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## Alan H. Schoen

Comments are welcome.

Triply-periodic minimal surfaces (TPMS)

This is an illustrated account of my amateur study of TPMS, aimed at both beginner and specialist. It contains

links to the contemporary mathematical and scientific literature. I describe some of the chance events in 1966

and 1967 that led to my three-year immersion in this study, in which I was guided by both mathematics and

physical experimentation. I benefited greatly from discussions with several mathematicians, some of whom

appear in photos below, but helpful above all else in my study of the geometry of periodic structures were the

two books 'Third Dimension in Chemistry', by A. F. Wells and 'Regular Polytopes', by H. S. M. Coxeter.

I owe special thanks to the architect Peter Pearce, who in 1966 introduced me to his concept of,saddle polyhedron

and to the physicist Lester C. Van Atta, who in 1967 offered me a research position I couldn't dream of refusing:

senior scientist at the NASA Electronics Research Center in Cambridge, Mass. The sculptor Harald Robinson

provided invaluable technical support during my time at NASA. Fashioning special tools for the fabrication of

plastic models of minimal surfaces was just one of several tasks he performed with unfailing skill and ingenuity.Other people who assisted me in various ways while I was at NASA/ERC (1967-1970) or afterward at California

Institute of the Arts (1970-1973) or at Southern Illinois University/Carbondale (1973-1983) include John Brennan,

Bob Davis, Jay Epstein, Bob Fuller, Thad Heckman, Rick Kondrat, Blaine Lawson, Randall Lundberg, John Milnor,

Bob Osserman, Kenneth Paciulan, Charlie Strauss, and Jim Wixson. I am extremely grateful to all of them. In

September 1999, I began a collaboration with Ken Brakke, who models minimal surfaces with breathtaking

precision with his Surface Evolver. Some results of our joint work are displayed at one of Ken's fabulous web

sites — Triply periodic minimal surfaces,The

gyroidsurface receives special emphasis here, primarily because in recent years it has become an

object of special interest for materials scientists. They have found, for example, that it serves as a

template for self-assembled periodic surfaces that separate two inter-penetrating regions of matter, with

unit cells that vary in size over several orders of magnitude, depending on the constituent materials.

§0. Eversion of the Laves graphThe Laves graph, also known as (10-3)-a, is triply-periodic (on a

bcclattice) and chiral.

It is of interest for a variety of reasons, not least because a left- and right-handed pair of

these graphs (anenantiomorphicpair) are theskeletal graphsof the two intertwined

labyrinths of the gyroid, a triply-periodic minimal surface or TPMS (cf.gyroid history,

images, analysis, below in §1). Because the Laves graph is one of the rare examples of

a triply-periodic graph on a cubic lattice that is symmetric, it can beevertedwithout any

change in the lengths of its edges. Each edge rotates through a half-turn in a particular one

of the three orthogonal coordinate planes.The video below is a [cross-eyed] stereoscopic animation of the eversion

of a finite piece of the Laves graph.(It is just 'eye candy' and has no

direct bearing on the relation between the Laves graph and the gyroid!)

The animation shows a single period of the eversion. At mid-period, the

handedness of each regular helical polygon in the graph has changed from

CW to CCW. At the end, the initial CW handedness has been restored.For additional details concerning the eversion of the Laves graph, see

Reflections Concerning Triply-Periodic Minimal Surfaces, pp.663-665,

a summary of my minimal surface research published in October, 2012

inInterface Focus, a journal of The Royal Society.

§1. The gyroid

All paired images — both computer drawings and photos — like those just

below are stereoscopic.They are arranged to be viewed with eyes crossed.

You can run this time-lapse sequence by using the

Page DownandPage Upkeys.

It shows how a sequence of rotations and rotatory reflections transforms a curved

triangularFlächenstück( 'asymmetric unit') of the gyroid into this skew hexagon,

which corresponds to a face of the Coxeter-Petrie map {6,4|4}. The infinite gyroid

could in principle be constructed by first attaching a second hexagon to any open

edge of an initial hexagon and then repeating the attachment process at every open

edge of the developing assembly. Each attachment is implemented by applying a

symmetry of the gyroid: ahalf-turnabout an axis normal to the surface through the

midpoint of each edge. Each of these 'C2 axes' is parallel to a line joining the

midpoints of opposite edges of a cube like the one shown in the illustration above.A more economical way to describe the construction of the entire infinite surface is

based on the translation symmetries of the surface. A connected assembly of eight

suitably located hexagons — no two oriented alike — defines a lattice fundamental

domain. The infinite surface is generated by appropriate translations of this domain.

(It is not essential that the domain be connected, but it is conventional for it to be.)

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

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

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

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

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

§2. My videos about soap films and TPMS1972 Part 1 (56:42)

1972 Part 2 (27:09)

1972 Part 3 (35:20)

1972 Part 4 (28:54)

1974 Part 1 (36:02)

§3.Interface Focussummary article (2012)Reflections Concerning Triply-Periodic Minimal Surfaces,

a summary of my minimal surface research, which appeared in

October 2012 inInterface Focus, a journal of The Royal Society

§4. TheP−G−Dfamily ofassociateminimal surfacesSchwarz's

PandDsurfaces and their associate surfaceG(the gyroid) are the topologically

simplest examples of embedded TPMS that havecubiclattice symmetry. They are related

by the continuous bending transformation described in 1853 by Ossian Bonnet. The mean

curvature (which is zero at every point), the Gaussian curvature, and the orientation of the

tangent plane do not change anywhere throughout the bending.PandDare calledadjoint

surfaces: straight lines in one surface are replaced by plane geodesics in the other, andvice.

versaGcontains neither straight lines nor plane geodesics. Among the countable infinity

of TPMS related toPandDby Bonnet bending,Gis the only example that is embedded,

i.e.,free of self-intersections.P,D, andGare each of genus three, which is the minimum

possible genus for a triply-periodic minimal surface.

_{ P }_{ G }_{ D }

_{ {6,4|4}P }_{ {6,4|4}G }_{ {6,4|4}D }

_{ sc lattice }_{ bcc lattice }_{ fcc lattice }

Each of the surfacesP,G, andDis shown here in a tiling pattern called{6,4|4}by Coxeter.

The prototile for each of these tilings is a specific variant ofhex_{90}, a generic regular skew

-gon6

with 90º face angles. There are

faces4

incident at each vertex andholeswith

-fold4

symmetry.The three variants of

hex_{90}forP,G, andDare calledPhex_{90},Ghex_{90}, andDhex_{90}, respectively.

In a 1970 NASA Technical Note Infinite Periodic Minimal Surfaces Without Self-Intersections (p.38

ff),

I described howskeletal graphscan be used to represent TPMS. More recently David Hoffman and Jim

Hoffman (no relation) have demonstrated in their Scientific Graphics Project that for the TPMSP,G,D,

and also for a fourth surface (I-WP) of genus 4, there is a striking connection between the skeletal graph

of the surface and a modified version of its level surface approximation. Perhaps similar matches will be

found for other examples of TPMS, including surfaces of genus greater than 4.

P

'P'stands forprimitif, the name assigned long ago by German crystallographers to crystal structures

that have the symmetry of a packing of congruent cubes. ThePsurface exhibits this symmetry. Each

of the two congruent inter-penetrating labyrinths into which space is partitioned by thePsurface may

be regarded as an inflated version of the skeletal graph withtubularedges that enclose the edges of a

packing of congruent cubes. The symbol for the space group ofP(No. 229) isIm3′m. (Alan Mackay,

the British physicist and crystallographer, has wittily dubbedP'the plumber's nightmare'.)

G

'G'stands forgyroid, a name I chose to suggest the twisted character of its labyrinths, which — unlike

the labyrinths ofPandD— are opposite mirror images (enantiomorphs). The skeletal graphs ofGare

dualLaves graphs. The symbol for the space group ofG(No. 230) isIa3′d.

D

'D'stands fordiamond. The congruent pair of skeletal graphs of its labyrinths are dual diamond graphs ,

whose edges correspond to the bonds between adjacent carbon atoms in diamond. The symbol for the

space group ofD(No. 224) isPn3′m.

I believe now that'reciprocal'might have been a more suitable name than'dual'for the relation between

the two triply-periodic graphs —e.g.,simple cubic, diamond, Laves, and others — of an intertwined pair.

(The two graphs in many of these pairs are non-congruent.)'Reciprocal'would have reduced the risk of

confusion with the accepted meaning of'dual'in the expression 'dual graph' which by convention refers

either toplanegraphs or to the graphs of edges of the triply-periodic Coxeter-Petrie polyhedra. In what

follows I will continue to use the word'dual'to refer to the vaguely defined butsymmetricalrelation

between pairs of triply-periodic graphs like those described here.

PandDwere discovered and analyzed by H. A. Schwarz in 1865. He derived an

explicit Enneper-Weierstrass parameterization for the surfaces, which morph into

each other via the Bonnet bending transformation. (Another well known example

of Bonnet bending is the helicoid-catenoid transformation.)

PandDcontain both embedded straight lines and plane geodesics. A straight line

in either surface morphs into a plane geodesic (a mirror-symmetric plane line of

curvature) in the other surface. Because every straight line embedded in a minimal

surface is an axis of 2-fold rotational symmetry, a half-turn about the line switches

the two sides of the surface and also switches the two inter-penetratinglabyrinths

into which space is partitioned. A TPMS is calledbalancedif its labyrinths are

congruent.PandDare both balanced surfaces.The Weierstrass integrals shown below define the rectangular coordinates of

Pand

Dand of countably manyassociatesurfaces.θis called theangle of associativityor

Bonnet angle. Forθ= 0, the equations describeD, and forθ=π⁄ 2, they describeP.

DandPare calledadjoint(orconjugate) surfaces.

_{ Enneper-Weierstrass equations for the embedded minimal surfaces D, G, and P, which are related by Bonnet bending }In the spring of 1966, I had never heard of the Schwarz surfaces

PandD. In fact I knew

next to nothing aboutanyminimal surfaces! But then I met Peter Pearce (cf.§31), who

showed me two plastic models of what he calledsaddle polyhedra. The faces of these

objects were skew polygons spanned by minimal surfaces. I decided to find out to what

extent I could use Peter's concept of saddle polyhedron to simplify a rule for defining a

dualityrelation I had been struggling to impose on certain pairs of triply-periodic graphs.Using a toy vacuum-forming machine, I made plastic replicas of several soap films that

span skew polygons. By a stroke of luck, two of these polygons, which happened to be

regular skew hexagons, turned out to be modules of Schwarz'sPandDsurfaces. I was

startled when I realized that these two simple and elegant surfaces must surely be widely

known (even though I had never heard of them). From a literature search in the UCLA

library I discovered that J.C.C. Nitsche was a renowned authority on minimal surfaces,

so I telephoned him to ask him to identify these two intriguing surfaces. He replied that

judging from my descriptions, they were almost certainly two famous minimal surfaces

analyzed about one hundred years earlier by H.A. Schwarz. He said that the coordinates

of these two surfaces are defined by three complex integrals derived in 1863 by Karl

Weierstrass, who — I learned later — was one of Schwarz's teachers.During the next few weeks, after a rather superficial study of what Schwarz wrote about

his two periodic surfaces, which I dubbedandD, I became convinced thatP, but forthere must

also exist a minimal surface with the symmetry and topology of— the gyroidG

almost two years I had no idea how to construct it. I knew that if it did exist, the skeletal

graphs ('backbones') of its two labyrinths must be an enantiomorphic pair of Laves graphs.In September, 1968, during a telephone conversation with Blaine Lawson about the gyroid

problem, it at last became clear to me that the gyroid is simply a(hybridcf.Eq. 1 in §5) of

the two Schwarz surfacesDandPthat happens to be embedded. (In §51, I summarize this

conversation with Blaine.) The minimal surfacesD,G, andPare all described by Enneper-

Weierstrass equations, and the coordinates of any point in a lattice fundamental domain of

Gare a linear combination of the coordinates of the corresponding points inDandP. The

Bonnet angleθ_{G}for the gyroid is given by the equation

θ_{G}= ctn^{-1}(K′ ⁄K)

≅ 38.014773989108068108º,

where

K=K(1/4) ≅ 1.6857503548125960429,

K′ =K(3/4) ≅ 2.1565156474996432354.

Kis a complete elliptic integral of the first kind withparameterm=1/4;K′ is its complement.

The images of

Dhex_{90}andPhex_{90}shown directly below in §5 demonstrate the transformation

under Bonnet bending of straight lines in each surface into plane geodesics in the other surface.

In contrast toDandP,Gcontains neither straight lines nor plane geodesics. Its labyrinths are

enantiomorphic (oppositely congruent).Gcan be regarded as a special kind of balanced surface.

Further details about how I discoveredGare described in §3, §5, §7, and §51.

§5. How I derived the value ofθ_{G}, the Bonnet angle for the gyroid

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

_{ The boundary curves for the Dhex90, Phex90, and Ghex90 faces of D (blue), P (red), and G (violet) }

_{ The trajectory of every point on the surface under Bonnet bending is an ellipse. }

_{ The trajectories for the six vertices of each face are shown here. }

_{ Stereo view of the outline of the surface patch Dhex90 (blue) and its adjoint image Phex90 (red) The [red] curve P1 P2 is the adjoint image under Bonnet bending of the [blue] curve D1 D2. }

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

A lattice fundamental domain ofDorPcan be tiled by eight differently oriented

replicas of one hexagonal surface patch. Outlines of two such replicas are shown

above in blue (Dhex_{90}) and red (Phex_{90}).Dhex_{90}is inscribed in a cube of edge

lengthλand_{D}Phex_{90}is inscribed in a cube of edge lengthλ. In 1866, Hermann_{P}

Schwarz proved that in order forPhex_{90}andDhex_{90}to have the same area (which

is required for Bonnet bending of either surface into the other), it is necessary that

λ⁄_{P}λ=_{D}K(3/4)/K(1/4) ≅ 1.2792615711710064662. For bothDandP, the normal

vectors at the centers of the eight hexagonal patches in a lattice fundamental domain

are all directed toward different corners of the circumscribing cube. For the patches

Dhex_{90}andPhex_{90}shown above, the normal vectors point in the (1,1,1) direction (x

is positive in the direction toward you,yis positive to the right, andzis positive up).If we fix point

Oat the center ofDhex_{90}, then asDhex_{90}morphs intoPhex_{90}by

Bonnet bending, the pointD_{1}onDhex_{90}moves along an elliptical trajectoryr_{1}(θ)

with center atO. The equation for this ellipse is

r_{1}(θ) =d_{1}cosθ+p_{1}sinθ(1)The orthogonal vectors

d_{1}andp_{1}are directed outward from the center of the ellipse

along its semi-minor and semi-major axes, respectively.If

θ=π/2, the images under bending ofd_{1}andd_{2}arep_{1}andp_{2}, respectively.Let us fix the scale by setting λ= 2. Then_{D}As stated above, H.A.Schwarz proved ( d_{1}= (− 1,1,1)

|d_{1}| = √3

p_{1}= (λ⁄_{P}λ) (0,− 1,1)_{D}

|p_{1}| = √2 (λ⁄_{P}λ). (2)_{D}

cf.his Collected Papers, vol. I, p. 88) that

λ⁄_{P}λ=_{D}K′(1/4) ⁄K(1/4) (≅ 1.2792615711710064662). (3)If we substitute for

d_{1}andp_{1}in Eq. 1 from Eqs. 2 and 3, we obtain

r_{1}(θ) = (−1,1,1) cosθ+ (K′ ⁄K) (0,−1,1) sinθ(4)If we define

θ=θfor the gyroid, then Eq. 4 becomes_{G}

r_{1}(θ) = (−1,1,1) cos_{G}θ+ (_{G}K′ ⁄K) (0,−1,1) sinθ(5)_{G}

The key to solving Equation 5 for θis found in the geometrical properties of the_{G}

manifoldM_{6}, an infinite regular skew polyhedron whose faces are regular skew

hexagons. Each of these hexagons can be inscribed in a truncated octahedron.

Below is an illustration of one hexagonal face ofM_{6}, with its central normal vector

(not shown) oriented in the (1,1,1) direction, followed by an illustration of a lattice

fundamental region composed of eight differently oriented replicas of this face.

_{ The edges of a skew hexagonal face of M6 with central normal vector in the (1,1,1) direction This face of M6 is inscribed in a truncated octahedron, a space-filling polyhedron. }

_{ x is positive toward the observer, y is positive to the right, and z is positive up. The coordinates of the six vertices of M6, in CCW order starting from G1, are (−1,0,2), (0,−2,1), (2,−1,0), (1,0,−2), (0,2,−1), and (−2,1,0), respectively. }

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

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

_{ The boundary curves for the Dhex90, Phex90, and Ghex90 faces of D (blue), P (red), and G (violet) }

_{ The trajectory of every point on the surface under Bonnet bending is an }

_{ ellipse. The trajectories for the six vertices of each face are shown here. }

Now suppose that each edge of

M_{6}is replaced by a quarter-pitch of a helical arc,

as depicted in the image above. The table below lists the direction of the axis of

the helix associated with each arc and also the sense (CWvs.CCW) of each arc.arc direction sense

G_{1}G_{2}yCW

G_{2}G_{3}xCCW

G_{3}G_{4}zCW

G_{4}G_{5}yCCW

G_{5}G_{6}xCW

G_{6}G_{1}zCCWLet us denote by

Ghex_{90}the minimal surface that spans this modified hexagon

with helical edgesG_{1}G_{2},G_{2}G_{3}, ...,G_{6}G_{1}, because — like each of the hexagonal

facesDhex_{90}andPhex_{90}ofDandP— its face angles are also equal to 90º. Since

the directions of the tangents to the pairs of edge curves that intersect atD_{1},P_{1},

andG_{1}are identical, the surface orientation is the same at these points.

Vector equation Eq. 5 is equivalent to the following three scalar equations, one

for each of the three components of the vectorr_{1}(θ):_{G}

x_{1}(θ) = −cos_{G}θ+ (0) (_{G}K′ ⁄K) sinθ(5_{G}')

y_{1}(θ) = cos_{G}θ− (_{G}K′ ⁄K) sinθ(5_{G}'')

z_{1}(θ) = cos_{G}θ+ (_{G}K′ ⁄K) sinθ(5_{G}''')

But

r_{1}(θ) =_{G}g_{1}

= (−1,0,2). (6)Hence

y_{1}(θ) = 0, (7)_{G}If now we substitute for

y_{1}(θ) from Eq. 7 in Eq. 5_{G}'', we obtaincos

θ= (_{G}K′ ⁄K) sinθ. (8)_{G}Therefore

θ_{G}= ctn^{-1}(K′ ⁄K). (9)Because of the 6-fold rotatory reflection symmetry of

Dhex_{90},Phex_{90}, andGhex_{90},

Eq. 9 could have been derived by considering any of the other five vertices of

Ghex_{90}instead of the vertexG_{1}.

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

_{ M4 was produced by executing an intermediate step of an empirically developed procedure that was aimed at deriving duals of triply-periodic graphs. The hexagons of M6 are the conventional duals of the quadrilaterals of M4: every vertex of M6 lies at the center of a face of M4, and every vertex of M4 lies at the center of a face of M6. The viewpoints are both in the (1,1,1) direction. }

The fortuitous failure of my dual graph recipeOn February 14, 1968, I constructed a model (

cf.photo at above left) of an infinite regular

skew polyhedron I callM_{4}. It appeared at the penultimate stage of a recipe (which I was still

testing) for what I called thedualof a triply-periodic graph. The graph I was testing on that

day is a triply-periodic symmetric graph of degree six that I calledBCC_{6}, because it can be

transformed intoBCC_{8}, the ordinarybccgraph of degree eight, by adding two additional edges

at each vertex. I call such a symmetric graphdeficient, because it remains symmetric

even when the number of edges incident at each vertex is increased. I had expectedM_{4}to be

finite, because this part of my recipe had never yet failed to yield one or more finite polyhedra.I had been trying to understand whether it is possible to define precisely those classes of

triply-periodic graphs for which my recipe would produce a unique pair of sets of saddle

polyhedra (cf.§27). I required that one of these two sets be composed ofinterstitial

polyhedra that fill thecavitiesof the graph. The other set was supposed to contain what I

callednodalpolyhedra (cf.§28) enclosing the graph'svertices. I did not rule out the

possibility that some or all of the faces of these saddle polyhedra might be flat.

My immediate reaction to

M_{4}was simply to be surprised that it was not finite. But I also

realized how naive I had been to imagine that anything as jerry-rigged as this dual graph

recipe would always perform the way I had planned. I was actually enormously relieved to

encounter this failure! For a year and a half, I had been testing lots of graphs in a search

for a 'counter-example', and I had finally found one. At last I could stop searching.All these thoughts were swept aside as soon as I examined

M_{4}carefully, because I

discovered that the skeletal graphs of its two enantiomorphic labyrinths weredual Laves. With mounting excitement I realized that I had at last found an object with the

graphs

same global combinatorial structure and symmetry as the elusive 'Laves periodic minimal

surface' I had hunted without success in the summer of 1966.I still had no idea how to transform

M_{4}into a single minimal surface, but I judged by eye that

M_{6}, itsdual— a tiling with four skew hexagons incident at each vertex — would be somewhat

lessbumpythanM_{4}. In any tiling of a minimal surface by replicas of a straight-edged polygon

spanned by a minimal surface (like the tiling of Schwarz'sDsurface by regular skew hexagons

with 90º face angles, or the tiling of Schwarz'sPsurface by regular skew hexagons with 60º

face angles), the two tiles of every adjacent pair are related by ahalf-turnabout their common

edge (Schwarz's reflection principle). If adjacent tiles were instead related by an angle (let's call

itθ_{dihedral}) that islessthanπ, there would be a kind of bump between them — a discontinuity in

the orientation of the tangent planes on the two sides of their common edge. A calculation

confirmed that theM_{6}bump is smaller than theM_{4}bump. InM_{4},θ_{dihedral}= cos^{−1}(−1/2) = 120º,

implying a bump equal to 180º − 120º = 60º, but inM_{6},θ_{dihedral}= cos^{−1}(−5/7) ≅ 135.585º, implying

a smaller bump of approximately 180º − 135.585º = 44.415º. (The face angles inM_{4}andM_{6}are

equal to cos^{−1}(1/3) ≅ 70.529º and cos^{−1}(−1/6) ≅ 99.594º, respectively.)

I decided that a reduction in bumpiness of ~15.6º (60º − 44.415º) was large enough to justify

making a vacuum-forming tool for the skew hexagons ofM_{6}.Three days later, I finished assembling the model of

M_{6}shown at the upper right.M_{6}really

did look somewhat smoother thanM_{4}. More significantly, by this time I had come to

realize thatM_{4}andM_{6}are described by Coxeter's dual regular maps {4,6|4} and {6,4|4}.

