacancy exchanges.
As an additional part of the fantasy, I imagined measuring the isotope effect for selfdiffusion in this hypothetical crystal.
I believe that the BardeenHering 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 onehalf, 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 onehalf.
But I realized that the fantasy was farfetched, because the interstices in
this hypothetical crystal would be so large that
selfdiffusion 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 ZnS (zinc sulfide), or InSb (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 selfdiffusion would provide evidence
for or against the hypothesis that selfdiffusion occurs by the vacancy mechanism.
Below is a stereo pair of recent photos of an ancient model of the two interpenetrating 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 'K_{4} crystal' (i.e., the Laves graph) emerges as the unique mathematical twin of the diamond crystal.
He proves that diamond and K_{4} are the only threedimensional crystals with the property he calls strong isotropy,
and also that the honeycomb (cf. graphene) is the only twodimensional 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 interpenetrating pairs of triplyperiodic 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'seye 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 vicepresident 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 spaceframes, based on the geometry of spacefilling tetragonal disphenoids.
Each column, which was clad in aluminum and stored in a flat collapsed configuration, was designed to be selfdeployed 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 threedimensional 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 manmade periodic structures.
He showed me many ballandstck 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 motordriven 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 triplyperiodic 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 triplyperiodic graph,
I did hope to find one that would work at least for every symmetric graph — a graph
which is both edgetransitive and vertex transitive, i.e., a graph in which all vertices are
symmetrically equivalent and all edges are symmetrically equivalent.
As explained in pp. 7685 of
Infinite Periodic Minimal Surfaces Without SelfIntersections),
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. E2.50E2.53 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 bodycentered 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 g_{interstitial}, 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 selfdual, 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 R^{3}. 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 spacefilling 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 triplyperiodic 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 FCC_{6}(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.

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

FCC_{6}(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.
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 selfdual.)
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 16face Voronoi cell for the vertices of the diamond graph
For a sharper [pdf] version of this image, look here.
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 IWP
(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 twentyfour edges (black) of ERO
For a sharper [pdf] version of this image, look
here.
Fig. E2.30
The 14face 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.
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 triplyperiodic 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.
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
dual graph

a nonsymmetric 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 FCC_{6}(I) graph (Z=6),
a locallycentered deficient symmetric graph (LCDSG)
The vertices of the FCC_{6}(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.
FCC_{6}(I) is described on pp. 4748 of
Infinite Periodic Minimal Surfaces Without SelfIntersections.
Fig. E2.47
Four vertices' worth of the FCC_{6}(I) graph
inscribed on the surface of the polyhedron VP_{diamond}
(Voronoi polyhedron for the diamond crystal structure)
Fig. E2.48
Five VP_{diamond}s' worth of the FCC_{6}(I) graph
Fig. E2.49
Five VP_{diamond}s' worth of the FCC_{6}(I) graph
Fig. E2.50
Doubly expanded tetrahedron ('DET'),
the interstitial polyhedron of the FCC_{6}(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 twentyfour 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 FCC_{6}(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 FCC_{6}(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
doublyruled surface.
Although it resembles the actual minimal surface, it is only an approximation.)
Below are four views of a portion of the compound [selfintersecting] 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
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 FRD
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
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 halfturn 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 halfturns produces
a single smooth, embedded infinite labyrinthine surface with the global topology and symmetry of a
CoxeterPetrie 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
CohnVossen's
'Geometry and the Imagination';
the authors write that
In this way, Neovius^{4} succeeded in constructing a minimal surface that extends over the entire space
without singularity or selfintersection and has the same symmetry as the diamond lattice (italics added).
^{4}E. 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 triplyperiodic 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 triplyperiodic surface of nonzero 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.), facecentered cubic (f.c.c.), and bodycentered 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
triplyperiodic 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 triplyperiodic graphs of cubic symmetry that are selfdual.
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 illdefined.
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
triplyperiodic graphs and saddle polyhedra.
On evenings and weekends throughout the spring and summer of 1966, I used a toy
vacuumforming machine and homemade 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 drawforming— pushing a tool
in the shape of a skew polygon outline against a transparent vinyl sheet
that had been softened by heating. I preferred vacuumforming 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 triplyperiodic 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 locallycentered, 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 locallycentered subsets of edges that contains at least three edges — that it is
impossible to construct a locallycentered 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 bodycentered cubic lattice.
The only example of a LCDSG that I examined in 1966 was a graph of degree six I call FCC_{6}(I)
(cf. Fgs. E2.45  E2.55b).
Its vertices are those of the facecentered 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 FCC_{6}(I) by removing a symmetrical set of six outofplane edges from each vertex,
leaving behind a flat sixedge cluster that occurs in each of the four possible [111] orientations. Although FCC_{6}(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 FCC_{6}(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 spacefilling 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 halfcube
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 triplyperiodic 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 spacefilling 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 nonunary 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 pseudogyroid, which is composed of hexagons with perfectly helical boundary curves.
Recall that FCC_{6}(I) is a locallycentered 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 FCC_{6}(I):
a locallycentered deficient graph on the vertices of the b.c.c. lattice.
(It's easy to prove that no locallycentered
deficient graph on the vertices of the s.c. lattice exists.)
Once I started looking, it didn't take me long to discover BCC_{6}, 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.
BCC_{6} proved to be the longsought counterexample to my 'duality rule'.
Its edges are those of the infinite regular warped polyhedron ('IRWP') that
I call M_{4} (cf. Fig. E1.16a and E1.20a to E1.20f).
In the case of BCC_{6}, 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 — M_{4} — appeared.
I shed no tears over this failure of the duality rule, because the twolabyrinth character of M_{4} 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 M_{4} were enantiomorphic Laves graphs.
This was potentially exciting, because it suggested that M_{4}
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 SelfIntersections
explains why I constructed M_{6} (in February 1968).
M_{6} was the result of an attempt to improve on its predecessor, M_{4}
(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 M_{4}.
The hexagons in M_{6} are the duals of the quadrangles in M_{4}.
In M_{4}, the dihedral angle between adjacent faces is 60°.
In M_{6}, it is only ~44.4°.
I hoped that this small reduction would enable
M_{6} to look at least a little more like a continuous minimal surface than M_{4} did.
As soon as I had constructed the model of M_{6} 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 M_{6} 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 pseudogyroid 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 CoxeterPetrie
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 pseudogyroid is a minimal surface.
Instead I sent a physical model of the pseudogyroid 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 nonselfintersecting 'infinite regular warped polyhedra' (and 'infinite
quasiregular 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 2dimensional 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 spaceframe 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 7^{min}00^{sec} 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. 8688 of
Infinite Periodic Minimal Surfaces Without SelfIntersections.
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 onequarter
or threequarters 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 spaceframe, 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 CoxetriePetrie 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
Onefourth 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 2dimensional 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 lowgenus embedded TPMS S_{1} and S_{2}
for which the respective adjoint surfaces S_{1}† and
S_{2}† have elementary patches
that are bounded by simplyconnected straightedged polygons
P_{1} and P_{2},
there must exist an embedded hybrid TPMS S_{h} whose adjoint surface
S_{h}† has an elementary patch with boundary polygon equal to
a linear combination of P_{1} and P_{2}.
(I defined a linear combination of two polygons P_{1} and P_{2} to be
a polygon interpolated between P_{1} and P_{2}.
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 genus14 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 P_{1} at the upper right are
12, 34, 45, 51.
Consecutive edges of the quadrangle P_{2} at the lower right are
12, 23, 34, 41.
The transformation of P_{1} into P_{2} 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
P_{1} and P_{2} require
trialanderror 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 straightedged 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
CostaHoffmanMeeks surfaces. The original
surface from that family is shown below, in a stereo image due to
Hermann Karcher.
The original CostaHoffmanMeeks 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
vacuumforming 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
IWP that I call O,CTO (cf. Figs. E2.79 to E2.81).
Its genus is only 10.
Then for the next fortytwo 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 longlasting [polyoxyethylene] soap film
spanned by a straightedged
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 longforgotten 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 onehalf 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,CTO
The 'bcc labyrinth surface' referred to at the top
of the page is the surface I later renamed IWP.
Fig. E2.75
O,CTO (genus 10), a hybrid of IWP (genus 4) and P (genus 3)
cubic unit cell
view: [100] direction
Fig. E2.76
O,CTO
unit cell
view: [111] direction
Fig. E2.77
O,CTO
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 nonbalanced.