This suggested a strong connection to Schwarz'sDandPsurfaces and to the infinite regular

skew polyhedra of Coxeter and Petrie, but I wasn't sure exactly what to make of all this.After long exposure to examples of triply-periodic graphs on cubic lattices, the only examples

of asymmetricgraph on a cubic lattice I had succeeded in identifying were the simple cubic,

diamond, and Laves graphs. The simple cubic graph is the skeletal graph of Schwarz'sP

surface, and the diamond graph is the skeletal graph of Schwarz'sDsurface. I couldn't help

wondering whether the Laves graph is the skeletal graph of athirdminimal surface. But I

couldn't imagine using the concept of skeletal graph to prove the existence of a minimal surface!Encountering this breakdown in my dual graph recipe had induced me to change direction. I

decided to stop trying to construct arigorousdefinition of the dual of a triply-periodic graph.

M_{4}andM_{6}had forced me at last to recognize the futility of trying to transform a bunch of

empirical relations between graphs into a comprehensive scheme with genuine predictive

value. I concluded that the notion of duality for triply-periodic graphs would never lead to

consistent results unless I could somehow tighten the definition of the classes of graphs to

which it is applied. I summarized my struggle with dual graphs in my 1970 NASA Technical

Note, Infinite Periodic Minimal Surfaces Without Self-Intersections.

Summarizing: the reason I suddenly lost interest in trying to formalize the concept of duality

for triply-periodic graphs is that I had noticed thatthe skeletal graphs of the two intertwined

labyrinths ofM_{4}are the Laves graphs.M_{4}andM_{6}both have the same topology and global

symmetry (space groupIa3′d, No. 230) as the gyroid, which I had tried unsuccessfully to

construct in the summer of 1966. The geometry ofM_{6}strongly suggested to me the

possibility of somehow constructing the gyroid out of hexagonal faceswhose vertices. Each face of

coincide with those ofM_{6}M_{6}, like the hexagonal faces ofDandP, is oriented

in one of the eight (±1,±1,±1) directions. They-coordinatesy_{1}(θ) and_{G}y_{4}(θ) of vertices_{G}

G_{1}andG_{4}(see images above), which are related by inversion in the center of a hexagonal

face, are both equal to zero. For each of the other seven orientations of the hexagonal face,

it is likewise true that one of the three rectangular components of the vectors that define

the positions of a pair of vertices ofM_{6}related by inversion is equal to zero.My model of

M_{6}demonstrated that the straight edges ofM_{6}defineinfinite regular helical, which are centered on lines parallel to the rectangular coordinate axes. This

polygons

suggested to me that if I were to replace the straight edges ofM_{6}by curved helicalarcs, the

'bump' along the edges might shrink almost to zero. This was a wild and woolly guess,

with absolutely no theoretical justification. I knew that there are perfect helices embedded

in the helicoid, but I also knew that no solution was known for a minimal surface bounded

by six helical arcs. (This whole idea actually turned out to be something of a red herring,

and it threw me off the trail of the gyroid! Although it is now known that the difference

between cylindrical helices and the spiralling geodesics in the gyroid surface that are

centered on lines parallel to the three coordinate axes of the surface is quite small, it is not

zero. The image in §9 shows the difference.)I later dubbed the object that is tiled by skew hexagons with strictly helical edges the '

pseudo-

gyroid'. It isnota triply-perodic minimal surface. Even if all of its separate faces were

minimal surfaces, they would not define a global surface that is continuous across the

boundary between adjacent faces. (Incidentally, it is still not known how to derive an

analytic expression for a simply-connected minimal surface spanning a 'Schwarz chain'

composed of consecutive helical arcs.)

Below are images of the Voronoi polyhedron of a vertex of

M_{6}.

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

§6. New models ofM_{4}andM_{6}constructed in 2011

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

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

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

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

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

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

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

§7. The relation betweenM_{6}and the gyroidIn March 1968 I played my hunch about helical edges by sending a purchase order to a

machine shop for a custom-designed brass tool for vacuum-forming plastic modules of

hexagons with edges in the shape of helical arcs of alternating handedness. Below is a

photo of the model of the pseudo-gyroid that resulted. It didlookexactly like a minimal

surface, but I had no idea how to derive equations for it. I sent a smaller version of this

model to Bob Osserman, a distinguished authority on minimal surface theory. Bob then

suggested to Blaine Lawson, his talented PhD student, that he look into this problem (cf.§5 and §51). I introduced myself by telephone to Blaine, who explained that he was in the

throes of writing the final part of his PhD dissertation and couldn't predict when he

would be able to start thinking about the gyroid problem. But I was relieved that my

puzzle was at last in expert hands.

_{ The pseudo-gyroid (1968) }By early summer one of the progress reports I was required to write about my research

had apparently disturbed somebody in NASA Headquarters. I was informed that some

officials there were concerned about my 'playing with soap bubbles'. I also learned that

Headquarters was thinking about having me transferred to a project more closely related

to NASA's mission (the mechanical support structure of the Hubble Space Telescope

was mentioned as an example of such a project). This news induced in me a state of mild

panic, and I abruptly switched my attention to a problem concerningcollapsing graphs,

thinking (not very rationally) that I might be able to stave off threatening catastrophe by

demonstrating how such graphs could be applied to the design of spaceframes that could

be stored compactly in a collapsed state and later deployed in an expanded state.

§8. A long summer distractionDuring the summer of 1968, as I waited to hear from either Bob or Blaine, I analyzed

the collapse kinematics of several triply-periodic graphs, while trying to put the gyroid

problem aside, still convinced that it was best left to experts. But as luck would have it,

the analysis of these collapsing graphs required that I consider in detail the orientation

of the surface normals onP,G, andD, and I was confronted daily by the most obvious

imaginable clues to the fundamental relation (the Bonnet bending transformation) that

connects these three surfaces.And yet I didn't recognize the clues!('... eyes have they,)

but they see not.'

§9. The cylindrical helix and theflattenedcylindrical helix

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

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

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

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

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

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

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

_{ If I hadn't discovered this puzzle set of tetrominoes and shared them with my friend and colleague Bill Van Atta, I would probably never have met his father, Lester C. Van Atta, who found the set fascinating and told Bill he wanted to meet me. We met for several hours, and the the invitation to join NASA/ERC was the result. }

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

My addiction to recreational mathematics, which worsened considerably

once I started playing with solid tetrominoes in 1965, was not helped by

exposure to the writings of Martin Gardner and Solomon Golomb. I am

extremely grateful to both of them.

Martin Gardner

10 October 1914 — 5 May 2010

Solomon Golomb

§11. Symmetrical lattice fundamental domains of the gyroidThe hexagonal faces of the gyroid Coxeter-Petrie map {6,4|4} can be joined

in a connected assembly in a variety of ways, each of which defines a lattice

fundamental domain (lfd) of the surface. Below are several examples oflfds

that are arranged to illustrate one or more of the symmetries of the surface.

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

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

_{ high resolution image }

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

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

_{ high resolution image }

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

_{ high resolution image }

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

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

_{ high resolution image }

_{ Top view of lfd2(G) }

_{ high resolution image }

_{ Bottom view of lfd2(G) }

_{ high resolution image }

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

_{ high resolution image }

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

_{ high resolution image }

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

_{ high resolution image }

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

_{ high resolution image }

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

_{ high resolution image }

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

_{ high resolution image }

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

_{ high resolution image }

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

_{ high resolution image }

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

_{ high resolution image }

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

_{ high resolution image }

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

_{ high resolution image }

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

_{ high resolution image }

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

_{ View along (111) direction }

_{ high resolution image }

§12. Polyhedral surrogates of TPMSIt is interesting to explore the relation between TPMS — especially those

of low genus, likeD,P, andG— and simpler structures that we will call

polyhedral surrogates with plane faces(). Each of these surrogates is aps

plane-faced triply-periodic polyhedron that is homeomorphic to a TPMS.

The most symmetrical examples have the same symmetry as the TPMS.

Both the TPMS and its surrogate can be represented by the same pair of

inter-penetratingskeletal graphs. Each edge in such a graph joins a pair

of vertices that lie at centers of symmetry of the TPMS and its surrogate.

The regular skew polyhedron {4,6|4} of Coxeter and Petrie is an example

of apsof a TPMS. Since it corresponds to Schwarz'sPsurface, we call it

{4,6|4}_{P}. This is what it looks like:

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

A second example of a regular

psofPis {6,4|4}_{P}:

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

An example of aregularpsof Schwarz'sDsurface

is the regular skew polyhedron {6,6|3}_{D}:

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

psof Schwarz'sDsurface is thequasi-regularskew polyhedron

(6.4)^{2}(below). Quasi-regular polyhedra are both edge-transitive and vertex-

transitive, but they are not face-transitive.

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

G, there exist no examples of either regular or quasi-regularps,

but there is an infinite uniform skew polyhedron that is apsofG. It is a kind of

snub polyhedron, but unlike the twelve examples offinitesnub polyhedra, which

are chiral and exist in two enantiomorphic forms, it — like the gyroid — has only

a single form. Its two labyrinths are enantiomorphic. It is vertex-transitive, but it

is neither edge-transitive nor face-transitive. It is called (6.3^{2}.4.3)_{G}. Two images

are shown below.

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

_{ high resolution image }

_{ (6.32.4.3)G, including its skeletal graphs }(6.3

^{2}.4.3)_{G}was discovered by Norman Johnson in 1969, when he derived

the combinatorial type of every possible example of a 'uniform tessellation

of the {4,6} family'. He proved that the enumeration is complete. Norman

Johnson is well known for his 1966 enumeration of the 92 Johnson solids,

later proved exhaustive by Victor Zalgaller. He is currently writing a book

about uniform polytopes inR^{3}andR^{4}.2.4.3)

_{G}polyhedron was independently rediscovered several years ago by

John Horton Conway, who named itmu-snub cube. ('mu' meansmultiplehere.)

The Voronoi polyhedron of a vertex of the union of the two dual skeletal

graphs of Schwarz'sD— or of (6.4)^{2}, itsps— is the truncated octahedron.

The boundary of eachchamberin (6.4)^{2}is a truncated octahedron from

which four of the eight hexagons have been removed.

_{ (6.4)2 In both labyrinths there are two differently oriented chambers. }The four faces incident at each vertex of (6.4)

^{2}are arranged cyclically: 6^{.}4^{.}6^{.}4.

Hence the name (6.4)^{2}. There are two differently oriented varieties of chambers

in each labyrinth. Chambers that share an open hexagonal face belong to the same

labyrinth and are oppositely oriented. Chambers that are separated by a square face

belong to different labyrinths and are identically oriented.

(I plan to post additional links here from time to time, but this

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

§13. BibliographyFor an introduction to the mathematics of minimal surfaces, I enthusiastically recommend The Parsimonious Universe: Shape and Form in the Natural World, by Stefan Hildebrandt and Anthony Tromba.

Below is a partial list of

sources (not limited toon-lineminimal surfaces): mathematics, images, videos, applications in physics, chemistry, biology, engineering,periodicetc.

- Ken Brakke:

Triply Periodic Minimal Surfaces

Comprehensive collection of illustrations of surfaces derived with Ken'sSurface Evolversoftware

- Ken Brakke:

2005 translation (by Brakke) of J. Plateau's monumental 1873 treatise:

Experimental and Theoretical Statics of Liquids Subject to Molecular Forces Only

This is Plateau's account of his pioneering investigations of capillarity phenomena. It includes a description (pp. 103-107) of H.A. Schwarz's then almost brand-newreflection principle, which defines the role of rotation and reflection symmetries in the analytic continuation of finite pieces of many examples of periodic minimal surfaces. The problem of proving that there exists a minimal surface spanning a given boundary is called Plateau's problem, because it was he who called special attention to the central importance of such a proof.

- Tomonari Dotera, Masakiyo Kimoto, and Junichi Matsuzawa

Hard Spheres on the Gyroid Surface

- EPINET:

Triply Periodic Minimal Surfaces

An encyclopedia of triply-periodic minimal surfaces that includes a description of related periodic graphs

- Steven Finch:

Soap film Experiments

A collection of both analytic and numerical computations of the surface area of examples of minimal surfaces — some known to Schwarz and others newly introduced here

- Steven Finch:

Gergonne-Schwarz Surface

TPMS, elliptic integrals, and elliptic functions

- Steven Finch:

Mathematical constants and functions

Although it is not relevant here, this collection is too fascinating to be left out!

- Shoichi Fujimori and Matthias Weber:

A Construction Method for Triply Periodic Minimal Surfaces

- Paul J. F. Gandy, Sonny Bardhan, Alan L. Mackay, and Jacek Klinowski:

Nodal surface approximations to the P, G, D, and I-WP triply periodic minmal surfaces,

- Darren Garbus:

Isoperimetric Properties of Some Genus 3 Triply Periodic Minimal Surfaces Embedded in Euclidean Space, M.S. Thesis, May 2010

- Chaim Goodman-Strauss and John Sullivan:

Cubic Polyhedra

- Wojciech Gòzdz and Robert Holyst:

High Genus Periodic Gyroid Surfaces of Non-Positive Gaussian Curvature

- David Hoffman and Jim Hoffman:

Geometry: Minimal Surfaces

- David Hoffman and Jim Hoffman:

The Lidinoid Surface

- Stephen Hyde, Christophe Oguey, and Stuart Ramsden:

Triply connected graph embeddings

- Stephen T. Hyde, Michael O'Keeffe, and Davide M. Proserpio:

A Short History of an Elusive yet Ubiquitous Structure in Chemistry, Materials and Mathematics

- Hermann Karcher and Konrad Polthier:

Touching Soap Films, An Introduction to Minimal Surfaces

- Hermann Karcher:

The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions, Manuscripta Math. 64, 291-357 (1989)

- Hermann Karcher and Konrad Polthier:

Exhibition of Historical Minimal Surfaces

(video)

- Hermann Karcher and Konrad Polthier:

list of references to materials that are.not online

- Jacek Klinowski:

Periodic Minimal Surfaces Gallery

- Elke Koch and Werner Fischer

3-periodic minimal surfaces at the '[Details]' hyperlink in theirMathematical Crystallography

- Elke Koch and Werner Fischer:

3-periodic surfaces without self-intersections

- Xah Lee:

Gallery of Famous Surfaces

- Eric Lord and Alan Mackay:

Slide Show of Triply Periodic Surfaces

- Eric Lord and Alan Mackay:

Periodic minimal surfaces of cubic symmetry

- William H. Meeks III:

Introduction to Minimal Surfaces

- Isabel Hubbard, Egon Schulte, and Asia Ivic Weiss:

Petrie-Coxeter Maps Revisited

- Alan H. Schoen:

Infinite Periodic Minimal Surfaces Without Self-Intersections,

NASA TN D-5541 (May 1970)

- Alan H. Schoen

Reflections Concerning Triply-Periodic Minimal Surfaces

A summary of my minimal surface research, published in October 2012 inInterface Focus, a journal of The Royal Society

- Gerd Schröder-Turk:

The bicontinuous fish tank

- Gerd Schröder-Turk:

Bonnet transformation between the D, G and P minimal surfaces (video)

- Gerd Schröder-Turk, Stuart Ramsden, Andrew Christy, and Stephen Hyde:

Medial Surfaces of Hyperbolic Structures

- Gerd Schröder-Turk, Andrew Fogden, and Stephen Hyde:

Localv/avariations as a measure of structural packing frustration in bicontinuous mesophases, and geometric arguments for an alternating Im3m (I-WP) phase in block-copolymers with polydispersity

- Gerd Schröder-Turk, Andrew Fogden, and Stephen Hyde:

Bicontinuous geometries and molecular self-assembly: comparison of local curvature and global packing variations in genus three cubic, tetragonal, and rhombohedral surfaces

- B. Schuetrumpf, M. A. Klatt, K. Iida, G. E. Schröder-Turk, J. A. Maruhn, K. Mecke, P.-G. Reinhard:

Appearance of the Single Gyroid Network Phase in Nuclear Pasta Matter

- Martin Steffens and Christian Teitzel:

Grape Minimal Surface Library

- Toshikazu Sunada:

Crystals That Nature Might Miss Creating

- Matthias Weber:

Gallery of Minimal Surfaces

- Matthias Weber:

Bloomington's Virtual Minimal Surface Museum

- Margaret and Christine Wertheim:

The Institute for Figuring (IFF)

An organization dedicated to the exploration of the relation between mathematical forms and natural objects, with particular emphasis on shapes defined by hyperbolic geometry

- Adam Weyhaupt:

New Families of Embedded Triply Periodic Minmal Surfaces of Genus Three in Euclidean Space

- Adam Weyhaupt:

Meet the Gyroid

- Adam Weyhaupt:

Deformations of the gyroid and Lidinoid minimal surfaces

- 3D-XplorMath-J Applets

- Wikipedia:

Calculus of Variations

- Wikipedia:

Gyroid

list will never be more than a tiny sample of what's out there.)

§14. Regular warped polyhedra based on the gyroid

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

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

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

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

(6. )4^{2}— alternatively written (6..6.4)4_{G}— is a triply-periodic

quasiregularwarped polyhedroncomposed of regular convex

hexagons and regular skew quadrilaterals that are spanned by

minimal surfaces. (The minimal surfaces are approximated here

by hyperbolic paraboloids.) (6..6.4)4_{G}has the same topological

structure and symmetry as the gyroid. We adopt the convention

that a triply-periodic polyhedron is calledif at least onewarped

of its face species is a skew polygon. Thein (6.4)4^{2}is written in

to indicate that its quadrilateral faces are skew.bold face italicsThe 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 (.4)6^{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 (.4)6^{2}. In the future I will post images of some examples.

§15. Regular tilings ofP,D, andG

PDG

{6,4|4}_{P}{6,4|4}_{D}{6,4|4}_{G}

latticesclatticefcclatticebcc

These three TPMS are tiled here in a pattern

called{6,4|4}. The prototile for each surface

is a particular variant ofhex_{90}, a regular skew

-gon6

with 90º face angles. In this pattern, there are

faces4

incident at each vertex and 'holes' with

-fold4

symmetry.Alternatively, these surfaces can be tiled in the

patterns{4,6|4}and{6,6|3}. The prototiles for

{4,6|4}are variants ofquad_{60}, a regular skew

-gon with 60º face angles. In this tiling, there4

arefaces incident at each vertex and holes6

with-fold symmetry.4The prototiles for

{6,6|3}are variants ofhex_{60},

a regular skew-gon with 60º face angles. In6

this tiling, there arefaces incident at each6

vertex and holes with symmetry of order.3

It is conventional to call skew polygons

regularif they are bothequilateral

andequiangular, irrespective of

whether their edges are

straight or curved.

The names {6,4|4}, {4,6|4}, and {6,6|3} are

Coxeter's modified Schläfli symbols for the

three infinite regular skew polyhedra. Each

of these three polyhedra ishomeomorphic

to — and has thesame symmetryas — one

of the two Schwarz surfaces. {6,4|4} and

{4,6|4} are homeomorphic toP, while

{6,6|3} is homeomorphic toD.

P,D, andGare related by Bonnetbending,

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 ofhex_{90}(D) ↔hex_{90}(G) ↔hex_{90}(P).

§16. The bending ofD→G→P

_{Sequence of Bonnet bending stages from D to G to P (stereo)}

_{ One lattice fundamental domain (plus one additional hexagonal face) }

0º D

~38.015ºG

90ºP

The portion of the surface shown here is a lattice fundamental domain plus one _{ The surfaces at all other angles between 0º and 90º are self-intersecting. }The images above are

eleven framesselected from a movie I designed in 1969

using a FORTRAN program written by the computer scientist Charles Strauss.

The cinematographer was Bob Davis of the MIT Lincoln Lab. He used a Bell

and Howell 35mm movie camera that he had modified so that it could accept

input data from ERC's PDP-11 computer. Ken Paciulan and Jay Epstein carried out

the data input tasks. The enthusiastic help of all of these wonderful guys is very

gratefully acknowledged.

hexagonal face — the topmost one in the image for 0º Bonnet angle. The three

dashed lines are a set of lattice basis vectors. But the surface is notperiodic(its

symmetry group is notdiscrete) unless the Bonnet angleθsatisfies the equation

θ_{p,q}= ctn^{-1}[(p/q)(K'/K)],

wherepandqare any two coprime positive integers.This obviously necessary restriction hadn't occurred to me before Blaine Lawson

mentioned it — more than once! — in September 1968. Thanks, Blaine!Note that in the Bonnet transformation of the complete surface

S_{1}intoS_{2}, not all

pairs of faces inS_{1}that share a common edge preserve this connection also inS_{2}.

To the time-lapse bending sequence above,animate

clickhere, 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.

§17. Poincaré's hyperbolic disk model of uniform tilingsThe combinatorial structure of the three tiling patterns

{6,4|4},{4,6|4}, and{6,6|3}is illustrated by

Poincaré's hyperbolic disk model of

uniform tilings in the hyperbolic plane.

(image by Eric W. Weisstein,

Wolfram MathWorld)

{4,6|4} {6,4|4} {6,6|3}

(images from Wikipedia)

§18. The three Coxeter-Petrie infinite regular skew polyhedraCoxeter and his friend Petrie long ago discovered the three triply-

periodic regular skew polyhedra called {p,q|r}. Their faces are

regularplanepolygons.

is the number ofpedgesof eachface,

is the number ofqfacesat eachvertex,

and

is the number ofredgesof eachhole.

{4,6|4}

{6,4|4}

{6,6|3}

{4,6|4}, including theskeletal graphof labyrinth A

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

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

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

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

§19. Views ofGtiled byhex_{90}Among the countable infinity of surfaces that are

associatesurfaces (cf.Fig. E1.2m) ofPandD,

the gyroidGis the only one that isembedded.

Unlike

PandD,Gcontains neither straight lines

nor plane geodesics. It has the same symmetry as

the union of its two enantiomorphic skeletal graphs

(Lavesgraphs). The lattice isbcc.Like

PandD,Gcan be tiled by

(i) regular skew hexagonshex_{90}({6,4|4} tiling) or by

(ii) regular skew quadrilateralsquad_{60}({4,6|4} tiling) or by

(iii) regular skew hexagonshex_{60}({6,6|3} tiling).

Below are views of

Gtiled byhex_{90}

in the Coxeter-Petrie {6,4|4} 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 half-turn about an axis

of type (110) perpendicular toGat the midpoint of

the shared edge.

(1)

(2) (3) (4) (5)(1) [100] orthogonal projection

(2) (100) viewpoint

(3) [100] orthogonal projection

(4) [111] orthogonal projection

(5) ~(110) viewpoint

§20. Views ofGtiled byhex_{60}Below are views of

Gtiled by the regular

skew hexagonhex_{60}, the prototile of

the Coxeter-Petrie {6,6|3} map.

hex_{60}is so named because it has 60º face angles.

It is related to six otherhex_{60}faces, with each of

which it shares an edge, by a half-turn about an

axis of type (110) perpendicular to the surface at

the midpoint of the shared edge.

Stereo image of a cubic unit cell ofG

tiled by eight replicas ofhex_{60}

This unit cell is comprised of

twolattice fundamental regions.

(100) viewpoint

High-resolutionversion

Stereo image of the cubic unit cell ofG

illustrated just above

(-1-11) viewpointHigh-resolutionversion

Stereo image of the

hexagonal facehex_{60}ofG

(111) viewpoint

High-resolutionversion

Stereo image of the

hexagonal facehex_{60}ofG

and an associated cuboctahedronThe midpoints of the six edges of

hex_{60}coincide with vertices

of the cuboctahedron.

(111) viewpointHigh-resolutionversion

Stereo image of the

hexagonal facehex_{60}ofG

(110) viewpointHigh-resolutionversion

Stereo image of the

hexagonal facehex_{60}ofG

(415) viewpointHigh-resolutionversion

§21. Views ofGtiled byquad_{60}

§22.Gyring Gyroid— a sculpture by Goodman-Strauss and Sargent

Gyroid sculpture

by Chaim Goodman-Strauss and Eugene SargentClick here for the authors' account of

how they implemented their design.

Here's their razzle-dazzle video

showing bits and pieces of its construction.

§23. Shapeways 3D-printed models of TPMSIf you're unfamiliar with TPMS, one place to begin looking

is the set of Shapeways models made by

and by others.

Shapeways displays a large collection

of gyroid and gyroid-related models here.

Bathsheba Grossman's model of

a kind ofinversionof the gyroid

bounded by an ellipsoid

Below are a few of the TPMS models

produced by Shapeways

forKen Brakkeand forAlan Mackay.

There's much more information about TPMS at

one of Ken Brakke's webpages,

including many illustrations he made with hisSurface Evolver.

PC(P)

(models byKen Brakke)

DC(D)

(models byKen Brakke)

I-WPF-RD

(models byKen Brakke)

Batwing

(model byKen Brakke)

GyroidDouble Gyroid

(models byAlan Mackay)

Sven Lidin's LidinoidSven Lidin's Lidinoid 222

(models byAlan Mackay)

NodalGyroid222aFluoritesolid7

(models byAlan Mackay)

SBA-1x50Batwing5x2

(models byAlan Mackay)

§24. The four founding fathers of triply-periodic minimal surfaces

Georg Friedrich Bernhard Riemann Karl Hermann Amandus Schwarz

(1826-1866) (1843-1921)

Alfred Enneper Karl Theodor Wilhelm Weierstrass

(1830-1885) (1815-1897)(In §E7 there are photos of a few

contemporaryexperts in this field.)

§25. Definition ofminimal surface

Mathematicians have studied minimal surfaces since 1762, when Lagrange derived the 'Euler-Lagrange equation', which is satisfied by the surface of least area spanning a given closed curve. Aside from the plane, which defines a trivial solution of this equation, the first surfaces found as solutions of the Euler-Lagrange equation were the helicoid and catenoid, both of which were discovered by Meusnier in 1776. Meusnier also proved that the mean curvature of every solution surface is equal to zero. Since some closed curves are spanned by more than one surface with zero mean curvature everywhere, a minimal surface is conventionally defined as

a surface with vanishing mean curvature at every point, rather than asa surface of least area. Of course every minimal surface islocallyarea-minimizing,i.e., the surface patch inside every sufficiently small closed curve enclosing any point of the surface has less area than any other surface bounded by that closed curve.Pioneering investigations of

triply-periodicminimal 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 so-called self-assembled structures that are studied by physical, chemical, and biological scientists. Below are links to a tiny sample of the relevant mathematical and materials science studies, but the sample is neither comprehensive nor up-to-date.I have attempted here to summarize my own study of TPMS, which began quite unexpectedly in the spring of 1966. I have included an account — warts and all — of some of the events that led to my involvement in this study, during which I frequently wandered down bypaths that were well off the main route. (I have long believed that such bypaths sometimes offer a more rewarding view than the main route. John Horton Conway has explained that he finds it fruitful to juggle several ostensibly unrelated problems at the same time, because one problem may turn out to be the key to the solution of another. My discovery of a precursor of the gyroid minimal surface in 1968 was for me a validation of Conway's truism, as explained below.)

§26 My stint at NASA/ERC (1967-1970)

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

Lester C. Van Atta

1905-1994

Associate Director, NASA Electronics Research Center

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

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

Before arriving at ERC in the fall of 1967, I decided that I would concentrate there on two areas of research: §27. Peter Pearce's concept ofsaddle polyhedra

(a) symmetric triply-periodic graphs and their

nodal and interstitial polyhedra(see explanation below), and

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

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

Peter Jon Pearce

Architectural designer

saddle polyhedron. These saddle polyhedra had straight edges, but each face was curved in the shape of a minimal surface.(All of the stereoscopic image pairs below are arranged for cross-eyed viewing.)

A portion of thediamondgraph The interstitial polyhedron

of thediamondgraph

A portion of thebccgraph The interstitial polyhedron

of thebccgraph

§28. Dual graphs andinterstitial vs. nodalpolyhedra

I was thunderstruck by Peter's two saddle polyhedra, because I understood at a glance that they were the critical ingredient missing from a scheme I had tried to develop for illustrating the relation between the combinatorial and symmetry properties of

crystal networks(triply-periodic graphs) andpolyhedral 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

fof faces as the numberZof 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

fof faces of the cube, which is the Voronoi polyhedron for a vertex of the simple cubic (sc) lattice, is six, which is also the numberZof edges incident at each node (vertex) of a conventional ball-and-stick model of the lattice, and (ii) the cube also has the same symmetry as the node with respect to the surrounding lattice.

A piece of the simple cubic (sc) graph,

for which theinterstitial polyhedronis the cube,

which is the Voronoi polyhedron for a vertex of the graph.A similar correspondence holds for the face-centered cubic (

The cube is also thenodal polyhedronfor a vertex of thescgraph

fcc) lattice if each vertex is enclosed by the rhombic dodecahedron, which is the Voronoi polyhedron for a vertex of this lattice.

The nodal polyhedron of thefccgraph is the

rhombic dodecahedron, the Voronoi polyhedron of a vertex of the graph.

The fluorite graph is thedualof thefccgraph.

Its nodal polyhedra are

the regular tetrahedron and

the regular octahedron.

Thefccgraph and the fluorite graph

It is convenient to define this pair of graphs asduals.

For the body-centered cubic (

bcc) lattice, however, the combinatorial part of this correspondence breaks down. Although there are only eight nearest neighbors of each vertex in this lattice, the Voronoi polyhedron of a vertex is the truncated octahedron, which has fourteen faces. The reason for this numerical disparity is hardly profound. It's just that thesecond-nearest-neighbor sites in thebcclattice 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 onlynearestneighbor sites into account. I had observed a similar mismatch for the diamond crystal structure: even though there are only four nearest neighbors for each site, the Voronoi polyhedron has sixteen faces.

The 14-faced Voronoi cell for a vertex of thebcclattice

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

bcclattice and thediamondcrystal structure. I described to Peter a space-filling eight-faced saddle polyhedron, composed of regular skew hexagons with 90º corners, that wouldenclose each vertexof thebccgraph. A few days later I dubbed it theexpanded regular octahedron, orERO(see stereo image below). I proposed calling vertex-enclosing polyhedranodal polyhedra, irrespective of whether they turn out to be saddle polyhedra or convex polyhedra. For about ten years, I had been calling the triply-periodic graph whose edges correspond to the edges of a packing of expanded regular octahedra theWPgraph, because it mimics the pattern of string tied around a wrapped cubic box (see image below).Curiously, the

EROwas about to introduce me to TPMS!

The eight faces of

The expanded regular octahedronERO,

which is thenodal polyhedronof thebccgraph

and theinterstitial polyhedronof theWPgraph

EROmatch the number of edges incident at each vertex of thebccgraph, and both the saddle polyhedron and a vertex of the graph have the same symmetry.

Thebccgraph (Z=8)

The edges of thetetragonal tetrahedron TT

are shown in blue.

TheWPgraph (Z=4)

The edges of theexpanded regular octahedron ERO

are shown in blue.

Thebccgraph (green vertices)

and its dual,

theWPgraph (orange vertices)

§29. Soap film interlude (2012)

When I started playing with soap films and minimal surfaces in May 1967, I was totally ignorant of the extensive literature on these subjects. I didn't know that there are boundary frames that can be spanned by more than one shape of minimal surface and that so-called

unstableminimal surfaces, which do not minimize area, can span those boundary frames. On one of his web pages, called 'Catenoid Soap Films', Ken Brakke illustrates what may be the first known example of this phenomenon, the existence of both stable and unstable versions of the catenoid spanning the space between two parallel circular boundary frames. I recall poring over James Clerk Maxwell's classical article in the legendary Eleventh Edition of the Encyclopedia Britannica, inherited from my father, in which the author analyses this behavior of of the catenoid.As I learned more about minimal surfaces, I eventually realized that in Peter Pearce's prescription of minimal surfaces for the faces of saddle polyhedra, it should probably be stated explicitly that the minimal surface is

area minimizing— and thereforestable. (In fairness to Peter, I suspect that he was already aware of these distinctions! I don't recall ever having discussed these questions with him.) I did wonder a little about the variety of shapes of saddle polyhedra that would result if there existed more than one minimal surface spanning a given circuit of edges in a triply-periodic graph.In June 1966, I undertook some soap film experiments in order to explore these questions. I found that the boundary curve

C_{0}(shown at left below), a simple closed curve in the shape of one of the several Hamilton cycles on the cuboctahedron, is spanned by at least two disk-type soap film surfaces of different shape. One of these two surfaces,S_{1}, is an area-minimizing surface ('least-area surface') and is calledstable. The other,S_{0}, isnota least-area surface. It is calledunstable, because it can be formed as a soap film onC_{0}only if one or more wires or threads are added toC_{0}along appropriate curves embedded inS_{0}—i.e.,curves that partitionS_{0}into an assembly of smaller surface patches, each of whose boundary curves is spanned by a uniquestableleast-area surface.

S_{1}can drape the boundary frameC_{0}in either of two positions. Let's call itS_{1a}if it's in one of these positions andS_{1b}if it's in the other position.S_{1a}andS_{1b}are related by a halfturn about the axisA_{1}A_{2}. (See Ken Brakke's computed images ofS_{1a}andS_{1b}below.) If a wire frame in the shape ofC_{0}is withdrawn from a solution of soap and water, it will be spanned by a soap film in the shape of one or the other (but not both) of these surfaces.

S_{0}, which drapes the wire frameC_{0}in only one position, forms as a soap film if threads or wires are incorporated inC_{0}along one or both of the linesA_{1}A_{2}orB_{1}B_{2}.

_{ C0, a curve that can be spanned by at least two differently shaped soap films, one stable and the other unstable }

_{ Cuboctahedron }

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

_{ The soap film S1a is an area-minimizing ('least-area') minimal surface. It is one of two differently oriented — but congruent — stable surfaces that span the boundary curve C0. }

Incorporating internal threads or wires along (a) either

_{ S1a S0 S1b }

_{ (Stable) (Unstable, unless string (Stable) }

_{ or wire is added along either A1A2 or B1B2 or both A1A2 and B1B2) }

A_{1}A_{2}orB_{1}B_{2}or (b) bothA_{1}A_{2}andB_{1}B_{2}in the wire frameC_{0}, which has twelve edges, partitionsC_{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 span a given boundary curve has been found to be extremely knotty, but it is known that there are two properties of a simple closed boundary curve

either of which guarantees that it is spanned by only one minimal surface of disk type:C(i) having a convex simple projection — whether central or parallel — onto a plane (Rado's 1932 theorem);

(ii) having total curvature less than 4π (Nitsche's 1967 theorem).Since the aforementioned 5-gons and 7-gons have total curvatures of only 2

^{1}⁄_{6}π and 3^{1}⁄_{3}π, respectively, it follows from Nitsche's theorem that each of them is spanned by only one minimal surface of disk type. This implies that incorporating a wire or thread alongeither or bothof the axesA_{1}A_{2}andB_{1}B_{2}will convert the bare frameC_{0}into a frame that is spanned by the surfaceS_{0}.

Note that the soap film in each of the photos below is the same piece of S_{0}.

_{wire added at A1A2}_{wire added at B1B2}_{wires added at A1A2 and B1B2}

In October 1967, three months after I joined NASA/ERC, I was a self-invited guest at the home of the late Hans Nitsche in Minneapolis. Although Hans showed considerable interest in my wire-frame model of

C_{0}and in my vacuum-formed plastic models ofS_{0}andS_{1}, he never even mentioned the ground-breaking paper in which he introduced and proved his 4π theorem, which was about to be published! Since I always found Hans to be both kind and modest, I later concluded that perhaps he thought I was so ignorant of the mathematics of minimal surfaces that he would only confuse and embarrass me if he discussed such a subtle problem.

.I extend my warm thanks to Ken Brakke for pointing out a serious elementary blunder, in an earlier version of this discussion, of the number of soap films spanning the frameC_{0}

Ever since 1954, when I began an informal study of polyhedral packings (triggered by my Ph. D. research, at the University of Illinois/Urbana-Champaign, on atomic diffusion in crystalline solids), I had ruminated from time to time over the relation between polyhedra and crystal structures. I became familiar with a variety of commonly known crystal structures, and I sawed wooden models of the Voronoi polyhedra that enclose the vertices of some of these structures. §30. Triply-periodic graphs

In 1956 I designed and ran a FORTRAN program that confirmed my hunch that for self-diffusion in

fcccrystals, the isotope effect and the Bardeen-Hering correlation factor are precisely equal. (The program modeled diffusion by an infinite random walk of a vacancy in a sequence of cubically symmetrical crystal volumes of increasing size.) This exact identity of the isotope effect and the correlation factor became the basis of the first experimental method — using radioactive tracers to sample the behavior of the diffusant atoms — of distinguishing between theinterstitialandvacancymechanisms of atomic self-diffusion in crystals.I modeled

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

ad hocmethods to identify dual pairs of triply-periodic graphs, I found that while the dual of thediamondgraph is also adiamondgraph, the dual of thefccgraph is thefluoritegraph and the dual of thebccgraph is theWPgraph. These relations are illustrated by the images shown above. But it soon became apparent to me that for many pairs of graphs, if one ignores the atoms in the crystals represented by the graphs there is no justification for labeling one graph substitutional and the other interstitial.

§31. My first encounter with TPMS

In April 1966, two days after meeting Peter Pearce, I made some examples of saddle polyhedra for myself, using the toy vacuum-forming machine I had bought for my children. My first model was the

bccnodal polyhedron, theexpanded regular octahedron EROillustrated above. But afterwards out of curiosity I joined two of its skew hexagonal faces byrotationinstead ofreflection. To my great astonishment, I found that if I continued to add faces in this fashion, the infinite smooth labyrinthine structure shown below began to emerge. (This vinyl model, as well as those shown in the next three images, are new ones I made the following year, after I had purchased a larger vacuum-forming machine.)

A piece of Schwarz'sDsurface

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

A portion of Schwarz'sPsurface

tiled by 60º skew hexagons

A transparent model ofPI had unwittingly stumbled onto the two classical examples of

adjoint(orconjugate) 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 themD(fordiamond) andP(forprimitif), after the crystal structures with matching topology and symmetry. I recognized that the chambers in the two complementary labyrinths ofPdefine the sites of the cesium and chlorine ions, respectively, in the ionic crystal Cs-Cl. Only after consulting a handbook of crystal structures did I learn that the atoms of sodium and thallium in the binary solid solution Na-Tl occupy sites that correspond to the symmetrical 'chambers' in the respective labyrinths ofD. 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

DandPin 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

Gare simplya linear combinationof the coordinates of corresponding points ofDandP,i.e.thatGisassociatetoDandP. A few days later Blaine Lawson pointed out to me that ifDandPare scaled so that the 90º hexagonsD_hex_{90}andP_hex_{90}that tile the map {6,4|4} in these surfaces are inscribed in a cubeof the same size, that linear combination becomes thearithmetic meanof the coordinates of corresponding points ofDandP!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

DandP. It seems very likely that if he had done so, he would have discovered the gyroid.

§32. Dual graphs and skeletal graphs

The concept of a dual relation for pairs of triply-periodic graphs had suddenly acquired new significance for me. I began to think of such graphs as potential

skeletal graphsof 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 besidesDandP, only three other examples of embedded TPMS —H,CLP, and Neovius's surface — had been known since 1883. But I found it hard to imagine that there were not others!

§33. Early hints of the existence of the gyroid

For the

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

diamondandbccinterstitial 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 anad hocrecipe for constructing both interstitial and nodal polyhedra that I hoped would be effective for every possible example of a triply-periodic graph. Although the recipe worked without a hitch for every graph I tested, I felt distinctly uneasy, because I suspected that there must exist cases for which it would be ineffective. Although I modified the recipe several times during the next several months, I was never able to give it a solid theoretical foundation.Beginning in June 1966, as a spare-time hobby I set out to discover a 'counterexample' — a graph for which the recipe fails to produce either interstitial or nodal polyhedra. I continued to test a variety of graphs, gradually accumulating a diversified collection of vacuum-formed interstitial and nodal polyhedra. I had the additional goal of finding a way to construct a hypothetical TPMS I originally named

L(for Laves). Here, however, I will refer to it asG(for gyroid), even though I didn't invent that name until almost two years later.It was obvious that

Gcouldn'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 inG, since by a theorem of H. A. Schwarz, a straight line embedded in a minimal surface is an axis of 2-fold rotational symmetry. This implies that a half-turn rotation ofGabout such an axis would interchange the two labyrinths ofG— and therefore also interchange the two skeletal graphs ofG. But that is impossible, since the two skeletal graphs are enantiomorphic. I couldn't imagine how to define the boundary curves of an elementary surface patch whose edges are neither straight line segments nor curved geodesics (mirror-symmetric plane lines of curvature). For each of the five examples of TPMS known before 1968, there exists a skew polygon surface patch with straight edges.

§34.Symmetric graphsI nevertheless had a strong conviction that

Gmust exist. The principal reason for my thinking so was that the skeletal graph of each labyrinth ofGshares with the skeletal graphs of Schwarz'sPandDsurfaces what I believed to be an exceptionally rare property: it is asymmetricgraph. A second reason I believed in the existence ofGwas derived from purely visual evidence: when I compared thetoy modelofGthat I had constructed out of stubby paper cylinders with the toy models I had made forPandD, I found that theGmodel was no less convincing than the other two. In all three models, when the cylinders are made as fat as they can possibly be (see the images just below), the total volume of the gaps between the two 'cylinder-ized' labyrinths is surprisingly small. I was startled to observe how snugly the cylinders of the intertwined labyrinths nestle against each other. When I compared the contours of my toy models ofPandDwith the smoother contours of my vacuum-formed models ofPandD, it seemed entirely plausible that the junctions between cylinders in the toy model ofGcould be flared and filleted so that the envelopes of the two labyrinths would coalesce into one single surface — a TPMS — just as they do forPandD. I understood, of course, that vague intuitive arguments like these are only rarely fruitful, but on the other hand I was not prepared to dismiss the arguments as worthless.

§35.Toy modelsofP,G, andG

P

D

G'Toy models' of

P,D, andG

The edges of the dual skeletal graphs

are represented as right circular cylinders.In each of the images at the right, the cylinder radius

is the maximum possible consistent with the requirement

that the cylinders intersect only at isolated points of tangency.

§36.BCCand_{6}M— hints of the gyroid's existence?_{4}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 long-sought 'counterexample' to my empirical recipe for deriving the interstitial and nodal polyhedra of a triply-periodic graph. (b) Although this graph failed spectacularly to yield a finite interstitial polyhedron, it pointed toward something much more interesting — aninfinitetriply-periodic saddle polyhedron that I callM_{4}. The symmetry and combinatorial structure ofM_{4}strongly strongly suggested to me that the hypotheticalGminimal 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 theGsurface and on searching for other new examples of TPMS.

A portion of the

deficientsymmetric graphBCC_{6}of degree sixI define a

deficientsymmetric 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 intoBCC_{8}, thebccgraph.)viewpoint: close to [100] direction

30 quadrangles ofM_{4}(stereo)

view: [111] direction

30 quadrangles ofM_{4}(stereo)

view: [110] direction

30 quadrangles ofM_{4}(stereo)

view: [110] direction

30 quadrangles ofM_{4}

view: [111] direction, backlit by summer sky

30 quadrangles ofM_{4}

view: [100] direction, backlit by summer sky

§37. A premature announcement of the gyroid

Abstract 658-30 submitted in summer 1968

to the American Mathematical SocietyThis was an awkwardly premature

announcement of the existence of

thegyroid, which I then calledL.

I had merelyconjectured, notproved,

thatLis a minimal surface.

(After I recognized that it isassociate

toPandD, I renamed itgyroid.)

§38. A botanical link to Schwarz'sPsurfaceMy 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

Psurface. (In 1971 Michael Berry [Berry, 1971] stated that Gunning and Jagoe later revised their analysis in favor of a network of tubules along the edges of the diamond graph instead of the simple cubic graph.)

§39. More about NASA/ERCRemarkably, Lester Van Atta, who had recruited me to work at NASA/ERC and was my immediate supervisor there, never interfered with my choices of what to work on. Since it was he who had invented the name 'Office of Geometrical Applications' for my 'administrative unit', I concluded that he did expect me to try to produce something of practical value for NASA. But he was never less than enthusiastic about my concentration on the study of periodic minimal surfaces.

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

ERC was famously top-heavy (or perhaps I should say bottom-heavy) with a bloated support infrastructure of low- and mid-level administrators, clerks, etc., many of whom were from the Boston area, hired in the early days before President Kennedy was assassinated. Soon after it opened, the hiring of scientists and engineers slowed down abruptly, and it appeared that the original plan to develop a well-rounded scientific and technical staff had been abandoned. It was our impression that Lyndon Johnson preferred to support NASA activities elsewhere, especially in Texas. However, according to the Wikipedia entry for ERC:

"Although it was the only Center NASA ever closed, ERC actually grew while NASA eliminated major programs and cut staff. Between 1967 and 1970, NASA cut permanent civil service workers at all Centers with one exception, the ERC, whose personnel grew annually."

Whatever the case, I arrived at ERC in July 1967 in a state of blissful ignorance. Only after I began work did I begin to learn from my colleagues about discrepancies between ERC's officially stated 'mission' and what seemed to be its actual potential for significant accomplishment.

§40. Geometers and other mathematical mentorsMy own position there was relatively comfortable, however, with one exception: the absence of in-house colleagues who shared my scientific and mathematical interests. I would have benefited from having someone close by for chit-chat about — and even collaboration in — those areas of research in physics and mathematics in which I had a special interest. There were spectacular compensations for this deficiency, however. For one, I was acquainted with a few extremely bright young mathematicians in the greater Boston area who showed a friendly interest in my work, and I benefited greatly from my few conversations with them. If only I had shown more initiative, I could have benefited even more from knowing them than I actually did. They included Thomas Banchoff (differential geometry), Norman Johnson (convex polytopes), Nelson Max (computer graphics), and Charles Strauss (computer graphics).