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 BaTiO_{3} is shown in Fig. E2.75 for comparison with Manta.
One can try to match the sizes and positions of the ions in
BaTiO_{3} 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 (BaTiO_{3})
Ba^{2+} red
Ti^{4+} green
O^{2−} 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 fatherinlaw)
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 triplyperiodic 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 C_{19}(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 triangularprismshaped Coxeter cell.
The nylon thread that is stretched horizontally across the interior
of the cell lies on
an axis of 2fold 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 vacuumformed 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 modelmaker who had recently started to work with me
parttime 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 onequarter of the tetragonal disphenoid. It contains only one internal axis of twofold 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 C_{9}(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
C_{9}(P) are merely the topologically simplest members of a
countably infinite sequence of embedded surfaces
of progressively higher genus:
P, C_{9}(P), C_{15}(P), C_{21}(P), ...
I obtained experimental evidence for the existence of C_{21}(P) by
producing the foursided Flächenstück of
C_{21}(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 C_{9}(P),
which are in stable equilibrium.
Although the C_{21}(P) soap film corresponds to a [mathematical] stationary state,
its area is larger than that of nearby lying [nonminimalsurface] 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 C_{21}(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 C_{9}(P)
Fig. E2.83b
Twelve elementary triangular Flächenstücke
of Neovius's selfintersecting surface C_{9}(P)†
In both C_{9}(P) and C_{9}(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 C_{9}(P) and C_{9}(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 highergenus variants of the P and D surfaces and a variety of noncubic TPMS. All of these experiments involved
modifying the shapes of stationarystate 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 highergenus variant surfaces.
I was unable to produce the genus15 soap film, but occasionally I succeeded with the genus21 case.
genus=9
genus=15
genus=21
genus=27
genus=33
Fig. E2.84
Ken Brakke's Surface Evolver solutions for the first few
highgenus variants of Neovius's C_{9}(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, C_{9}(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 p_{k} of the k^{th} surface
M_{k} in the family {M_{k}}
is defined as p_{k} = p_{0} +
k gap (k = 0, 1, 2, ...);
the values of p_{0} and the positive integer
gap are characteristic of the family.
These families include — but are not limited to — highgenus
complements
of P and D.
For the P and D families, for example,
p_{0}=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 2fold symmetry axes of the enclosing polyhedral cell.
The curved soap film boundary edges lying in face planes of the cell are 'mirrorsymmetric 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. 4546 of
Infinite Periodic Minimal Surfaces Without SelfIntersections).
In Infinite Periodic Minimal Surfaces Without SelfIntersections,
I included only two examples of hybrid surfaces — C(H) (genus 7) and O,CTO (genus 10),
because at the time of writing, these were the only examples of such surfaces for which I had already constructed and photographed vacuumformed plastic models.
(In a footnote on p. 46, I mentioned a third example, of genus 5, that I called gg'.
I soon renamed that surface gW, 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 gW.
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 SelfIntersections.
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 [firstorder] complement of Schwarz's H surface
unit cell
view: caxis
Fig. E3.3
C(H)
view along caxis
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 (2fold axis embedded in the surface)
Note the infinitely long straight tunnels with pointy oval crosssection
Fig. E3.6
C(H)
view: caxis, 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
gW ("graphitewurtzite") (genus 5)
oblique view
Fig. E3.8
gW
oblique view
Fig. E3.9
gW and C(H)
oblique view
Fig. E3.10
H''R (genus 5)
view: caxis
Fig. E3.11
H''R (genus 5)
view: caxis, 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 (2fold axis embedded in the surface)
Note the infinitely long straight tunnels with pointy oval crosssection
Fig. E3.15
H'T (genus 4)
view (stereo): caxis
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
 Ken Brakke's
Triply Periodic Minimal Surfaces
 EPINET's
Triply Periodic Minimal Surfaces
 Steven Finch's
Soap film Experiments
 Shoichi Fujimori and Matthias Weber's
A Construction Method for Triply Periodic Minimal Surfaces
 Paul J. F. Gandy, Sonny Bardhan, Alan L. Mackay, and Jacek Klinowski's
Nodal surface approximations to the P, G, D, and IWP triply periodic minmal surfaces,
 Darren Garbus's
Isoperimetric Properties of Some Genus 3 Triply Periodic Minimal Surfaces Embedded in Euclidean Space,
M.S. Thesis, May 2010
 Chaim GoodmanStrauss and John Sullivan's
Cubic Polyhedra
 Wojciech Gòzdz and Robert Holyst's
Geometry: Minimal Surfaces
 David Hoffman and Jim Hoffman's
The Lidinoid Surface
 Stephen Hyde, Christophe Oguey, and Stuart Ramsden's
Triply connected graph embeddings
 Stephen T. Hyde, Michael O'Keeffe, and Davide M. Proserpio's
A Short History of an Elusive yet Ubiquitous Structure in Chemistry, Materials and Mathematics
 "Touching Soap Films, An Introduction to Minimal Surfaces",
by Hermann Karcher and Konrad Polthier.