In 1966, a year before I joined NASA/ERC, Norman Johnson introduced me to the analysis by Coxeter and Moser of the infinite regular maps {6,4|4}, {4,6|4}, and {6,6|3} (

cf.the book by these authors that is cited below, following Fig. E1.1k). These three regular maps describe the combinatorial structure of the flat-faced Coxeter-Petrie infinite regular skew polyhedra. But they also describe the combinatorial structure of H. A. Schwarz'sPandDsurfaces, the canonical 19th century examples of triply-periodic minimal surfaces, as well as that of their only embeddedassociatesurface, the gyroidG(which I nearly discovered in February 1968, when by chance I found a doppelganger that isspookily similar).(

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

H.S.M. "Donald" Coxeter

Thomas Banchoff at Berkeley in 1973

photo by George Bergman

Charles Strauss (seated) and

Thomas Banchoff

at Brown University in 1979

Norman Johnson

Nelson Max

MIT was directly across the street from ERC, and Harvard was only a 20-minute walk away. ERC staff members had unrestricted borrowing privileges at the MIT library — an enormous convenience. A few prominent members of the MIT math faculty indulged me now and then when I had a pesky mathematical question, but for the most part, I was hesitant about bothering them, partly because I held them in such awe but also because I knew that for them many of my questions would turn out to be extremely elementary, if not downright trivial.A few months after I arrived at ERC, I was visited by Harald ('Hal') Robinson, a sculptor, designer, master machinist, and model-maker who lived in a nearby suburb. We hit it off immediately. After examining my plastic minimal surface models, Hal easily convinced me that he could make more accurate and more durable vacuum-forming tools than I could. Dr. Van Atta was acquainted with Hal's father, an engineer who was president of High Voltage Engineering Corp., the manufacturer of Van de Graaff generators. I persuaded Dr. Van Atta to hire Hal in a flexible part-time arrangement so that he could fabricate vacuum-forming tools for me. Hal wan't interested in a fulltime job, since he had other clients, and in any event I expected to have only enough projects to keep him busy intermittently. From then until the end of my stay at ERC about thirty months later, Hal was my invaluable collaborator.

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

Dr. Van Atta also hired James Wixson, an experienced applied mathematician and computer programmer. Jim helped me with a variety of chores. One of his several accomplishments was the invention and programming of a computer algorithm for generating every possible skew quadrangle that serves as a module for a

compoundperiodic minimal surface on a cubic lattice — an assembly of finite surface patch modules whose four straight edges include at least one edge along a [111] direction, coincident with an axis of 3-fold rotational symmetry. Decades later, these solutions have become of some interest as models for structures investigated by physicists and chemists who are soft matter specialists.I recall now with some embarrassment that during my first week at ERC, I visited the Harvard mathematics department and stopped by the offices of one after another member of the faculty to ask naive questions about the rather prosaic problem of how to go about enumerating those examples of triply-periodic graphs that are

symmetric. Professors Zariski and Ahlfors were both polite, but it was clear that my questions held little interest for them, and the interviews were mercifully short.

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

If I had known then that (a) Andy Gleason and I both graduated from high schools in Westchester County, N.Y. (he in Yonkers and I in Mount Vernon), (b) he graduated from Yale in 1942, the year I entered Yale, and (c) we were both in Naval Intelligence during WWII (he helping to crack the Japanese code and I passively studying the Japanese language), I would undoubtedly have attempted some small talk about these coincidences, but that would hardly have advanced our discussion of mathematics!

As a federal civil service employee, I had unfettered access to the WATS government long-distance telephone line. I made good use of it now and then, including having several fruitful conversations about the

stabilityof minimal surfaces — beginning in 1968 — with Fred Almgren at Princeton. I first met him face-to-face in September 1969, when we arranged to have side-by-side seats on a flight to the USSR. For a week Fred was my roommate at the Hotel Iberia in Tbilisi, Georgia, while we were attending a conference on minimal surfaces. Afterward Fred went on to St. Petersburg for an extended sabbatical visit.

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

Robert Osserman

H. Blaine Lawson, Jr.

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

Stefan Hildebrandt at Berkeley (1979)

photo by George M. Bergman

©George M. Bergman

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

A.H.S. and the gyroid at the Courant Institute, 1968

Photo by Stefan Hildebrandt

§42. The end of NASA/ERCAt ERC I buried myself in my research with little thought about the future. Dr. Van Atta provided even more support for my work than I ever asked for.

On December 30, 1969, when we were first notified about the impending shutdown, Richard Nixon had been president for almost a year. According to an account published in the New York Times, a prominent science journalist overheard some interesting remarks in the White House by the physicist Lee DuBridge, the former CalTech president who was Nixon's scientific advisor. DuBridge was alleged to have said that the president's decision to close ERC was prompted by his wish to damage the presidential aspirations of the senior senator from Massachusetts, Teddy Kennedy. (Kennedy was widely regarded at the time as Nixon's most formidable potential rival.) NASA had been funneling about $60 million annually into Massachusetts, and a significant fraction of those funds supported ERC, with substantial collateral benefits to the state economy.

Beginning in January 1970, ERC director James Elms made frantic efforts to find another federal agency to occupy the new $40 million building into which we had moved a week or so before the announcement of the shutdown. By late spring 1970, it was decided that a handful of members of the technical staff— mostly engineers and a few applied mathematicians — would be retained to work for a newly minted federal agency that would be called the Transportation Systems Center, as part of the U. S. Department of Transportation. The rest of us were told, "Good luck!" (Thanks to the good offices of Peter Pearce, I had already been invited to teach at the about-to-be-formed California Institute of the Arts, in Valencia, California, so my distress over the demise of ERC was somewhat less acute than that of many of my colleagues.)

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

Dr.Van Atta then telephoned NASA headquarters and almost overnight obtained a grant of $25,000 to support the project. My friend Keto Soosaar, an expert structural engineer at MIT, introduced me to his colleague Jeannie Freiburghouse, an experienced Fortran programmer, and I immediately hired her to use the Weierstrass integrals to compute the coordinates of 8000 points on a hexagonal patch of the gyroid that corresponds to a face of the Coxeter-Petrie regular map {6,4|4}. I arranged for ERC to award a contract to the Gurnard Engineering Corporation of Beverly, Massachusetts to manufacture two CNC-milled aluminum dies for vacuum-forming two kinds of thin zinc-alloy modules — one in the shape of the hexagon of {6,4|4} and the other in the shape of the quadrilateral of {4,6|4}, its dual. These two kinds of module were to be joined by epoxy, in a face-to-face, overlapping arrangement, resulting in a design in which one side of the gyroid surface is tiled by hexagons while the other side is tiled by quadrilaterals. Such a design avoids the need for unsightly connectors along module edges. (I recognized that it would be more efficient to use a single module shape based on the hexagon of the

self-dual map {6,6|3}, which is twice as large as that of {6,4|4}, but I didn't pursue this idea, because I knew from experience that the larger negative draft angle of such a big curvaceous module would probably prevent successful vacuum-forming, by introducing ugly ridge-like 'wrinkles' in each module.)In the spring of 1970, NASA funding for the project was abruptly subjected to a special kind of 'mid-course correction': it was cancelled. On investigation, I was informed that at a retirement party — presumably well lubricated — for the senior ERC comptroller responsible for my MoMA account, someone had 'accidentally hit the wrong key on his computer', with the result that the money still left in the account was sent back to Washington (

i.e.,NASA headquarters). The new heir to the comptroller's office told me that there was no way to recover this money. Having had earlier experience with bureaucracies, I recognized that the project was finished. (But see Figs. E1.18a-d below.)Dr. Van Atta resigned from ERC in the autumn of 1969 to become research vice-president of the University of Massachusetts/Amherst. Months later my colleagues and I guessed that he might have received early warning signals about the impending demise of ERC. His successor, Lou Roberts, an able electrical engineer and administrator, generously arranged for the last remaining technical typist in our division to be assigned the single task of typing my technical note, but the deadline was so tight that much of what I wrote was litle more than a first draft, since I had no opportunity for either proper editing or for review by another person. (Personal computers had not yet been invented. If computer work stations that allowed for some kind of word-processing existed in those days, I never heard of them.) Immediately after I submitted my manuscript to NASA, I handed in a list of typos and other errors for final corrections. Although I was promised that they would be dealt with, they were not. The one hundred complimentary copies I was promised turned out to be

three copies. We all had the feeling that we were now ancient history, and nobody much cared. (Perhaps that is the way it always is with institutions that are in their death throes.)

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

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

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

When I landed at Vnukovo International Airport in Moscow about ten days later, I was welcomed by Revaz Gamkrelidze, but I still had to get my bulky collection of plastic models of TPMS through customs. When a stolid Ukrainian customs agent showed signs of balking at the sheer number of boxes I had brought with me (I suppose he suspected they contained contraband), Gamkrelidze put on an impressive show of commanding authority. He announced in a magisterial voice (in Russian) that the boxes contained "mee

'-nee-mal soor'-fa-cez". The agent echoed in a bewildered voice, "Mee'-nee-mal soor'-fa-cez?" Gamkrelidze replied with great emphasis that the conference would be impossible without them, and that was that.After the conference began, I recognized with dismay that it would have been more sensible to bring fewer models. A few of the very dignified Russian and Western European mathematicians at the conference appeared to be somewhat offended by the sheer quantity of models I had brought. In any event, by the end of the four-day conference, several of the larger models had magically disappeared from the locked auditorium storage room in which they were kept overnight after each day's session, thereby lightening my load on the trip home.

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

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.

Lev Pontryagin Ilia N. Vekua

Here is the program of the 1969 Tbilisi conference:

page 1

page 2

page 3

page 4

page 5

page 6

page 7

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

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

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

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

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

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

Here is the program of the Ledgemont conference: Prof. Koptsik showed special interest in my model of the minimal surface C(

page 1

page 2

page 3

page 4

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

§45. Ken Brakke and hisSurface Evolver

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.

Kenneth Brakke

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

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

Stephen T. Hyde and Gerd E. Schröder-Turk have effectively summarized the state of our understanding in 2012 of the role of triply-periodic minimal surfaces in chemistry and biology in their introductory review article, Geometry of interfaces: topological complexity in biology and materials, which was published in the Royal Society's

§46. Research conferences

Interface Focus(2012)2, 529-538. Here is the Table of Contents for that journal volume.In October 2012, my wife Reiko and I attended the Primosten, Croatia conference, organized by Hyde and Schröder-Turk, on which these conference papers are based. In my own presentation, entitled Reflections concerning triply-periodic minimal surfaces, I described how I came to be involved in the investigation of minimal surfaces, beginning in 1966. (There is considerable overlap between some parts of this account and the material on this website.)

§47. Examples of dual skeletal graphs

Skeletal graphs of the two inter-penetrating labyrinths of a TPMS

and Voronoi polyhedra that enclose the vertices of the graph are useful

for representing the symmetry and topology of some examples of TPMS,

especially when the graph edges for each labyrinth are symmetrically equivalent.

Of course these geometrical constructions do not yield analytic solutions for the surfaces.

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

47.1 Dual diamondskeletal graphs

Fig. E1.0a

A dual pair ofdiamondskeletal graphs

47.2 Dual simple cubicskeletal graphs

Fig. E1.0b

A dual pair ofprimitive cubicskeletal graphsCuriously, 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

areidentical.

The lattice for this set is b.c.c (body-centered cubic).The Voronoi polyhedron for a vertex in this set is the truncated octahedron.

I know of no other example of a pair of

directly congruentdual graphs with cubic lattice symmetry.

47.3. Dual graphs (skeletal graphs of thediamondDsurface)

A triply-periodic minimal surface (TPMS) that is embedded,i.e.,free of self-intersections, partitions space into a pair of disjointlabyrinths. If the labyrinths are congruent — either directly or oppositely — the surface is calledbalanced. TheD(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'sDsurface

(genus 3)For some purposes it is convenient to represent a TPMS of low genus by a

surrogatewith plane faces — a triply-periodic polyhedron with the same symmetry and topology as the surface. The two labyrinths of the TPMS and of its surrogate may be represented by a pair of triply-periodicskeletal graphsthat 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

Dsurface is the triply-periodicquasi-regular skew polyhedron(6.4)^{2}(cf.Fig. 1.1b), which is derived from the Coxeter-Petrie regular skew polyhedron {6,4|4} (cf.Fig. E1.35c). Quasi-regular polyhedra are edge-transitive, but not face-transitive.

Fig. E1.1b

(6.4)^{2}, a surrogate of Schwarz'sDsurface (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

chambersof the respective labyrinths. Each chamber in (6.4)^{2}is a truncated octahedron from which a tetrahedrally arranged subset of four hexagons has been removed. Hence the boundary of (6.4)^{2}is composed of four regular hexagons and six squares (cf.Fig. E1.1c).

Fig. E1.1c

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

The four faces incident at each vertex of (6.4)

^{2}are arranged in cyclic order 6^{.}4^{.}6^{.}4 — hence the name (6.4)^{2}. In each of the two labyrinths, there are two differently oriented varieties of chambers. They are related by a quarter-turn about any one of the three Cartesian axes. In each labyrinth, adjacent chambers related by a translation of type [111] are of opposite variety.

Fig. E1.1d

Fig. E1.1e

Fig. E1.1f

Fig. E1.1g

Like (6.4) ^{2}, the Coxeter-Petrietriply-periodic regular skew{6,6|3} (

polyhedroncf.Figs. E1.2a,c) has the same topology

and symmetry as Schwarz's diamond surfaceD(cf.Fig. E1.2b).

In this example, in contrast to the case of (6.4)^{2}, the

unit cell has only one orientation (cf.Fig. E1.2c),

since it is a translation fundamental region.

Fig. E1.2a

A lattice fundamental region of {6,6|3}

The lattice isfcc.

Fig. E1.2b

A lattice fundamental region of Schwarz's diamond surfaceD

Fig. E1.2c

Thirteen lattice fundamental regions of {6,6|3}

Dual pairs of diamond graphs are shown below in both 'medium thick' and 'thick' versions. In the thick version ( cf.Figs. E1.3e-h), the diameterdof the cylindrical tubes is the largest possible, consistent with the requirement that the dual graphs not overlap. Overlap occurs when the ratiod/e≥ 2^{1/2}/2 (~.707), whereeis the edge length of an ideally thin skeletal graph.

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

diamond graphsMedium thick

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

diamond graphsThick

view along ~(111) direction

Fig. E1.3e Fig. E1.3f Fig. E1.3g

graph 1 graph 2 graphs 1 and 2

Fig. E1.3h

Orthogonal projection of graphs 1 and 2 on [111] plane

Fig. E1.3i

(6.4)^{2}— a triply-periodicquasi-regular polyhedron

that has the same topology and symmetry as

Schwarz's diamond surfaceD

Fig. E1.3j

(6.4)^{2}with embedded [skinny] dual graphs

Fig. E1.3k

(6.4)^{2}with embedded [fat] dual graphs

47.4. Dual graphs (skeletal graphs of thesimple cubicPsurface)

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

simple cubic graphsMedium thick

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

simple cubic graphsThick

oblique view

Fig. E1.4e Fig. E1.4f Fig. E1.4g

graph 1 graph 2 graphs 1 and 2

Fig. E1.4h

Orthogonal projection of graphs 1 and 2 on [100] plane

Fig. E1.4i

The Coxeter-Petrietriply-periodic regular skew polyhedron{6,4|4},

which has the same topology and symmetry as

Schwarz's primitive surfaceP

Fig. E1.4j

{6,4|4} with embedded [skinny] dual graphs

Fig. E1.4k

{6,4|4} with embedded [fat] dual graphs

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

47.5. Dual graphs (skeletal graphs of theLavesGsurface)

Dual pairs of Laves graphs are shown below in 'thin', 'medium thick', and 'thick' versions. In the thick version (

The two intertwined skeletal graphs of the gyroid,

in a Shapeways 3D printed version designed byvirtox.Click here for a video of 'Bones',

an animated view of these graphs,

also made at Shapeways byvirtox.

cf.Figs. E1.5i-l), the diameterdof the cylindrical tubes is the largest possible, consistent with the requirement that the dual pair of tubular graphs not overlap. Overlap occurs when the ratiod/e≥ 3^{1/2}/2 (~.866), whereeis the edge length of an ideally thin skeletal graph.The fact that the

d/eratio 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 theintersectionsof 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].

Laves graphsThin

[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

Laves graphsMedium thick

[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

Laves graphsThick

[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

Laves graphsThin

[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]

Laves graphsMedium thick

[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

Laves skeletal graphsThick

[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]

Laves skeletal graphsThin

[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]

Laves skeletal graphsMedium thick

[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]

Laves skeletal graphsThick

[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 theGsurface

Fig. E1.9

The 'counter-clockwise' Laves graph,

skeletal graph of the other labyrinth of theGsurface

Fig. E1.10

The enantiomorphic skeletal graphs of the two disjoint labyrinths of theGsurface

Fig. E1.11

Orthogonal projection on [100] plane of the enantiomorphic Laves graphs

Fig. E1.12

Another view of the 'counter-clockwise' Laves graph

47.6. Other pairs of dual skeletal graphs

The edges of the blue graph are all symmetrically equivalent, but the edges of the orange graph are clearly

Fig. E1.13a

Fig. E1.13bThe dual skeletal graphs of a hypothetical but

nonexistentembedded TPMS calledTO-TD(

TOstands fortruncated octahedron,

the interstitial cage of the blue graph.

TDstands fortetragonal disphenoid,

the interstitial cage of the orange graph.)blue graph: degree 4

orange graph: degree 14

The blue graph issymmetric.

The orange graph isregularbut notsymmetric.

notall symmetrically eqivalent. They are not even all of the same length. IfTO-TDexisted, it would be anon-balancedTPMS—i.e.,its two labyrinths would benon-congruent. Hence there could be no straight lines embedded in the surface, since such lines are c2 axes and would have the effect of interchanging the two labyrinths. Instead, the surface would be tiled by replicas of a patchSbounded bycurved geodesicsthat — because of the reflection symmetries of the union of the blue and orange graphs — are mirror-symmetric plane lines of curvature.In 1974, I tested for the existence of

TO-TDby using a laser-goniometer method I had devised in 1968 at NASA/ERC. This extremely tedious method requires the construction of a set of several straight-edged boundary frames of various proportions. The laser is used to measure the orientation of the normal to the surface of a [long-lasting] polyoxyethylene soap filmS'bounded by each of these frames at many points that are as close as possible to the edges of the frame. EachS'is a candidate for the surfaceadjointtoS. The adjoint curves computed for the edges ofSdemonstrated that it is impossible to 'kill the periods' and therefore thatTO-TDdoes not exist.In 2001, Ken Brakke used his Surface Evolver program to confirm this conclusion with enormously greater speed and accuracy than is possible with the soap film-laser technique.

Fig. E1.13c

The dual pair of skeletal graphs for another

hypothetical butnonexistentembedded TPMSThe green vertices define the sites of the Cu atoms,

and the blue vertices define the sites of the Mg atoms

in the binary alloyCu_{2}Mg, which has the structure called

Cubic Laves phase C15.

I call the Cu graphFCC_{6}(II). The Mg graph is the diamond graph (cf.Fig. E1.3d). Both graphs aresymmetric. The interstitial cavities in the Cu graph are of two kinds: smalltetrahedralcages and largetruncated tetrahedralcages. All the interstitial cavities in the Mg graph are identical: theexpanded 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

nottrue that for both graphs, every interstitial cavity of the graph contains a vertex of the dual graph. (The small tetrahedral cages of the Cu graph do not contain any vertex of the Mg graph.)

§48. TPMS mathematical backgroundThe Gauss map

Fig. E2.1

(stereo image)

Curved triangularFlächenstückABC

of Schwarz'sDiamondsurface

Fig. E2.2

(stereo image)

Riemann sphere

(unit sphere)The elementary minimal surface

FlächenstückABCshown in Fig. E2.1 is mapped

onto the spherical triangleABCon theRiemann sphereshown 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 pointsA,B, andCindicate

the directions of the surface normal vectors.There are

twelvereplicas of theFlächenstückABCin the skew

hexagonal faceEDAEof Schwarz's'D'A'Dsurface (cf.Fig. E2.1), but

there are onlysixcorresponding spherical triangles in the large spherical triangle

AEDon the Riemann sphere (cf.Fig. E2.2). The twoFlächenstückeABCandA,'B'C

for example, are both mapped onto the same spherical triangle. An entire lattice fundamental

region covers the Riemann sphere twice. As a consequence the mapping defines a two-sheeted

Riemann surface, with branch points at the eight 'cube corner' points likeC.

Fig. E2.3a

Stereographic projection onto the complex plane

of the elementary triangularFlächenstückABC

of Fig. E2.2

Fig. E2.3b

Triply periodic minimal surfaces are infinitely-multiply-connected, but it is nevertheless easy to characterize the topological complexity of every example of such a surface by computing the genuspof 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 anembeddedsurface,i.e., one that is free of transverse self-intersections.Since the smallest posssible value for the genus is three, Schwarz's

PandDsurfaces 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, theGauss-Bonnet theorem, which links thetopologyand thegeometryof a surface. One can very crudely express the essence of the Gauss-Bonnet theorem in this context by saying that the larger the value of the integrated Gaussian curvature for one lattice fundamental region of the surface, the steeper the saddle-like surface contours, and — therefore — the larger the number of tubular 'handles' in the surface as it 'bends around' this way and that.On p. 233 of the 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 numberFof faces, the numberEof edges, and the numberVof vertices of a convex polyhedron to prove that adding a handle to an orientable surface reduces the Euler-Poincaré characteristicΧ= 2 − 2pby 2 and therefore increases the genuspby 1. The proof simply updates the values ofF,E, andVafter two differentn-gons of a map on the surface are joined by a 'bent prism' (which is a convenient device for representing a handle).Fis increased byn− 2,Eis increased byn, andVremains unchanged. Since

X=F−E+V,

Χis reduced by 2.

Fig. E2.4

Recipe for calculating thegenus

of one lattice fundamental domain of aTPMS,

applied to Schwarz'sPsurfaceAnother way to calculate the genus is to

substitute fordfrom Eq. 5 in the equation_{G}p=1−d._{G}

|d| is equal to the number of times the Gauss map_{G}

of the minimal surface ('Gauss image') covers the Riemann sphere.

For Schwarz'sPsurface,d= − 2._{G}

The sign ofdfor surfaces of negative Gaussian curvature,_{G}

like minimal surfaces, is negative because thesense

of a geodesic edge-circuit on the surface isopposite

to that of its Gauss image on the Riemann sphere.If you're not familiar with the

Gauss map, look here.For discussion of the

Euler-Poincaré characteristic, look here.Χ=2 − 2pFor information about the

Gauss-Bonnet theorem, look here.

§49. TheD-G-Pfamily of associate minimal surfaces

Fig. E3.1a

A page from Schwarz's Collected Works

49.1 H. A. Schwarz'sdiamondsurfaceD

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 theDsurface in in hisGesammelte Mathematische Abhandlungen, Springer Verlag, 1890.

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

Fig. E3.1d

Fig. E3.1e

Stereo view of thelinear asymptotics

embedded in the 'crossed trianglesD-catenoid'

wire-frame of Schwarz's diamond surfaceD

(cf.Fig. E3.1d)

The ratio 2h/λof the triangle separation 2h

to the triangle edge lengthλis 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 [DandP]

cannot be expressed as elliptic functions of the coordinates.

Fig. E3.1f

Orthogonal projection of thelinear asymptotics

embedded in the 'crossed trianglesD-catenoid'.

Fig. E3.1g

Four translation fundamental domains of Schwarz'sDsurface

Each face is one of thehex_{90}faces shown below in Fig. E3.1h.

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

Fig. E3.1h

The hexagonal facehex_{90}ofDis

defined by the Coxeter map {6,4|4}.

Its face angles are 90º, and its area is

halfthe area of the hexagonal face

hex_{60}defined by the Coxeter map {6,6|3},

shown below in Fig. 3.1i.

Fig. E3.1i

The hexagonal facehex_{60}ofDis

defined by the Coxeter map {6,6|3}.

Its face angles are 60º, and its area is

twicethe area of the hexagonal face

hex_{90}defined by the Coxeter map {6,4|4},

shown above in Fig. 3.1h.

Fig. E3.1j

A rhombic dodecahedraltranslation fundamental domain

of Schwarz'sdiamondtriply periodic minimal surfaceDAs a toy model for generating the 'pipejoint' module of

Din 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 ofDin Fig. 3.1j

larger image

Fig. E3.1l

A translation fundamental domain ofDon which

approximations to closed geodesics (the red curves)

are inscribed. These geodesics are not plane curves.

larger image

Fig. E3.1m

A regular skewcurvilinearhexagon ofD,

which is a face of theregular map with holes{6,6|3}

larger imageThe inscribed regular skew hexagon with

straightedges

is a face of {6,4|4},

one of the threeregular maps with holesdescribed in

Generators and Relations for Discrete Groups,

H.S.M. Coxeter and W.O.J. Moser, Springer-Verlag, New York, 1965

and in

Infinite Periodic Minimal Surfaces Without Self-Intersections,

NASA TN D-5541, p. 49.

49.2 H. A. Schwarz'sprimitivesurfaceP

In 1966 I named this surface primitivebecause its two interwined labyrinths,

which are congruent, each have the shape of an inflated tubular version

of the familiarprimitive(orsimple cubic) graph (cf.Figs. E3.3d to E3.3k).

DandPareadjointsurfaces: each surface can be mapped into the other by anisometry

(the Bonnet transformation). Straight lines in one surface are mirror-symmetric

plane lines of curvature (plane geodesics) in the other.

Fig. E3.2a

Stereoscopic image of

the linear asymptotics (blue)

and plane geodesic curves (green)

in the 'square catenoid' ofP(cf.Fig. 1.2d, e, f)

Fig. E3.2b

Stereoscopic image of the linear asymptotics

embedded in the 'crossed trianglesP-catenoid'

wire-frame of Schwarz's primitive surfaceP

(cf.Figs. 1.2c, d, e)Here the triangles are only half as far apart as

the crossed triangles in Schwarz'sDsurface

(cf.Figs. 1.1d, e, f). The ratioh/λof the triangle

separationhto the triangle edge lengthλis equal

to √ 6 / 12 (~.204).The

PandDsurfaces are related by a dilatation

along any of the four [111] directions.An annular 'crossed-triangles catenoid' (CTC) minimal surface exists

for every value ofh/λless than some allowed maximum value (h/λ)_{max},

but there are embedded straight lines only in the CTCs ofDandP. The

proof depends in part onSchoenflies's theorem(cf.Fig. E2.1), which

proves that there are only six skew quadrilaterals, spanned by minimal

surfaces, that generate TPMS by half-turn rotations about their edges,

i.e.,by repeated applications of Schwarz's reflection principle.

Fig. E3.2c

Orthogonal projection of thelinear asymptotics

in the 'crossed trianglesP-catenoid' of Fig. E3.2b

Fig. E3.2d

A cubically symmetricaltranslation fundamental domain

of theprimitivetriply periodic minimal surfaceP

discovered and analyzed by H. A. Schwarz

in 1866 together with itsadjointsurface

D(cf.Fig. E3.1h).

Fig. E3.2e

A translation fundamental domain ofP

stereo image

Fig. E3.2f

Six translation fundamental domains of Schwarz'sPsurface

The lattice for

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

Pis the unstablestationary stateof an inflatedjungle-gym-like soap film. Any finite portion of such a soap film can be made stable if threads are stretched along a sufficient number of the embedded straight lines ('linear asymptotics').

As a sort of metaphor for the 'pipejoint' module of

Pin 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,−zsimultaneously, forming six cylindrical tubules attached symmetrically to the inside of the cube faces at their centers.Ppartitions R^{3}into two congruent inter-penetratinglabyrinths. Theskeletal graph(NASA TN D-5541, pp. 38-39) of each labyrinth is the graph of degree 6 whose edges are those of a packing of congruent cubes.In Figs. E1.4a-h below are images of tubular simple cubic graphs shown in both 'medium thick' and 'thick' versions. In the thick version, the diameter

dof the cylindrical tubes is the largest possible, consistent with the requirement that the dual pair of tubular graphs not intersect. Intersection occurs when the ratiod/e≥ 1/2, whereeis the edge length of the [thin] skeletal graph.

49.3 The gyroid surfaceG

Fig. E3.3a

A lattice fundamental domain of the gyroidG

(100) viewpointThe Coxeter-Petrie {6,4|4} map defines the

arrangement of thehex_{90}hexagonal faces.

Ghas the same symmetry as that of the union

of its two enantiomorphic skeletalLavesgraphs.

The lattice isbcc.

Gis the onlyembeddedsurface among the countable infinity of surfaces

that areassociates(cf.Fig. E1.2m) of Schwarz'sPandDsurfaces.

Gcontains 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 half-turn about an axis

of type (110) perpendicular toGat the midpoint of

the shared edge.

Fig. E3.3b

Stereo view of the lattice

fundamental domain of

Gshown in Fig. E3.3a

higher resolution image

(100) tunnels inG

Fig. E3.3c

A ninthhex_{90}face has been added here at the

top of the piece ofGshown in Figs. E3.3a,b.This orthogonal projection of

Gonto the (100) plane shows that the

projected outlineof the spiralling geodesic that bounds eachS

(100)-type tunnel inGisapproximately circular.For a higher-resolution version of this image, look here.

For astereoscopic perspectiveview, look here.

(111) tunnels inG

Fig. E3.3d

This orthogonal projection onto the [111] plane of the piece

ofGin Fig. E3.3a demonstrates that the (111) tunnels are

fatter than the (100) tunnels and not so nearly circular.For a high-resolution view, look here.

For a stereoscopic perspective view, look here.

Fig. E3.3e

The hexagonal tilehex_{90}ofG

(front view)The regular skew hexagon

hex_{90}is a face of the

Coxeter-Petrie {6,4|4} map. Its face angles

are 90º, and its area isone-halfthe area

of the hexagonal facehex_{60}defined

by the Coxeter-Petrie {6,6|3} map.(The hexagon

hex_{60}is shown in Figs. 3.3m, n.)

Fig. E3.3f

The hexagonal facehex_{90}ofG

(back view)

The front and back surfaces are not the same!

Fig. E3.3g

The hexagonal facehex_{90}ofG

(side view)

Fig. E3.3h

This image suggests thathex_{90}ofGcan be inscribed

in a truncated octahedron, but that is impossible.

Although the vertices ofhex_{90}coincide with six

vertices of the truncated octahedron, its edges are

not plane curves. Each edge approximates the shape

of a quarter-pitch of a helix. One half of each edge

lies inside the truncated octahedron, and the other

half lies outside. Alternate edges are curves of

opposite handedness.

Gcontains one replica ofhex_{90}in three of every four

truncated octahedra in a packing of truncated octahedra.

Fig. E3.3i

The quadrangular tilequad_{60}ofG,

a regular skew polygon

(front view)

Fig. E3.3j

The quadrangular tilequad_{60}ofG

(back view)

Fig. E3.3k

The quadrangular tilequad_{60}ofG

(side view)The tile

quad_{60}is a face of the Coxeter-Petrie {4,6|4}

map, which is thedualof {6,4|4}. Its face angles are

60º, and its area is equal totwo-thirdsthat ofhex_{90}.Every

quad_{60}face is related to each of four faces with

which it shares an edge by a half-turn about an axis of

type (110) perpendicular toGat the midpoint of the

shared edge.Each of these midpoints is also the midpoint of an

edge of adualhex_{90}face (cf.text below Fig. E3.3a).(Fig. E3.3l illustrates a well-known property of dual regular tilings

of the plane: the midpoints of edges of dual polygons coincide. Not

surprisingly, this property holds for dual regular polyhedra as well.)

Fig. E3.3l

A pair of dual regular graphs in the plane

Points likePlie at the coincident midpoints of a

pair of triangle and hexagon edges that intersect.

Fig. E3.3m

Asemi-regularskew 12-gon composed

of six replicas ofquad_{60}ofG

Its face angle sequence is

..., 60º, 120º, 60º, 120º,...Unlike the three

regularskew polygons

quad_{60},hex_{90}, andhex_{60},

this 12-gon does nottileG.

For a high-resolution view, look here.

Fig. E3.3n

The hexagonal facehex_{60}ofGis defined

by theself-dualCoxeter map {6,6|3}.

Its face angles are 60º, and its area is

twicethe area of the hexagonal face

hex_{90}shown above in Figs. 3.3e,f,g.

Fig. E3.3o

The hexagonal facehex_{60}ofG

(side view)

Every

hex_{60}tile is related to each of six faces with

which it shares an edge by a half-turn about an axis

of type (110) perpendicular toGat the midpoint of the

shared edge.

Fig. E3.3p

Assembly of approximately octahedral shape

tiled by the hexagonal faceshex_{90}ofG

view: (100) direction

Fig. E3.3q

Another view of the model ofGshown in Fig. E3.3p

view: (111) direction

§50. More images of the three Coxeter-Petrie infinite regular skew polyhedraAfter I met Norman Johnson in June, 1966, I realized

that each of these three polyhedra ishomemorphicto

— and has thesame symmetryas —

either the Schwarz surfaceD

or the Schwarz surfaceP.

{6,6|3} is homeomorphic toD, while

{6,4|4} and {4,6|4} are homeomorphic toP.

Regular skew polyhedron {6,6|3}

(homeomorphic toD)

Fig. E3.2g

Translation fundamental domain

The lattice of the oriented surface isfcc

Fig. E3.2h

Assembly of thirteen

translation fundamental domains

Regular skew polyhedron {6,4|4}

(homeomorphic toP)

Fig. E3.2i

Translation fundamental domain

The lattice of the oriented surface issc

Fig. E3.2j

Assembly of twenty-seven

translation fundamental domains

Regular skew polyhedron {4,6|4}

(homeomorphic toP)

Fig. E3.2k

translation fundamental domain

The lattice of the oriented surface issc

Fig. E3.2l

Assembly of twenty-seven

translation fundamental domainsFor a discussion of the Coxeter-Petrie regular skew

polyhedra, which includes animated graphics, see this

Wikipedia article.

§51. My early interest in TPMS (1966-1970)In the spring of 1966, I accidentally 'discovered' the Schwarz surfaces

PandDand then observed their close connection to the Coxeter-Petrie regular skew polyhedra. In order to represent the symmetry and combinatorial structure of both the surfaces and their flat-faced relatives, the Coxeter-Petrie polyhedra, I employed the metaphorical device ofdual skeletal graphs, which I'll callg_{1}andg_{2}. These are triply-periodic graphs regarded as lying centered in the interiors of the two intertwined labyrinths of these structures. In the discussion that follows, it is assumed (although not stated!) thatg_{1}andg_{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 soap-film-like material, and that the space inside the entire connected network of these tubes defines a single hollow — but shrunken — labyrintht_{1}. (Assume thatt_{1}has no self-intersections. Remember: this is ametaphoricalconcept, not a rigorous mathematical construction.)Here is how I described the relation between

g_{1}andg_{2}, the two skeletal graphs of a TPMS in 1970, on p. 79 of Infinite Periodic Minimal Surfaces Without Self-Intersections:

"Assume that the skeletal graph is given for one labyrinth of a given intersection-free TPMS. Let each edge of the skeletal graph be replaced by a thin open tube, and let these tubes be smoothly joined (without [self]-intersections) around each vertex so that the whole

tubular graphforms a single infinitely multiply-connected surface, which contains the skeletal graph in its interior. Such a tubular graph is globally homeomorphic to the corresponding minimal surface. If the tubular graph is sufficiently "inflated", it becomes deformed into adualtubular 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

inflatingt_{1}so that at itssummit,

(i) in its interior it contains exactly half of all space,

(ii) its surface has zero mean curvature everywhere,

(iii) it has the same symmetry as the configuration of the two dual skeletal graphsg_{1}andg_{2}, and

(iv) it has no self-intersections.

At the inflation summit, the surfacet_{1}is an embedded TPMS. Until it reaches the summit,t_{1}exhibits the symmetry ofg_{1}. At the summit, it has the symmetry of the union of the [intertwined]g_{1}andg_{2}. As the inflation proceeds beyond the summit, the surface exhibits the symmetry ofg_{2}. It eventually shrinks down to the thin tubular grapht_{2}, which has the symmetry ofg_{2}.Curiously,

the outside of.t_{1}is transformed into the inside oft_{2}In early 1969, the distinguished topologist Dennis Sullivan

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

_{ Dirk J. Struik 1894-2000 In 1967, I had the rare privilege of becoming acquainted with Dirk Struik. We both enjoyed hiking along the nature trails in Concord near my home. When Prof. Struik was more than 100 years old, I attended his lecture on the history of mathematics at an AMS meeting in Cincinnati. He was in top form. (He lived to be 106 years old.) }who had shown him a draft copy of my Infinite Periodic Minimal Surfaces Without Self-Intersections technical note. Dennis invited me to his office to explain that the transformation of the tubular graphs

g_{1}andg_{2}is an example of the classical Alexander-Pontryagin duality (which I had never heard of before).For the Schwarz surface

P, the skeletal graphsg_{1}and g_{2}are identical to the 6-valent simple cubic graph defined by the edges of an ordinary packing of cubes. For the Schwarz surfaceD, bothg_{1}and g_{2}are copies of the 4-valent diamond graph, whose edges correspond to the nearest neighbor links in the diamond crystal structure. Both of these graphs aresymmetric,i.e.,there is a group of symmetries that is transitive on all of its edges and on all of its vertices.I knew of only one other example of a symmetric triply-periodic graph

on a cubic lattice— the Laves graph. In the spring of 1966, I was seized by the notion that there must exist a TPMS with [enantiomorphic] Laves graphs for its skeletal graphs. I called it the Laves surfaceL. Unlike the skeletal graphs ofPandD, however, the configuration of two dual Laves graphs has no reflection symmetries, and its axes of rotational symmetry lie in directions that I determined could not possibly correspond to lines embedded in the surface I was seeking. As a consequence, I had no idea how to generate a surface patch bounded by either straight line segments or plane geodesics a ("Schwarz chain").By the time I moved to NASA in July 1967, I had made a reasonably thorough study of selected parts of Schwarz's writings on periodic minimal surfaces, and I understood the Bonnet associate surface transformation that defines the relation between

PandD(and also the relation between the catenoid and the helicoid). The brightly colored plastic models ofPandDI 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

. The models illustrated in Figs. E1.2k, E1.2l, and E1.17 show the final steps in the procedure that led to this pseudo-gyroid. The resemblance between this virtualpseudo-gyroiddoppelgängerand 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 toPandDthat happens to be free of self-intersections. In those days there were not yet any known examples of embedded TPMS derived by examining intermediate stages of the 'morphing' transformation that bends one minimal surface into its adjoint surfaceviathe 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

Hand its self-intersecting adjoint surface, and they found exactly one, which is now known as the lidinoid:

The lidinoid

(which was originally dubbed 'the HG surface'

by its Swedish discoverer, Sven Lidin)

Schwarz'sHsurfaceTo 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

PandDsurfaces — is not merelyregular(all vertices are of the same degree), but alsosymmetric(it is both vertex-transitive and edge-transitive). InHandCLP, the two other examples of Schwarz's TPMS, which — likePandD— 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 althoughsomeof the few known examples of symmetric triply-periodic graphs are skeletal graphs of labyrinths of such surfaces, by no means all of them are.)During the spring and summer of 1968, I concentrated on the writing of a so-called preliminary report (an internal NASA document, not intended for general circulation), entitled "Expansion-Collapse Transformations on Infinite Periodic Graphs", NASA/Electronics Research Center Technical Note PM-98 (September 1969), draft versions of two patent applications, and computer graphics animations of collapsing graphs. The considerably less time-consuming one of the two patent drafts was eventually entitled, "Honeycomb Panels Formed of Minimal Surface Periodic Tubule Layers".

I had discovered no useful ideas about how to prove that the pseudo-gyroid (

cf.Fig. E1.17) was the basis for abona fideminimal surface. Blaine Lawson told me in early August that he too had made no progress toward figuring out how to prove that a skew hexagonal face of the pseudo-gyroid, with its strictly helical edges, could somehow be analytically continued to generate an embedded periodic minimal surface.But ever since my first phone conversation with Blaine in late spring, I had found it extremely helpful to discuss with him a variety of questions concerning minimal surfaces other than the gyroid. I used him as a sounding board on some of my still tentative ideas about how to derive new examples of embedded TPMS by

(a) enumerating all the ways of constructing a Schwarz chain as a connected sequence of arcs on thenfaces (onearc on each face) of what is now called a 'Coxeter cell' — a convex polyhedral space-filler related to each of its neighbors by reflection in a face, and, somewhat later,

(b) 'hybridizing' two TPMS (cf. Fig. E2.71).In early September 1968, I returned to Cambridge from an AMS summer meeting at Madison, Wisconsin, where I had used the pseudo-gyroid model shown in Fig. E1.17 to illustrate my 15-minute talk (

cf.Fig. E2.10). I was still calling the surface the 'Laves surface' in those days.

A souvenir postcard

Lake Mendota,

from the Wisconsin Union Boat House

Madison, Wisconsin

(1968)

It was at this Madison summer meeting that I met several mathematicians

of my father's generation who knew something about minimal surfaces.

I particularly enjoyed meeting Wolfgang Wasow, who is

shown below in a 1952 photo with Magnus Hestenes.

(I met Hestenes in 1969 in Tbilisi, Georgia.)

Wolfgang Wasow (left) and Magnus Hestenes (right)

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

H. Blaine Lawson, Jr.

To change the subject, I told Blaine about the graph collapse transformation I had discovered, and how it could be 'run backwards' to provide the basis for the design of expandable space-frames. I had investigated the transformation for graphs associated with the

P,G, andDsurfaces (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 anellipsenot only in the associate surface transformation of Bonnet but also in the totally unrelated graph collapse transformation. I emphasized that there is no fundamental connection between these two transformations. I described how I had found that of the triply-periodic graphs that are associated withPandD, either as embedded graphs or skeletal graphs, those that have reflection symmetries are not candidates for expandable space-frames because of pairwise collisions of edges (calledwebsorstrutsby space-frame engineers) that occur early in the collapse. In contrast to this behavior, for all of the twisted graphs derived from the pseudo-gyroid, including the Laves graph, no such collisons of edges occur. The only collisions are the ones that would occur in actual physical spaceframes, in which struts collide somewhat before the 'complete collapse' stage because of their finite thickness.I had not previously even mentioned graph collapse to Blaine, and it's hardly surprising that he didn't seem to understand the details of what I said to him. It was obvious that I hadn't explained the elliptical trajectories coincidence very well, because Blaine's response was something like:

"Are you saying that the gyroid is associate to Schwarz's

PandDsurfaces?"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. 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.PandDthat happens to be embedded (free of self-intersections)Because I had spent the summer analysing the details of graph collapse transformations on

P,D, andG, I was aware that the surface orientation at the vertices of the hexagonal faces of the Coxeter-Petrie map {6,4|4} on the pseudo-gyroid is identical to the surface orientation at the corresponding vertices ofPandD. 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 toPandD! 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 inPandDcountless 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

shouldbe obvious are sometimes anything but obvious, I nevertheless felt stupid when I realized that I had posed the wrong question to Bob Osserman back in March. when I asked him whether there might be a way to derive the Weierstrass parametrization for a Schwarz chain composed of six helical arcs. I had mistakenly assumed that the edges of the hexagonal faces of the {6,4|4} map on the gyroid were perfect helices.Immediately after Jim Wixson joined NASA/ERC in January 1968, I asked him to write a FORTRAN program for calculating — from Schwarz's equations — the coordinates of a set of closely spaced points on the equatorial geodesic of the 'square catenoid' in

P. A simple soap-film demonstration suggests that although this curve appears to be approximately circular, it cannot be a circle. Consider its shape in the limit of very small separation of the boundary squares of the 'square catenoid'. In that limit it can be described roughly as a square with slightly rounded corners. As the separation of the boundary squares is increased, the curve looks more and more like a circle, but I found it impossible to imagine that it becomes exactly circular when the separation becomes equal to its value in thePsurface. When I plotted the points computed by Jim, I found that the equatorial geodesic departs from perfectly circular shape by slightly less than 0.5%.In October 1968, after understanding at last that the coordinates of every point in a lattice fundamental region of

Gare a simple linear combination of the coordinates of a pair of corresponding points inDandP, I plotted a graph of the orthogonal projection of the quasi-helical imageinS_{G}Gof the equatorial geodesicinS_{P}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' ofP(cf.Fig. 1.2d, e, f)

Fig. E3.2b

Orthogonal projection on the [100] plane

of the quasi-helical geodesicinS_{G}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

in Schwarz'sS_{P}Psurface 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

PandD. He had been justifiably confused by my rambling description of those irrelevant elliptical trajectories. When I explained to him the evidence for the associate surface relationship, he agreed that it was a reasonable idea. I immediately proposed that we publish together an announcement about the gyroid. He courteously refused, explaining that his crucial question to me was prompted by a misunderstanding of what I was saying. But I insisted that if he had not asked me that question in precisely those words, it would have been impossible to say how long it might have taken me to understand what was going on. He then reluctantly agreed to collaborate on a paper about the gyroid.Two days later, Dr. Van Atta returned to ERC from an out-of-town trip. He had been following my struggles with the pseudo-gyroid for months. As soon as I told him my exciting news, including my plan to co-publish with Blaine Lawson, he scolded me in no uncertain terms! He insisted that I phone Blaine and explain that I had made a serious error, and that I must publish alone. (It was the only time Dr. Van Atta ever displayed anger or impatience toward me.) Blaine was courteous when I relayed my new message to him, but I realized that my vacillation must have offended him.