Besides providing a wealth of information in text form, it includes some beautiful videos.
 Hermann Karcher and Konrad Polthier's
list of references
to materials that are not online.
 Hermann Karcher's
The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions,
Manuscripta Math. 64, 291357 (1989)
 Hermann Karcher and Konrad Polthier's
Exhibition of Historical Minimal Surfaces
(video)
 Jacek Klinowski's
Periodic Minimal Surfaces Gallery
 Elke Koch and Werner Fischer's
3periodic minimal surfaces at the '[Details]' hyperlink in their Mathematical Crystallography
 Xah Lee's
Gallery of Famous Surfaces
 Eric Lord and Alan Mackay's
Slide Show of Triply Periodic Surfaces
 William H. Meeks III's
Introduction to Minimal Surfaces
 Isabel Hubbard, Egon Schulte, and Asia Ivic Weiss's
PetrieCoxeter Maps Revisited
 Alan Schoen's
Infinite Periodic Minimal Surfaces Without SelfIntersections,
NASA TN D5541 (May 1970)
 Gerd SchröderTurk's
The bicontinuous fish tank
 Gerd SchröderTurk's
Bonnet transformation between the D, G and P minimal surfaces
(video)
 Gerd SchröderTurk, Stuart Ramsden, Andrew Christy, and Stephen Hyde's
Medial Surfaces of Hyperbolic Structures

Gerd SchröderTurk, Andrew Fogden, and Stephen Hyde's
Local v/a variations as a measure of
structural packing frustration in bicontinuous mesophases,
and geometric arguments for an alternating Im3m (IWP) phase
in blockcopolymers with polydispersity
 Gerd SchröderTurk, Andrew Fogden, and Stephen Hyde's
Bicontinuous geometries and molecular selfassembly:
comparison of local curvature and global packing variations in genus three cubic, tetragonal, and rhombohedral surfaces
 Martin Steffens and Christian Teitzel's
Grape Minimal Surface Library
 Toshikazu Sunada's
Crystals That Nature Might Miss Creating
 Matthias Weber's
Gallery of Minimal Surfaces
Be sure not to miss Matthias's 'Archive' animations!
 Adam Weyhaupt's
New Families of Embedded Triply Periodic Minmal Surfaces of Genus Three in Euclidean Space
 Adam Weyhaupt's
Meet the Gyroid
 Adam Weyhaupt's
Deformations of the gyroid and Lidinoid minimal surfaces

3DXplorMathJ Applets

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 (19252006)
photo by Ludwig Danzer
©Ludwig Danzer
Source: Mathematisches Forschungsinstitut Oberwolfach gGmbH
Robert Osserman (19262011) at Berkeley in 1979
photo by George M. Bergman
©Mathematisches Forschungsinstitut Oberwolfach gGmbH
Raymond Redheffer (19212005)
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 SchröderTurk at ErlangenNürnberg
Isaac Van Houten at Carbondale (2008)
photo by the author
Matthias Weber at Oberwohlfach (2009)
photo by Renate Schmid
©Mathematisches Forschungsinstitut Oberwohlfach gGmbH
Adam Weyhaupt (right) at Edwardsville
with two of his students — Darren Garbuz and Caroline Coggeshall
photo by the author
Back to GEOMETRY GARRET
  