Gradually I succeeded in feeling very slightly less stupid than I had at first, after reflecting on the fact that neither Schwarz, Riemanm, Weierstrass, nor any of their successors seem to have suspected the existence of an embedded surface associate to

PandD, 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 ontoM_{4}andM_{6}, theprecursorsof 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 thePandDsurfaces. In January 1968, because I was curious about the precise shape and arc length of the quasi-circular edges of a {6,4|4} hexagon ofP(cf.Fig. E1.2c), I sketched the outline of a computer program for getting answers to these questions. My colleague Jim Wixson coded the program in FORTRAN and ran it on ERC's PDP-11 minicomputer. The output of Jim's program, combined with Schwarz's analysis, provided the required clues to the value of the angle of associativity ofG(cf.Fig. E1.23). These results demonstrated that the departure from perfect circularity of the quasi-circular holes in a pipejoint unit cell ofPis in the range of approximately ± 0.5% of the hole's mean radius, implying a comparably small departure from perfect helicity of their image curves in the gyroid.Not only did the quasi-helical curves in the pseudo-gyroid (

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

In late 1968, I decided that I must somehow force a nodal polyhedron for

BCCinto being, and by trial-and-error I produced the space-filling saddle polyhedron shown In Figs. E2.70a, b, and c. It is described in Infinite Periodic Minimal Surfaces Without Self-Intersections._{6}

Fig. E2.70a

BCCPinwheel polyhedron:_{6}

the 6-facednodalpolyhedron

of the deficient symmetric graphBCCof degree 6_{6}

(stereo pair)

The vertices of the graph are the complete set of vertices of thelattice.bcc

BCCis described on p. 82 of Infinite Periodic Minimal Surfaces Without Self-Intersections._{6}

Fig. E2.70b

An oblique view of theBCCPinwheel polyhedron_{6}

(stereo pair)

Fig. E2.70c

TheBCCgraph (orange) and its dual graph (black)_{6}

(stereo pair)The edges of the black graph are the edges

of theBCCPinwheel polyhedron._{6}A property of the black graph that I regard as not strictly kosher is that

itintersectsthe orange graph. (Perhaps one can find a 'nicer' example

of an improvised nodal polyhedron for theBCCgraph.)_{6}

Fig. E2.10

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

announcing the discovery of the gyroid

I referred to the gyroid here as 'L' (for 'Laves').

A few weeks later I renamed it 'gyroid'.

This announcement was slightly premature! I mailed in the abstract in Fig. E2.10 a month or so before the Madison summer meeting of the AMS, even though I had not yet succeeded in proving that the surface represented by the pseudo-gyroid (

cf.Fig. E1.4c) is a single continuous minimal surface. I had naively assumed that any expert on minimal surfaces would be able to construct such a proof. A few months earlier, I had sent a plastic model of the pseudo-gyroid to Bob Osserman, who passed it along to his PhD student Blaine Lawson, Blaine promised to think about the problem in his spare time, even though he was already fully occupied with his dissertation research.It wasn't until about ten days after the Madison meeting that at last I understood — thanks to a fruitful phone conversation with Blaine — that the gyroid is an associate surface in the Schwarz

P−Dfamily. (The details are described below in §51.) By means of drawings based on hand calculations, I confirmed that there are no other intersection-free associate surfaces betweenPandD. A few months later, the differential geometer Tom Banchoff introduced me to Charlie Strauss, the mathematician/computer graphics expert who is his friend and collaborator. I hired Charlie to write a computer graphics program for producing stereoscopic perspective animations, and I used one such animation sequence (cf.Fig E1.21) to strengthen the evidence that every other associate surface is self-intersecting. Twenty-eight years later, this claim was at last proved rigorously by Karsten Grosse-Brauckmann and Meinhard Wohlgemuth, in their article, 'The gyroid is embedded and has constant mean curvature companions', Calc. Var. Partial Differential Equations 4 (1996), no. 6, 499-523.The precise version of the surface that I had in mind at the time of the Madison meeting had a fatal flaw: the boundary of each of its hexagonal faces is a chain of one-quarter pitches of

, alternately right-handed and left-handed (circular helicescf.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 hexagonslooks likea single infinitely-connected minimal surface, it isnotone. To explain why I had chosen these helical curves for the surface patch boundary, I summarize below — in approximately chronological order — the tangled sequence of events that culminated in the construction of the model ofL.In February 1968 I found my first promising lead in the hunt for the gyroid — a pair of related surfaces I named

M_{4}andM_{6}. One might call themwrinkledversions 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 ofM_{4}andM_{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 pseudo-gyroid, which is shown in Fig. E1.17. The details of how I got from the pseudo-gyroid to the gyroid are described in §7 and §8.

By September 1968 I had concluded that the gyroid is the only embedded surface among the countably many surfaces associate to

Fig.E1.17

The pseudo-gyroid

PandD, but my 'proof' was based on (a) physically bending assemblies of plastic replicas of surface modules and on (b) computer graphics animation of that bending (cf.Fig. E1.21b). In 1996, Karsten Grosse-Brauckmann and Meinhard Wohlgemuth published a rigorous proof that the gyroid is embedded (free of self-intersections) and contains neither straight lines nor reflection symmetries, inThe gyroid is embedded and has constant mean curvature companions, Calc. Var. 4, 1996, 499-523.It seems likely that before 1968, no one had ever bothered to look at any of the countably many intermediate surfaces associate to Schwarz's

PandDsurfaces to determine whether any of them were embedded. I confess that before 1968 it had never occurred to me to look there (even though itshouldhave!).The gyroid has received much attention from physicists, chemists, and biologists since the early nineties, because it has been found to be — at least approximately — a kind of geometrical template for a great variety of self-assembled bicontinuous structures, both natural and synthetic. I first announced its existence at an AMS meeting in August, 1968 (

cf.abstract in Fig. E2.10), in the mistaken belief that the pseudo-gyroid is abona fideminimal surface. (It is remarkablycloseto one!) The actual gyroidGis described on pp. 48-54 of Infinite Periodic Minimal Surfaces Without Self-Intersections, NASA TN D-5541 (May 1970), and also here in Fig. E1.23.

Weierstrass parametrization ofG

Fig. E1.2m

The rectangular coordinates ofG,

defined by Schwarz's solution (Weierstrass-Enneper parametrization)

for the entire family of surfacesassociatetoPandD

Ifθ_{G}in the terme^{iθG}is replaced by zero, the coordinates are those ofD.

Ifθ_{G}is replaced by π/2, the coordinates are those ofP— theadjointofD.(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 Schwarz-Christoffel mapping,

in Adam Weyhaupt's 2006 PhD thesis (pp. 115-116).

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

Fig. E1.2p

TheDsurface

Fig. E1.2q

TheGsurface

Fig. E1.2r

ThePsurfaceFor high-resolution pdf versions of these three images, look

here (D),

here (G),

and

here (P).

These pdf images will probably load slowly, because they are large (7 to 10 Mb). For maximum image clarity, zoom in to make the width of the image almost equal to the screen width. The associate surfacesDandPare called. Theadjointangle of associativity(cf.Fig. E1.23) by which they are relatedviaBonnet's bending transformation is π ⁄ 2.Like all of the other intermediate surfaces associate to the helicoid and the catenoid,

Gcontains neither straight lines nor plane lines of curvature.The round tunnels centered on [100], [010], and [001] axes, which are arranged in square checkerboard arrays, are bounded by approximately helical curves of opposite handedness in the two intertwined labyrinths of the surface. The outermost curved edges of the eight congruent hexagonal faces in Fig. E1.3a correspond to the edges of the

regular map with holes{6,6|3}. If you examine these edges closely, you will see that they are not quite plane. Each of them lies half inside and half outside the enclosing cube.

Let us call a geodesic curve on a triply-periodic minimal surface a

if it is either a straight line or a mirror-symmetric plane line of curvature, and atrivial geodesicotherwise. Since there are neither straight lines nor mirror-symmetric plane lines of curvature in the gyroid, all of its geodesic curves are non-trivial.non-trivial geodesicIf you apply rubber bands to physical models of the three surfaces

P,G, andD(as I have done), you will easily discover a variety ofperiodicgeodesic curves, bothandclosed. I will eventually show here images of several non-trivial geodesics, both closed and unbounded, on these three surfaces.unboundedFor a discussion of geodesics on multiply-connected surfaces, see Steven Strogatz's 2010 NYTimes essay on

geodesics, with links to Konrad Polthier's videos on this topic.

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

The embedded surfaces

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

Fig. E1.15d Fig. E1.15e

F-RDF-RD(r)

Fig. E1.15f Fig. E1.15g

I-WPI-WP(r)

A spectacular large model of Gis shown in the

YouTubevideo,

Gyroid playground climbing structure at the San FranciscoExploratorium

and also in Figs. E1.18a-d (below).This structure was designed and built by a team that included

Thomas Rockwell, Paul Stepahin, Eric Dimond, and John Kinstler.

Kinstler describes here how the plastic modules of the

Exploratorium gyroid were designed and fabricated.Here is a photo of a more recent

laminated plywoodgyroid sculpture at the Exploratorium.

The photographer was the formidable polymath Jef Poskanzer, who has informed me that

the young man inside the sculpture is his nephew Henry.

Fig. E1.18a

Exploratorium gyroid

photo courtesy of Amy Snyder

High-resolution photos of the Exploratorium gyroid

(courtesy of Amy Snyder) are at:

E1.18a

E1.18b

E1.18c

E1.18d

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

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

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

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

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

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

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

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

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

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

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

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

Cubic unit cell of G

_{ Fig. E1.22 Cross-eyed stereogram of a gauzy cubic unit cell of G G is associate to P and D, i.e., its rectangular coordinates are a linear combination of those of P and D (cf. Fig. E1.23). This figure shows twice as much of G as is contained in one translation fundamental domain. The lattice for G is body-centered cubic (bcc). Among the countable infinity of surfaces that are associate to P and D, G is the only embedded surface. A translation fundamental domain has genus 3. }

Cubic unit cell of G

The sequence of three stereo images immediately below illustrates the application of Ossian Bonnet's 1853

_{ Fig. E1.24 A view from a different corner (the lower right rear corner) of the unit cell of G shown in Fig. E1.22 The skeletal graphs of the two enantiomorphic labyrinths of G are enantiomorphic Laves graphs of girth ten (and degree 3). Since the lattice for G is bcc, a truncated octahedron is a reasonable choice for a translation fundamental domain. This cube unit cell has volume equal to the volume of two translation fundamental domains. }bendingtransformation — with no stretching or tearing — to the'morphing'of Schwarz'sDsurface into Schwarz'sPsurface. The orientation of the tangent plane at each point of the surface remains fixed throughout the bending. TheGsurface shows up a little more than one-third of the way (cf.Fig. E1.23) through the transformation. A simply-connectedtranslation fundamental domainof genus 3 is shown in each of these figures. Each such domain is composed of eight congruent regular skew hexagonal faces, which are defined by Coxeter'sregular map with holes{6,4|4} (cf.Fig. E1.4).The lattices for the three [oriented] surfaces are face-centered cubic (

fcc) forD, body-centered cubic (bcc) forG, and simple cubic (sc) forP.

Stereo view of G

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

A hexagonal face of G

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

Two hexagonal faces of G

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

David Hoffman and Jim Hoffman (not relatives!) have made this animation of the associate surface transformation P→G→Dfor a single translation fundamental domain.

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

The

"level-set"surfaceG*, which very closely resemblesG, is defined by the equation

cos xsiny+ cosysinz+ coszsinx= 0.David Hoffman and Jim Hoffman have illustrated the striking resemblance between

GandG*.

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
squarearray of quasi-helical tunnels. Adjacent tunnels spiral alternately CW and CCW.

- (g) view of the assembly in (c) from the (1,1,1) direction, showing the
kagoméarray of quasi-helical tunnels. Adjacent tunnels spiral alternately CW and CCW.

- (h) oblique view of the assembly shown in (g)
The quasi-helical curves ('flattened helices') in

G* that are centered onc4 axes coincide with right circular cylindrical helices of radiusπ/4 at just four points in each period, at exactly quarter-period intervals. At all other points helices fail to satisfy the level surface equation forG*. Halfway between each pair of consecutive points of coincidence, the radius of each flattened helix has a local minimum ≈π/4 − 0.015941383804981744. Hence this reduced radius is approximately 2.02972 % smaller thanπ/4.

L_{1}V_{17}, the chiral Voronoi polyhedron of a single Laves graphL

_{1}V_{17}is the Voronoi polyhedron of a Laves graph vertex. It is

shown below and also in Fig. II-2h, p. 90 in NASA TN D-5541.

_{ L1V17 has 17 faces: 2 hexagons, 3 octagons, 6 quadrangles, and 6 pentagons Like the Laves graph itself, L1V17 is chiral. }

_{ Here's a list of the coordinates of the 30 vertices of L1V17: }

_{ Below are three perspective views of L1V17 }

_{ The isometries of L1V17 include an axis of 3-fold rotational symmetry through the centers of its opposite hexagonal faces and axes of 2-fold rotational symmetry through the centers of each of its three octagonal faces. }

_{ Note that although L1V17 itself lacks reflection symmetry, each of its faces has reflection symmetry. The quadrangles and pentagons have d1 symmetry, the octagons have d2 symmetry, and the hexagons have d3 symmetry. (L1V17 is also shown in Fig. II-2h on p. 90 of Infinite Periodic Minimal Surfaces Without Self-Intersections.) }

_{ The images below show assemblies of two, three, four, and sixteen specimens of L1V17. }

_{ Two contiguous L1V17 cells Every cell is related to each of its three nearest neighbor cells by a halfturn about either of the two orthogonal C2 axes that bisect the octagonal face they share. These axes also bisect an edge of the associated Laves graph that is orthogonal to that face. }

_{ Three contiguous L1V17 cells }

_{ Four contiguous L1V17 cells This assembly of four cells is a translational fundamental domain of an infinite packing of these polyhedra. }

_{ Sixteen contiguous L1V17 cells (Four translational fundamental domains) }

_{ Evolute (aka 'net') of L1V17 }

_{ Schlegel diagram}_{ of L1V17 }

_{ A small portion of the triply-periodic surface composed of a space-filling assembly of L1V17 cells from each of which the three octagonal faces have been removed. }

_{ Orthogonally projected end view of the edges of L1V17 If the colors are ignored, this image has d3 symmetry. }

_{ If this Voronoi polyhedron were opaque, the green edges would be visible only from the front, the red edges would be visible only from the rear, and the black edges would be visible from both front and rear. }

L_{1}V_{20}, chiral Voronoi polyhedron of a

singlepartially collapsedLaves graphL

_{1}V_{20}is the Voronoi polyhedron of a vertex of a

partially collapsed Laves graph. It is shown below

and also — at a slightly more advanced stage of

collapse — in Fig. II-2g, p. 90 in NASA TN D-5541.

_{ L1V20 has 20 faces: 2 hexagons, 6 decagons, and 12 triangles Like L1V17 (cf. above), L1V20 is chiral. }

_{ Here's a list of the coordinates of the 36 vertices of L1V20: }

_{ Here's a view of a collapsing Laves graph at an earlier stage than the one that defines L1V20. }

_{ Below are several perspective views of L1V20, plus two views of the associated Laves graph at the stage of collapse that defines L1V20. }

_{ Above is a close-packed assembly of six translucent specimens of L1V20, each enclosing a vertex (blue sphere) of this partially collapsed Laves graph. The three green line segments incident at each vertex are edges of the graph. The shorter red line segments (which are not edges of the graph) have been inserted to demonstrate that some pairs of vertices that were separated before collapse by a distance greater than the edge length are now separated by a distance that is smaller than the edge length. }

_{ This is an optically exact Schlegel diagram of L1V20 produced by a generic notebook I wrote in Mathematica. }

_{ Here is a combinatorially correct but optically inexact Schlegel diagram of L1V20 that I designed by eye before I wrote the Mathematica notebook mentioned just above. }

L_{2}V_{17},chiralVoronoi polyhedron of the vertices

of the union of two dual Laves graphsL

_{2}V_{17}is the chiral Voronoi polyhedron whose two enantiomorphic

versions enclose the vertices of a pair of properly intertwined

dual Laves graphs. It is shown below (and also in Fig. II-2a, p. 89

in NASA TN D-5541). An infinite assembly of replicas of either

enantiomorph, in which adjacent polyhedra sharedecagonsonly,

fills just half of space. In apacking(space-filling assembly), which

is populated by equal numbers of the two enantiomorphs,hexagons

andquadranglesare shared by oppositely congruent enantiomorphs.

_{ Three views of L2V17, the chiral Voronoi polyhedron of a Laves graph vertex L2V17 has 17 faces: 2 hexagons, 12 quadrangles, and 3 decagons. }

_{ Schlegel diagram}_{ of one of the two enantiomorphs of L2V17 }

_{ ABCE, one of the twelve quadrangle faces of L2V17 }

_{ The isometries of L2V17 are identical to those of L1V17. They include an axis of 3-fold rotational symmetry through the centers of its opposite hexagonal faces and an axis of 2-fold rotational symmetry through the center of each of its three decagonal faces. }

_{ Furthermore, just like L1V17, L2V17 lacks reflection symmetry. Although its decagons have d2 symmetry and its hexagons have d6 symmetry, its quadrangles are asymmetrical. However, L2V17 has a curious combinatorial property that is distantly related to a property of Solomon Golomb's rep-tiles: each quadrangle can be tiled by three similar isosceles triangles. (Each of the two large triangles is exactly 50% greater in area than the small triangle.) }

The Voronoi cell of a vertex of a collapsing graph

On pp. 85-90 of Infinite Periodic Minimal Surfaces Without Self-Intersections,

I introduced the concept of thecollapseof a triply-periodic graph, and I included

images of the Laves graph at five stages of collapse. Computer animations of the

collapse of the Laves graph and also of the collapse of three other symmetric triply-

periodic graphs are included in my 1972 video Part 4, starting at time code 7^{m}04^{s}.

Also included is an animated sequence of views of the evolution of the Voronoi

polyhedron of a graph vertex. (Some other examples of the animation of Voronoi

polyhedron evolution were unfortunately lost, because of the confusion and disorder

that prevailed at NASA/ERC during its closing months in the spring of 1970, when

these computer movies were made. However, a few stills from these movies are

shown on p. 90 of Infinite Periodic Minimal Surfaces Without Self-Intersections.)

_{ Rotation of the edges of the Laves graph during the collapse transformation }

_{ The red edges and the vertices A0, B0, and C0 belong to the Laves graph before collapse. The blue edges and the vertices V, A1, B1, and C1 define the stage of collapse that produces the Voronoi polyhedron L1V20 shown above. }On pp. 662-664 of Reflections Concerning Triply-Periodic Minimal Surfaces, I

summarize some of the properties of the collapse transformation (which is applicable

only to triply-periodic graphs that aresymmetric). In the image shown above,

the vertex V at the center of the cube is a vertex of the Laves graph. The three red

line segments VA_{0}, VB_{0},, and VC_{0}represent the three edges of the graph incident

at V. From the vantage point of an observer fixed at V, each of the three red edges

rotates 90º in the course of the graph collapse. The continuous change in the relative

positions of the graph vertices producescontinuouschanges in theshapeof the

Voronoi polyhedron, andintermittentchanges in itscombinatorial structure.In the terminal [collapsed] state of the graph, each edge is mapped onto one of the

six edges (shown above in green) of a regular tetrahedron. For the configuration

of blue edges illustrated above, the edge rotation angleθ_{1}= cos^{-1}[3/√10] (≅18.4349º).

This configuration appears shortly after the first change, but before the second change,

in the combinatorial structure of the Voronoi polyhedron. If V is at the origin, then

A_{0}= (0,1,-1)/√2,

B_{0}= (-1,0,1)/√2,

C_{0}= (1,-1,0)/√2.

A_{1}= (0,1,-2)/√5,

B_{1}= (-2,0,1)/√5,

C_{1}= (2,-1,0)/√5.

In 1969, with the help of the computer graphics expert Charles Strauss and the camera engineer Bob Davis, I made 35mm movies of the collapse of several examples of triply-periodic graphs ( cf.NASA TN D-5541, pp. 85-88). Several of these movies, including the one in my 1972 video "Shapes of Soap Films Part 4", which is now on YouTube, showed the continuous transformation of the Voronoi cell that encloses each vertex of a collapsing graph. However, the only one of these movies showing Voronoi cell transformations that survives today is the one currently on YouTube. (I am planning to remake these missing movies.)During the transformation of the Voronoi polyhedron that encloses each vertex of a collapsing graph (a symmetric graph of degree

Z), if a particular vertexVof the graph is regarded as fixed, then theZvertices that are endpoints of the edges incident atVall travel along circular arcs centered onV. In its collapsed state, the infinite graph degenerates into the edge complex of a single convex polyhedron (cf.Reflections Concerning Triply-Periodic Minimal Surfaces for a few explanatory details, including illustrations). I plan to add here a detailed analysis of graph collapse, including a derivation of the surprisingly simple algebraic relation between (a) thefictitiouslinear displacementof vertices that would occur if instead of rotating edges, one translated vertices, allowing edges to stretch without limit and (b) theactualrotationof edges that occurs with edges treated as unstretchable.The sequence of Voronoi cells displayed in the 1972 video cited above demonstrates that the rearrangement of vertices produced by graph collapse has an interesting consequence: at discrete intervals, both the combinatorial structure and the symmetry of the Voronoi cell undergo an abrupt change. This can be seen in the 1972 video in the scenario entitled

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

_{2}V_{17}suddenly appears with its enlarged complement of symmetries at approximately 10 min:33 sec after the beginning of the scenario, at the very center of a symmetrical sequence of combinatorial types of polyhedra. But it is immediately preceded and followed by a continuous sequence of polyhedra of the same combinatorial type but lower symmetry.As stated above, L

_{2}V_{17}is the chiral Voronoi cell of a vertex of one enantiomorph of the Laves graph in the presence of its enantiomorphic dual. But the 1972 video demonstrates that L_{2}V_{17}is also the Voronoi cell of a vertex of the partially collapsed graph of degree 6 with the combinatorial structure and symmetry of the 'deficient' graphBCC_{6}— the graph composed of the edges and vertices ofM_{4}(cf.[cross-eyed] stereo image just below).Fig. E1.32 shows a set of screen-captured images from the 1972 video, extracted from the continuous sequence of geometrical transforms of the initial Voronoi cell, a truncated octahedron. Polyhedra of three combinatorial types appear in this sequence. L

_{ 30 skew quadrangles of M4 (stereo) view: [110] direction }_{2}V_{17}, at the center of the sequence, is the polyhedron labeled '10:32'. The arrangement of thecombinatorial typesin the sequence is symmetrical with respect to the center of the sequence, but theorientationof each polyhedron that precedes L_{2}V_{17}is related to the orientation of the matching polyhedron following L_{2}V_{17}by inversion in the center of the polyhedron, followed by rotation through a quarter-turn. The approximate value of the video time code for each polyhedron is listed directly below its image, and the combinatorial type of each polyhedron is illustrated by its Schlegel diagram, shown below its time code. The values of the time code indices demonstrate that the transformation is accelerating at a significant rate.The rearrangement of graph vertices produced by graph collapse is vaguely reminiscent of the atomic rearrangements that occur in

polymorphic transformationsof the crystallographic structure of many metals.NOTE:

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

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

_{ Top row: Screen-captured video images of Voronoi cells of the vertices of the partially collapsed M4 graph }

_{ The Voronoi cell at 10:32 is L2V17! }

_{ Bottom row: Schlegel diagrams of the Voronoi cells in the top row. }

_{ Each of these Schlegel diagrams displays the combinatorial structure of the polyhedron displayed above it, but the only optically exact projection is that of L2V17. Moreover, some of the diagrams have reflection symmetries not present in the corresponding polyhedron. }

VoronoiG, a polyhedral toy model

of the gyroid derived from L_{2}V_{17}In 1934, the German mathematician Berthold Stessmann published an article in Voronoi Gis a triply-periodic polyhedron with plane faces that roughly approximates the gyroid surface (cf.Figs. E1.34c to E1.34g below). It is derived by removing the three decagonal faces from each polyhedron in an infinite packing composed of the two enantiomorphs of the convex polyhedron L_{2}V_{17}(cf.Figs. E1.34a to E1.34b below).

_{ Fig. E1.34a Fig. E1.34b Stereoscopic views of the two enantiomorphic shapes of the chiral Voronoi cell L2V17 of a vertex of one Laves graph in the presence of its dual graph (This polyhedron is depicted in Fig. II-2a on p.89 of Infinite Periodic Minimal Surfaces Without Self-Intersections.) Fig. E1.34c Voronoi G, viewed in the [100] direction (stereoscopic pair) (For a higher-resolution [pdf] version of the above image, look here.) Fig. E1.34d Orthogonal projection of Voronoi G onto [100] plane (cf. Fig. E1.34c) Fig. E1.34e Voronoi G, viewed in the [110] direction (stereoscopic pair) (For a higher-resolution version of the above image, look here.) Fig. E1.34f Orthogonal projection of Voronoi G onto [110] plane This image demonstrates that the gyroid is symmetrical by a half-turn about each of its six axes of type [110]. Fig. E1.34g Orthogonal projection of Voronoi G onto [111] plane (For a higher-resolution [pdf] version of the above image, look here.) Compare the shapes of the tunnel holes with the corresponding holes in G shown in Fig. E1.19. Fig. E1.34h One chamber of Voronoi G, which is L2V17 minus its three decagon faces. All but six of its thirty vertices lie on the sphere of radius r1 shown here. The six outliers — shown as red dots — lie on a slightly larger sphere (not shown) of radius r2. The ratio r2/r1 ≅ 1.029. Uniform polyhedra based on G Figs. E1.35a,b The uniform gyroid (6.32.4.3), an infinite uniform polyhedron The faces of (6.32.4.3) are all regular plane polygons, and its symmetry group is transitive on both faces and vertices. (For a pdf version of the image in Fig. E1.35a, look here.) The combinatorial structure and symmetry of (6.32.4.3) are defined by the Poincaré hyperbolic disk model of uniform tilings in the hyperbolic plane. Fig. E1.35c The Poincaré hyperbolic disk model of 6.32.4.3) (Wikipedia image) Let me define two objects to be homologous if they have the same symmetry and the same topology. Since April 1966, I had been mulling over the homologous relations between the two Schwarz minimal surfaces P and D and the Coxeter-Petrie regular skew polyhedra {6,4|4}, {4,6|4}, and {6,6|3}, whose faces are regular plane polygons, My first model of the gyroid Fig. E1.4i A Voronoi surrogate of P — the Coxeter-Petrie triply-periodic regular skew polyhedron {6,4|4}, which is homologous to P. And here is one D surrogate: Fig. E1.2c A Voronoi surrogate of D — the Coxeter-Petrie triply-periodic regular skew polyhedron {6,6|3}, which is homologous to D. After I create an image of {4,6|4}, I will post it here. (cf. Wikipedia's animated images of the three Coxeter-Petrie regular skew polyhedra.) }

Both {6,4|4} and its dual {4,6|4} are homologous toP, and the self-dual {6,6|3} is

homologous toD. In 1968 I derived an example of a triply-periodic polyhedron with

plane faces that is homologous toG: VoronoiG(cf.Figs. E1.34c-g). It was known

from the work of Coxeter and Petrie that there exists noregulartriply-periodic skew

polyhedron with plane faces that is homologous toG. I wondered, however, whether

there exists auniformtriply-periodic polyhedron with plane faces that is homologous

toG. The abstract in Fig. E1.35c is a condensed summary of the results of my search

forquasi-regulartilings ofP,G, andD.Many of my notebooks, files, and physical models related to these tilings were destroyed

when an intruder ransacked offices at Southern Illinois University in 1982. I plan to

reconstruct this work and to post here images of more examples to supplement the single

example, (6.4)^{2}, shown in §14.After I showed my abstract (

cf.Fig. E1.35c) to Norman Johnson in 1969, he supplemented

it by deriving the combinatorial structure (although not the geometry) of every quasi-

regular or uniform tiling in the {4,6} family. When I investigated thegeometryof his

examples, I discovered that (6.3^{2}.4.3) is the only uniform tiling of the gyroid byplane

polygons in this family. The (6.3^{2}.4.3) tiling was later discovered independently by John

Horton Conway, who named itmu-snub cube. ('Mu' is an abbreviation formultiplehere.)Norman Johnson is well known for his enumeration of the 92 Johnson solids, which Zalgaller

later proved is exhaustive. He is currently writing a book on uniform polytopes inR^{3}andR^{4}.

(For decades, I have tried — without success — to persuade Norman to publish anNorman's list of the tessellations of the {4,6} family, which

account of his enumeration.)

he proved is complete, is shown (without Norman's permission) in Fig. E1.35b.

Fig. E1.35b

Norman Johnson's 1969 enumeration of

the uniform tessellations of the {4,6} family

(unpublished)

Fig. E1.35c

A summary of my enumeration of the

quasi-regular tessellationsof the {6,4} family

(abstract published by the American Mathematical Society in 1969)Fig. E1.35d (below) shows AMS Abstract 658-30 (1968),

cited in the first line of the Abstract in Fig. E1.35c (above).

Fig. E1.35d

Abstract 658-30, submitted in summer 1968

to the American Mathematical Society for the

Madison meeting in August

I gave a 15-minute oral presentation there that was based on this work.

Uniform polyhedron models of P

Fig. E1.36

(6.4^{3}), an infinite uniform polyhedron

model constructed by my Cal Arts student Bob Fuller in 1971

Fig. E1.37

(6.4^{3}), an infinite uniform polyhedron

model constructed by Bob Fuller in 1971

Fig. E1.38a

(6.4^{3}), an infinite uniform polyhedron

model constructed by Bob Fuller in 1971

Fig. E1.38b

Stereo photo of Bob Fuller's model of (6.4^{3})

Isaac Van Houten's bronze sculpture of G

Fig. E1.39a

Isaac Van Houten's 2006 bronze sculpture of the gyroidG

Fig. E1.39b

Template used by Isaac to cast his sculpture ofG

Bathsheba Grossman's 3D-printed sculpture of G

Fig. E1.41

Two views of one of Bathsheba Grossman's printed models ofG

Mathematische Zeitschrift38, 1934 (414-442) entitled "Periodische Minimalflächen".A book by Stessmann, also entitled " Periodische Minimalflächen ", was published by J. Springer in 1934.

* * * * I have been unable to learn what became of Stessmann in the WWII period. (Perhaps there are some leads in here.) If you discover any information about him, please email me.

* * * * On the first page of Stessmann's article there are drawings of the six skew quadrilaterals that Arthur Moritz Schoenflies proved in 1890 are the only skew quadrilaterals, spanned by minimal surfaces, that generate TPMS by half-turn rotations about their edges,i.e.,by repeated applications of Schwarz's reflection principle. These six Schoenflies quadrilaterals are reproduced here in Fig. E2.1.

The six Schoenflies quadrilaterals

Fig. E2.1

The six Schoenflies quadrilateralsOf the six Schoenflies quadrilaterals, only I and III are patch boundaries for

embeddedsurfaces — Schwarz'sDandP, respectively (cf.Figs. E1.2a and E1.3a). By applying Schwarz's reflection principle, it is easy to demonstrate that the other four quadrilaterals — II, IV, V, and VI — define surfaces with transverse self-intersections.

But what about theadjoints(cf.text below Fig. E1.23) of II, IV, V, and VI?

It is a fundamental property of any two adjoint minimal surfacesS_{1}andS_{2}that if a boundary edge inS_{1}(say) is a straight line segmentE_{1}, then its image inS_{2}is a segment of a plane line of curvatureC_{1}(a plane geodesic) that lies in a plane perpendicular toE_{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 nameC_{9}(P) — or justC(P) — because it has a remarkable property that I believe was never mentioned in writing either by Neovius or Schwarz: it has exactly the same embedded straight lines asPand has genus 9. (The 'C' inC_{9}(P) stands for 'complement'.)

C_{9}(P) is illustrated in Figs. E2.2a,b,c, and its self-intersecting adjoint surfaceC_{9}(P)† is illustrated in Fig. E2.2d.Edvard Neovius (1851-1917), who became Professor of Mathematics at the University of Helsinki, was a cousin of Ernst Lindelöf and uncle of Rolf Nevanlinna. (For further information, see this short article about the history of the Finnish Mathematical Society.)

On June 27, 2012, one day after my wife Reiko and I visited the magnificent

The embedded Neovius surfaceC_{9}(P) and its self-intersecting adjoint surfaceC_{9}(P)†

(a) (b)

Figs. E2.2a and b

One lattice fundamental region of

the embedded Neovius surfaceC_{9}(P) (genus 9)

The images are copied from Tafeln II and III of Neovius's 1883 dissertation.

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 ofeightlattice fundamental regions of the embedded Neovius surfaceC_{9}(P)

and

(d) an assembly ofsevenlattice fundamental regions of its self-intersecting adjoint surfaceC_{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 ofC_{9}(P)

The 'see-through' tunnels are aligned in [110] directions.

Fig. E2.3b

A later model ofC_{9}(P)

The possibility that there exists an embedded , a TPMS that is

Fig. E2.4a

Announcement in a February, 1969 abstract published by the American Mathematical Society

describingC(D)complementaryto Schwarz'sDsurface"Abstract 658-30" mentioned in the third line

and shown here as Fig. E2.10, refers to the gyroid,

which I originally named "L" (forLaves) in 1968.

counterpartofC(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'sPsurface. 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

PandDcan be regarded as [infinite]regular polyhedra, it is plausible that ifoneof them (P) has an embedded companion surface that contains the same set of straight lines, then theother(D) probably does too. I concluded that if I could confirm the existence of such a surface forD, 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

Dto recognize instantly that the embedded surfaceC(D) exists. A few months later, I realized that higher order complements of bothPandDprobably exist too (cf.discussion of 'Notched adjoints' following Fig. E2.82).The MIT librarian told me that no one had borrowed Neovius's thesis during the fifty years since the library had acquired it. It was printed in large folio format, and as a result she had to cut all the pages into quarters (which took her a good while!) before she could make a photocopy for me. Perhaps that's a good indication of how interested mathematicians were in periodic minimal surfaces in those bygone days.

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

C(D)

Fig. E2.4b

C_{19}(D) (genus 19)

Right: One translation fundamental domain

Left: One-fourth of a translation fundamental domain

C_{19}(D) is called the [first-order]complementofD,

because it has exactly the same embedded straight lines asD.

Ken Brakke's images of a sequence of

higher-order complements ofC_{19}(D)

of genus 35, 51, 67, ..., which belong to two families,AandB,

are shown

Fig. E2.4c

C_{19}(D)

Two translation fundamental domains

Fig. E2.4d

C_{19}(D)

One-fourth of a translation fundamental domain,

cut from two different parts of the surface

Fig. E2.4e

What's left in 2011 of the brittle 1970 model ofC_{19}(D) shown in Fig. E2.4b

view: [111] direction

Fig. E2.4f

C(D)

view: [-1-1-1] direction

Fig. E2.4g

C(D)

view: [110] direction

What about the adjoints of Schoenflies IV, V, and VI? When (in 1968) I first considered these three quadrilaterals, I was startled to realize that although both the triply-periodic surface derived from Schoenflies IV and the one derived from its adjoint are obviously self-intersecting (both of them have a 120º corner and therefore a branch-point), it seemed highly likely that the adjoints of both Schoenflies V and Schoenflies VI are patches for

embeddedsurfaces! I developed persuasive experimental evidence in support of this conjecture by (a) blowing on soap films inside tetrahedral cells to model the adjoint patches and (b) performing Bonnet bending of physical replicas of Schoenflies V and Schoenflies VI made from vacuum-formed plastic sheet material. I chose the nameF-RDfor the embedded adjoint of Schoenflies V (which has genus 6) and the nameI-WPfor the embedded adjoint of Schoenflies VI (which has genus 4). My naming conventions are explained on p. 38 of Infinite Periodic Minimal Surfaces Without Self-Intersections.Andrew Fogden confirmed that

F-RDis embedded, in his 1992 article entitled "A systematic method for parametrizing periodic minimal surfaces: the F-RD surface", Journal de Physique 2 (1992) 233-239. Fogden succeeded in deriving an implicit equation (fifth degree polynomial) for theF-RDWeierstrass polynomial, a task that was begun by Berthold Stessmann in 1934 but not completed (cf.Stessmann B., Periodische Minimalflächen, Mathematische Zeitschrift 33 (1934) 417-442). Because it is impossible to solve for theF-RDWeierstrass polynomial in explicit form, its surface coordinates can be computed only numerically.In my 1997

F-RDposter essay, I summarize some of Fogden's remarkable results forF-RDandI-WP.In a 1992 collaboration between the versatile mathematician Djurdje Cvijović and the Cambridge University chemist Jacek Klinowski, the authenticity of

I-WPwas rigorously established and several of its quite surprising mathematical properties were derived (Cvijović, D. and Klinowski, J.: The computation of the triply periodic l-WP minimal surface, Chemical Physics Letters 226 (1994) 93-99). The analysis ofI-WPturned out to be far simpler than that ofF-RD.In every known example of an embedded TPMS that contains no straight lines, the two labyrinths are found to be non-congruent. The versatile crystallographers Elke Koch and Werner Fischer have classified such surfaces as

and surfaces that contain straight lines — likenon-balancedPandD— as. BecausebalancedI-WPandF-RDcontain no straight lines, they are non-balanced. These two surfaces are illustrated in Figs. E2.5 and E2.6.

F-RDandI-WPwere the first examples of non-balanced TPMS to be identified, but in recent years many additional examples of such surfaces have been discovered.

I-WP

Fig. E2.5a

I-WP(genus 4)

oblique view

_{ Fig. E2.5b I-WP Modeled by Ken Brakke, using his Surface Evolver software oblique view }

Fig. E2.5c

I-WP

view: [111] direction (approximately)

Fig. E2.5d

I-WP

view: [100] direction

Portions of two non-trivial geodesics (cf.discussion below Fig. E1.24) are shown.

The one is at the left is closed, and the triply-periodic one at the bottom is unbounded.

Fig. E2.5e

I-WP

view: [100] direction

Fig. E2.5f

I-WP

view: [100] direction

Fig. E2.5g

I-WP

oblique view

F-RD

Fig. E2.6a

F-RD(genus 6)

view: [111] direction(See my 1997

for details.F-RDposter essay)I am indebted to

John BrennanandRobert Fuller, two of my truly outstanding students

at California Institute of the Arts, for volunteering in 1971 to complete this large model of

F-RD. In 1983, when I returned to Southern Illinois University/Carbondale from an extended

trip, I was informed by two junior teachers that a marauder (whose identity I never learned)

had recently trashed several of our offices and laboratories, including mine. They also said

they had already 'cleaned up the mess' in order to prevent an official investigation. To my

horror, I found that several of my research notebooks and almost all of my models, including

F-RDand several others made by John and Bob, were nowhere to be found in either my

office or my lab. (I was informed by the same two junior teachers that it was too late for me

to inspect any of the remains, because all the trash had long since been carted off to the dump

and buried. I was also told that it had been decided not to report what had happened to the

university police, because the attendant publicity would reflect badly on the Design

Department. To my great regret, I acceded to the demands made by these two young men.)

Fig. E2.6b

F-RD

view: [111] direction

Fig. E2.6c

F-RD

view: [110] direction

Fig. E2.6d

F-RD

view: [100] direction

Fig. E2.6e

F-RDtranslation fundamental domain

view: [111] direction

Fig. E2.6f

F-RD

One-half of a translation fundamental domain

Fig. E2.6g

F-RD

One translation fundamental domain (stereo view)

Fig. E2.6h

F-RD

One-and-a-half translation fundamental domains

MY MEANDERING ODYSSEY

A more or less chronological account

of some of my physics research

before I became interested

in minimal surfacesI 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

thisstory. So much of what I did depended on chance events that it may sometimes seem to the reader like a sort of random walk. It was more than a decade after my journey began that I first had an inkling of where it might lead. For those readers who make it all the way to the end, I can only say, "Mazel Tov!"

1953-1957

Fig. E2.10

The author in 1954 sectioning a single crystal of silver at the jeweler's lathe

used for radioactive tracer studies of atomic diffusion in metals and alloys.

This apparatus — and most of the other equipment we used —was designed

and/or assembled by Prof. David Lazarus, assisted by Carl T. Tomizuka, my

distinguished predecessor at the University of Illinois in Urbana/Champaign.

At the University of Illinois in the 1950s, research in condensed matter physics was heavily weighted toward the study of point defects in metals, semiconductors, and alkali halides. For my PhD research in David Lazarus's group, I made radioactive tracer measurements of atomic diffusion coefficients as a function of temperature and alloy composition in single crystals of Ag-Cd and Ag-In, using experimental techniques developed by Dave, his post-docs, and the students (principally Carl Tomizuka) who preceded me. Although I enjoyed this work at first, my progress was slow, and as I looked in awe at the accomplishments of some of my classmates, I gradually began to question whether I was temperamentally suited for a career as an experimental physicist. Besides, it seemed to me that my thesis topic did not have much scientific significance, and I saw little prospect of making any fundamental advance in the field of diffusion.I was fascinated, however, by the mathematics of random walks on lattices — a famous example of Brownian motion. This fascination eventually led me to my first significant discovery in physics — that by measuring the isotope effect for self-diffusion in an elemental crystal, one could distinguish between 'substitutional' diffusion and 'interstitial' diffusion. This had not previously been possible. How this came about is described below.

One day in the spring of 1957, I read a 1952 paper by John Bardeen and Conyers Hering entitled, "Diffusion in Alloys and the Kirkendall Effect". Appendix A of that paper is an analysis by Hering of the difference between the diffusion coefficient of a

vacancy(vacant atomic site) and that of anatom. It had long been widely accepted that the mechanism for self-diffusion in noble metals, for example, is the exchange of an atom with an isolated vacancy. The concentration of vacancies was known to be relatively dilute, even at the elevated temperatures required for observing self-diffusion. As a consequence, after an atom has exchanged positions with a particular vacancy, it is somewhat more than randomly likely that the next jump of that atom will be an exchange with thesamevacancy. 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 acorrelatedrandom walk. An atom in aninterstitialposition (e.g.a lithium atom in a germanium crystal), on the other hand, is believed to hop from one interstitial site to another with no correlation between the directions of consecutive jumps. Its diffusion is characterized as a strictly random walk.Hering proved that if atoms diffuse by the vacancy mechanism ('substitutional' diffusion) and the vacancies are relatively dilute, then the diffusion coefficient for an atom is smaller than the diffusion coefficient for a vacancy by a fractional amount that depends on the coordination number (number of nearest neighbors of an atom) of the host crystal. The smaller the coordination number, the larger this fraction. In a crystal with cubic symmetry in which a vacancy jumps a distance

awith frequencyΓin anuncorrelatedrandom walk, the diffusion coefficient for thevacancyis given byD_{vacancy}=a^{2}Γ/ 6. (E2.11)

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

D_{atom}=fa^{2}Γ/ 6, (E2.12)where the

correlation factorfis given by

< cos f= (1 +<cosθ>_{Av}) / (1 −<cosθ>_{Av}); (E2.13)

θ>_{Av}is the average value of the cosine of the angle between two consecutive jumps of an atom.Since the diffusion of an atom in an interstitial position does not involve an exchange with a vacancy, the directions of its consecutive jumps are uncorrelated.

I had a hunch that in crystals of cubic symmetry, Bardeen-Hering correlation would reduce

boththeself-diffusion coefficientand theisotope effectfor self-diffusion by the same fractional amount. This turned out to be the case. LetDand_{α}Dbe the self-diffusion coefficients of isotopes_{β}αandβof massmand_{α}mand correlation factors_{β}fand_{α}f, respectively. I defined the_{β}

isotope effect = (( D/_{β}D) − 1) / ((_{α}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. If this conjecture were correct, one could distinguish between interstitial self-diffusion and self-diffusion by the vacancy mechanism in cubic crystals simply by measuring the isotope effect for self-diffusion. I first estimated the influence of Bardeen-Hering correlation on the isotope effect by using an approximate model of correlation published by Alan LeClaire and Alan Lidiard in_{β}Phil. Mag.,1, 518 (1956). This rough estimate appeared to confirm my hunch that in crystals with the required symmetry,the isotope effect is equal to the correlation factor.

1958-1964In 1958, in order to refine my calculation of the influence of correlation on the isotope effect, I designed a Fortran program for extending it to a higher order of approximation. In this program, I modeled the infinite crystal by a sequence of four cubically symmetrical sub-crystals of successively larger volumes, each centered at the initial site of the diffusing atom.

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

Fig. E2.14

The four cubically symmetrical sub-crystals in my

Fortran program for the random walk of a vacancyThe top row lists the number of atomic sites in each sub-crystal.

The vacancy is located at the center of each sub-crystal.

The four sub-crystals contain

nearest neighbors,

2^{nd}nearest neighbors,

3^{rd}nearest neighbors,

and

4^{th}nearest neighbors

of the vacancy, respectively.

Phil. Mag.,4, Issue 44, 1959, pp. 899-906).

In 1960, I derived complicated combinatorial expressions for the the Bardeen-Hering correlation factor for self-diffusion by the vacancy mechanism on the sites of four 2-dimensional and four 3-dimensional crystal structures. A few weeks later, while I was visiting the University of Chicago, I showed these expressions to the theoretical physicist Junjiro Kanamori, who immediately described for me a widely used method of transforming such combinatorial expressions into multiple integrals. Thanks to Kanamori's help, Robert W. Lowen, Jr. and I were then able to evaluate correlation factors for seven of these eight structures. (I don't recall whether we ever completed our calculations for the eighth structure, the face-centered cubic lattice, for which Hering had obtained a value of 0.78.) We reduced the correlation factors for the other three 3-dimensional cases to triple elliptic integrals and published our results in the Bulletin of the American Physical Society, April 1960,

Junjiro Kanamori

4, No. 5, p. 280 (cf.Figs. E2.15a,b).

Fig. E2.15a

Computed values of Bardeen-Hering correlation factors (APS abstract)

After 1959, I tried — with limited success — to invent a systematic

STRUCTUREZ− < cosθ>_{Av}f

Fig. E2.15b

linear chain 2 1 0 honeycomb layer 3 1/2 .333333 square layer 4 1 − 2/ π.466942 triangle layer 6 5/6 − √ 3/ π.566057 diamond 4 1/3 .500000 simple cubic 6 .209841 .653120 body-centered cubic 8 1 − Γ^{4}(1/4)/8π^{3}− 8π/Γ^{4}(1/4).727194

Correlation factors for seven crystal structures

dualityrecipe for associating infinite periodic graphs in pairs to represent plausible geometrical pathways for diffusing atoms in bothvacancy-exchangediffusion andinterstitialdiffusion. It never occurred to me to imagine a surface of some sort separating such pairs of graphs until 1964, when I learned about the Coxeter-Petrie[infinite] regular skew polyhedra, (Only in 1966 did I realize that these three infinite polyhedra are 'flattened and folded' incarnations of Schwarz'sPandDsurfaces. When I met Donald Coxeter for the first time at a 1966 geometry conference in Santa Barbara, he told me that he had never heard of the Schwarz surfaces. He looked thoroughly startled when I showed him my plastic models ofPandD, which he examined closely for a couple of minutes or more before saying a word.)I catalogued a variety of examples of crystal structures that could be neatly partitioned into two disjoint substructures, and I computed the shapes of the Voronoi cells for many examples of unary, binary, and ternary crystal structures. From time to time I made wooden models of many of these polyhedra and used some of them as nodes for ball-and-stick network models of crystal structures. An especially useful resource for me in those days was 'Third Dimension in Chemistry', by Alexander F. ('Jumbo') Wells (

cf.John Tanaka's oral history interview of Wells). It was from this book by Wells that I learned, in 1958, of the existence of the Laves graph.A graph is called

symmetricif all of its vertices are symmetrically equivalent and all its edges are symmetrically equivalent. Another way of saying this is: Asymmetric graphis one that is bothedge-transitiveandvertex-transitive. Aregular graph, on the other hand, is one in whichevery vertex has the same degree. Hence every symmetric graph is regular, but not every regular graph is symmetric.I know of only three symmetric graphs on cubic lattices: the simple cubic graph (degree six), the diamond graph (degree four), and the Laves graph (degree three). I imagined in fantasy an elemental crystal whose atomic sites correspond to the vertices of a single Laves graph, with self-diffusion occurring by means of atom-vacancy exchanges. As an additional part of the fantasy, I imagined measuring the isotope effect for self-diffusion in this hypothetical crystal. It seems that the Bardeen-Hering correlation factor (

cf.Eq. E1.3) has never been computed for the Laves graph, but it is likely to have a value of less than one-half, since the degree of the Laves graph is smaller than that of the diamond graph. (The data in Fig. E2.15b suggest the possibility that this correlation factor may be exactly 1/3. I may calculate it some day, just for fun!) Consequently the isotope effect, which theory says is equal to the correlation factor, would also be less than one-half. But I realized that the fantasy was far-fetched, because the interstices in this hypothetical crystal would be so large that self-diffusion would not necessarily occur by a simple vacancy exchange mechanism.On the other hand, suppose there exists a strongly ordered binary intermetallic compound in which the atoms of the two elements sit on the respective sites of two dual Laves graphs. Such a structure would be analogous to Zn-S (zinc sulfide), or In-Sb (indium antimonide), but with coordination number (degree) of each subgraph equal to three, not four. In a hard sphere model of such a structure, the interstitial cavities would be of modest size if the sphere radii for the two species were not grossly different. Radioactive tracer measurements for each species of the isotope effect for self-diffusion would provide evidence for or against the hypothesis that self-diffusion occurs by the vacancy mechanism.

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

_{ A pair of dual Laves graphs Every node (wooden triangle) in each graph is joined by a wooden dowel not only to its three nearest neighbors in the graph, but also to its two nearest neighbors in the dual graph. }

UPDATE:

_{ Fig. E2.16 Toshikazu Sunada }

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

- On the K
_{4}Crystal- Diamond's Chiral Chemical Cousin.
- my 2008 letter to AMS Notices about the Laves graph

1964 − April 1966In July 1964, after spending a few stimulating months consulting for a new division of Beckman Instruments on the design of apparatus for measurements of the Mössbauer effect, I joined the Physics Research Laboratory of Space Technology Laboratories (STL) in Los Angeles. Within a few months, STL was acquired by TRW and changed its name to TRW Systems. I continued — from time to time — to ponder the question of how to develop a 'partitioning algorithm' for inter-penetrating pairs of triply-periodic graphs.

April 1966 −April 1967One day in April, 1966, in a hallway of the TRW Physics Research Laboratory, I noticed an engineer who was drawing polygons on a large plastic sphere. When I [politely] asked him what he was doing, he replied with some impatience that he was trying to model a fly's-eye lens by arranging a few hundred hexagons on a sphere but was having some difficulties. (In 'Ernst Haeckel (1843 − 1919) is still a problem',

Eclectica(2009), Alan Mackay describes a similar error made — and subsequently corrected — by Ernst Haeckel.)Trying not to sound patronizing, I suggested to the engineer that he might consider including some pentagons, and I explained why the patterm he was searching for didn't exist. I told him the famous story about Euler and the bridges of Königsberg, and I explained that Euler had derived a simple equation that accounts for every possible combination of polygons that tile the sphere. Because it was obvious that he was somewhat less than pleased by my butting in, I decided not to pursue the matter further. However, I did casually mention the incident to my supervisor.

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

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

Two of Peter's models each contained a specimen of what he called

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

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

- each vertex of the graph
gis joined by an edge to every one of theZnearest neighbor vertices (the graphgis described as beingof maximum degreewith respect to the vertices.

- each vertex lies at the centroid of the positions of the
Znearest neighbor vertices (the graphgis described as beinglocally centered).

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

One (

cf.Fig. E2.20) of Peter's two saddle polyhedra filled an interstitial cavity of thediamondgraph, a symmetric graph of degree four on the vertices of thediamond crystal structure, while the other (cf.Fig. E2.25) filled an interstitial cavity of thebody-centered cubic(bcc) graph, a symmetric graph of degree eight on the vertices of the. Each of these saddle polyhedra is called thebcclatticeinterstitial polyhedronof the graphgand has the following properties:

Because the diamond graph happens to be self-dual, if every vertex of the graph is enclosed by a replica of the interstitial polyhedron, the assembly of such polyhedra — just like the assembly of interstitial polyhedra that occupy the interstitial cavities — define a packing of

- (a) the interstitial polyhedron and the graph
ghave the same point groupsymmetrywith respect to the center of the cavity;- (b) the number of
facesof the interstitial polyhedron is equal to the degreeZ(number ofedgesincident at each vertex) of a second [dual] graphg, in which there is a vertex_{interstitial}vat each cavity center andZedges — incident atv— that protrude through the faces of the interstitial polyhedron. Each of theseZedges is incident also at a vertexvof one of theZadjacent interstitial polyhedra.

R^{3}. In this role, these saddle polyhedra are callednodal polyhedra. The nodal polyhedron of thebccgraph 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

Fof faces of the space-filling Voronoi polyhedron that encloses each vertex is greater thanZ. The simplest example of a pair of symmetric graphs that illustrate the duality expressed by properties (a) and (b) is a pair of simple cubic (sc) graphs (cf.Fig. E2.58). My goal was to incorporate the concept of saddle polyhedron in a procedure that defines this duality in a systematic way.Among the many triply-periodic graphs that exhibit both properties (a) and (b) defined above are the seven symmetric graphs listed in the table below. The

fccgraph and theFCC_{6}(I) graph are the only examples among these seven for which thedualgraphs are not also symmetric. For thefccandscgraphs, both the nodal and interstitial polyhedra happen to beconvex.

GRAPHZFRELEVANT FIGS.

Fig. E2.17

Laves 3 17 E2.41, E2.42 WP4 12 E2.36 diamond 4 16 E2.19c sc6 6 E2.58 − E2.60 FCC_{6}(I)6 12 E2.46 − E2.49 bcc8 10 E2.24b, E2.36 fcc12 12 E2.56d − E2.56f

Examples ofsymmetricgraphs on cubic lattices

= degree of the graph (coordination number).Z

= number of faces of the Voronoi polyhedronF

associated with each vertex.

graphDiamond

Fig. E2.18

dual graph diamond graph nodal polyhedron expanded regular tetrahedronERTinterstitial polyhedron expanded regular tetrahedronERT

Fig. E2.19a Fig.2.19b

Schwarz'sDsurface

(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 theexpanded regular tetrahedron ERT(cf.Fig. E2.20),

interstitial polyhedron of the diamond graph,

are shown in blue.

Fig. E2.20

Theexpanded regular tetrahedron ERT,

interstitial polyhedron of the diamond graph

It is also the nodal polyhedron of the diamond graph.

(The diamond graph is self-dual.)

ERTis 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 angleθ= cos^{-1}(− 1/3) =~109.47°.

Fig. E2.21

The regular tetrahedron and the edges (black lines) ofERT

Fig. E2.22

The 16-face Voronoi cell for the vertices of the diamond graph

For a sharper [pdf] version of this image, look here.

graphbcc

Fig. E2.23

dual graph WPgraphnodal polyhedron expanded regular octahedronEROinterstitial polyhedron tetragonal tetrahedronTT

Fig.2.24a

A cubic unit cell ofI-WP

The skeletal graphs are

thebccgraph − aka theIgraph −and theWPgraph.

Fig. E2.24b

Thebccgraph (Z=8)

The edges of thetetragonal tetrahedron TT(cf.Fig. E2.25)

are shown in blue.

Fig. E2.24c

Thebccgraph (green vertices) and its dual,

theWPgraph (orange vertices)

Fig. E2.25

Thetetragonal tetrahedron TT,

interstitial polyhedron of thebccgraph

TTis the saddle polyhedron Peter Pearce constructed in

an interstitial cavity of thebccgraph (cf.Fig. E2.24).

Each face is a regular skew quadrangle

with face angleθ= cos^{-1}(1/3) =~70.13°.

Fig. E2.26

The eight edges (purple) ofTT

For a sharper [pdf] version of this image, look here.

Fig. E2.27

Theexpanded regular octahedron ERO,

nodal polyhedron of thebccgraph

Fig. E2.28It 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 NeoviusC(P) family. (I

first observed this curious result in 1975 and described it in a

letter to the physicist Tullio Regge in response to some questions

from him about minimal surface soap film experiments. I had met

Tullio a few weeks earlier at a Providence conference hosted by

Tom Banchoff. Tullio was a formidable expert in differential

geometry. He had studied the work of the pioneering

19th century Italian masters of the subject (Bianchi

et al) when he was a young student.)

Fig. E2.29

The twenty-four edges (black) ofERO

For a sharper [pdf] version of this image, look here.

Fig. E2.30

The 14-face Voronoi cell for the vertices of the b.c.c graph

For a sharper [pdf] version of this image, look here.

Fig. E2.31 Fig. E2.32

Fig. E2.33 Fig. E2.34The pair of surfaces illustrated in Figs. E2.31 and E2.33 are portions of

F—RDandI—WP, respectively.

Note that each of the two surfaces shown in Figs. E2.32 and E2.34 can be regarded, approximately, as

the image of the surface at its left after rotation through 45 degrees about a vertical axis. I obtained

experimental confirmation of the existence of the surface in Fig. E2.32 in 1975, using a laser

to measure the surface normal orientation for a set of hypotheticaladjointsoap films

near their boundaries. This method can be described as an extremely tedious (and

far from precise) way tokill periods. In 2001, Ken Brakke accomplished the

same task with enormously greater precision using his Surface Evolver

and also confirmed the existence of the surface in Fig. E2.34.

(These four images were all made by Ken Brakke.)

graphWP

Fig. E2.35

dual graph bccgraph − aka theIgraphnodal polyhedron tetragonal tetrahedronTTinterstitial polyhedron expanded regular octahedronERO

Fig. E2.36

TheWPgraph (Z=4)

The edges ofERO(cf.Figs. E2.27, E2.37) ,

interstitial polyhedron of this graph, are shown in blue.

Fig. E2.37

Theexpanded regular octahedron ERO,

interstitial polyhedron of theWPgraphIn 1966, while assembling the faces of this model,

I discovered that if adjacent hexagons are related

byrotationinstead ofreflection,

the result is an infinite smooth surface — Schwarz'sDsurface.

(That was my introduction to triply-periodic minimal surfaces.

A few minutes later, I replaced the 90° hexagons by 60° hexagons

and obtained Schwarz'sPsurface.)

Fig. E2.38

Thetetragonal tetrahedron TT,

nodal polyhedron of theWPgraph (cf.Fig. E2.36)

Fig. E2.39

Theexpanded octahedron EO,

the Voronoi cell for the vertices ofWP

For a sharper [pdf] version of this image, look here.

graphLaves

Fig. E2.40

dual graph enantiomorphic Lavesgraphnodal polyhedron trigonal trihedronTinterstitial polyhedron trigonal trihedronT'(enantiomorph ofT)

Fig. E2.41

A portion of the Laves graph (Z=3)

The edges of thetrigonal 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 ofTT

from a direction different from that in Fig. E2.41

Fig. E2.43

Thetrigonal 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 ofTT

FCC_{6}(I) graph

Fig. E2.45

dual graph a non-symmetric graph of degree 10 nodal polyhedron pinwheel polyhedronPPinterstitial polyhedron doubly expanded tetrahedronDET

Fig. E2.46

Four vertices' worth of theFCC_{6}(I) graph (Z=6),

a locally-centereddeficientsymmetric graph (LCDSG)The

verticesof theFCC_{6}(I) graph are those of thefcclattice.

At each vertex, six of the twelve edges

of the standardfccgraph are omitted.

FCC_{6}(I) is described on pp. 47-48 of

Infinite Periodic Minimal Surfaces Without Self-Intersections.

Fig. E2.47

Four vertices' worth of theFCC_{6}(I) graph

inscribed on the surface of the polyhedronVP_{diamond}

(Voronoi polyhedron for the diamond crystal structure)

Fig. E2.48

FiveVPs' worth of the_{diamond}FCC_{6}(I) graph

Fig. E2.49

FiveVPs' worth of the_{diamond}FCC_{6}(I) graph

Fig. E2.50

Doubly expanded tetrahedron ('DET'),

theinterstitialpolyhedron of theFCC_{6}(I) graph

Six faces are regular skew quadrangles,

and four faces are regular skew hexagons.

Fig. E2.51

Two views ofDET

(stereo)

Fig. E2.52

The Doubly Expanded Tetrahedron is so named because

its twenty-four edges are produced by reflecting each edge of every face

of a regular tetrahedronin each of theof that face.twoother edges

(Every edge of the tetrahedron is reflectedfourtimes,

since it is incident attwofaces.)

Fig. E2.53

When an infinite set ofDETs is assembled by gluinghexagonalfaces together in pairs,

thequadrangularfaces remain exposed and define Schwarz'ssurface.D

Fig. E2.54

When an infinite set ofDETs is assembled by gluingquadrangularfaces together in pairs,

thehexagonalfaces remain exposed and define Schwarz'ssurface.P

Fig. E2.55a

Pinwheel polyhedronPP,

thenodalpolyhedron of theFCC_{6}(I) graph

Fig. E2.55b

Pinwheel polyhedronPP,

rendered in Mathematica

Fig. E2.55c

The red edges are the edges of

the nodal polyhedronPPof theFCC_{6}(I) graph.

It has the same volume as the

Voronoi polyhedron (rhombic dodecahedron).

Fig. E2.55d

Partial packings ofPP, rendered in Mathematica

(The quadrangular surface patch module is rendered here as adoubly-ruled surface.

Although it resembles the actual minimal surface, it is only an approximation.)

Below are four views of a portion of the compound [self-intersecting] surface

composed of replicas of just one of the two enantiomorphous quadrangular surface

patches that make up the pinwheel polyhedronPP(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

FCCgraph

Fig. E2.56a

dual graph fluorite graph nodal polyhedron rhombic dodecahedroninterstitial polyhedra regular tetrahedron, octahedron

Fig. E2.56b Fig. E2.56c

Two views ofF-RD

Image at left courtesy of Ken Brakke

Fig. E2.56d

The nodal polyhedron of theFCCgraph

is the Voronoi polyhedron (rhombic dodecahedron).

Fig. E2.57

The fluorite graph is the dual of theFCCgraph.

Its nodal polyhedra are

the regular tetrahedron and

the regular octahedron.

Fig. E2.58

TheFCCgraph and the fluorite graph

scgraph

Fig. E2.59

dual graph scgraphnodal polyhedron cubeinterstitial polyhedron cube

Fig. E2.60

Thescgraph,

skeletal graph of one labyrinth of Schwarz'sPsurface

Fig. E2.61

Thescgraph, which is also the

skeletal graph of theotherlabyrinth of Schwarz'sPsurface

Fig. E2.62

The congruent skeletal graphs of the two disjoint labyrinths of Schwarz'sPsurface

(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'sDsurface) or the 60° regular skew hexagon (a module for Schwarz'sPsurface), if two specimens of the saddle polygon are related by ahalf-turnabout a common edge, instead of bymirror reflectionin a plane containing that edge (which is the arrangement in most, although not all, of the saddle polyhedra I had explored by then), not only does the junction between the two polygons appear to be perfectly smooth, but an endless sequence of these half-turns producesa single smooth, embedded infinite labyrinthine surfacewith the global topology and symmetry of aCoxeter-Petrie regular skew polyhedron!I had accidentally stumbled onto two examples of the application of Schwarz's reflection principle, his

PandDsurfaces − 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 thatsomecontemporary 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 —HandCLP, both also of genus 3 — and I made plastic models of them too.I soon noticed that on p. 271 of Hilbert and Cohn-Vossen's 'Geometry and the Imagination'; the authors write that

In this way, Neovius ^{4}succeeded in constructing a minimal surface that extends over the entire space without singularity or self-intersectionand 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'sDsurface. Eighteen months later, I learned that Neovius had treated an entirely different surface of genus 9.I invented a naive scheme for identifying and labeling these surfaces, each of which I regarded as lying between the two triply-periodic graphs of a dual pair. I named these pairs of graphs 'skeletal graphs', because I thought of them as the skeletons of their respective hollow labyrinths. I found it helpful to regard the skeletal graph edges as thin hollow tubes that could be enlarged by inflating them until the whole graph was transformed into the TPMS. Then if the tubes were

overinflated, the graph would eventually shrink down into thedualgraph! I imagined that for at least a portion of this inflation cycle, the surface of the graph would define a triply-periodic surface of non-zero constant mean curvature.As I began my admittedly superficial study of the mathematical underpinnings of these surfaces, beginning with the two Schwarz reflection principles, I couldn't help wondering what other examples of embedded 'TPMS' might exist. In particular, I wondered whether there was a TPMS whose skeletal graphs were the enantiomorphous pair of Laves graphs I had learned about seven years earlier in 'Third Dimension in Chemistry', by Alexander F. Wells. A pair of enantiomorphic Laves graphs that are related by inversion has

bcc.translation symmetry. It struck me as curious that of the three different cubic lattice symmetries — simple cubic (sc), face-centered cubic (fcc), and body-centered cubic (bcc) —bccwas missing from the inventory of cubic lattice symmetries for known examples of TPMS of ultimately simple topology (genus three).A second reason for my focus on the Laves graph was that it was apparently the only other example of a triply-periodic graph — besides the simple cubic and diamond graphs, which are the skeletal graphs of

PandD, respectively — in which congruentregularpolygons are incident at each edge. In the Laves graph, there are precisely two regular polygons incident at each edge. They happen to beinfinitehelicalpolygons, 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 areflexiveregular 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

symmetrictriply-periodic graphs of cubic symmetry that areself-dual. Even though I knew of no theoretical justification for claiming that such graphs — regarded asskeletalgraphs of embedded TPMS — play a unique role in defining embedded TPMS of cubic symmetry, I neverthelessbelievedthat they must play such a role! I was aware of the fact that the concept of skeletal graph was itself somewhat ill-defined. It seemed to me to be a very 'natural' construct when applied to the then known examples of embedded TPMS, but I had no idea how to prove that for every possible example of an embedded TPMS there is a unique pair of skeletal graphs.All of these considerations at times seemed to me to smack more of

theologythan ofmathematics. (I am reminded that the young Riemann, who was probably the first to solve the equations for what we now call Schwarz'sPandDsurfaces, as a young man abandoned the study of theology (his pastor father's choice) for a career in mathematics!I resolved to learn more about TPMS, which I recognized as far more interesting objects than saddle polyhedra, but in the meantime I was determined to continue exploring the relation between triply-periodic graphs and saddle polyhedra. On evenings and weekends throughout the spring and summer of 1966, I used a toy vacuum-forming machine and home-made moulds cast from polyester resin poured against a thin stretched rubber membrane to make dozens of saddle polyhedra of different shapes, all of which I shared with Peter Pearce. He preferred to make his saddle polygons by

draw-forming— pushing a tool in the shape of a skew polygon outline against a transparent vinyl sheet that had been softened by heating. I preferred vacuum-forming with solid moulds, but it was clear that Peter's method also worked well. It has the advantage of not requiring the extra labor involved in making a mould, but the disadvantage is that it cannot replicate the shape of a minimal surface as well as a carefully crafted mould can.During these months of experimenting, I found no counterexample to my improvised duality rule, even for triply-periodic graphs that are

notsymmetric. 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 graphAderived from a second symmetric graphBby omitting some of the edgesbut none of the verticesofB. InA, not every pair of nearest neighbor vertices is joined by an edge. I required that every deficient symmetric graph belocally-centered,i.e.,that every vertex lie at the centroid of the vertices with which it shares an edge.For the

simple cubic lattice, it's easy to prove — simply by enumerating each of the possible locally-centered subsets of edges that contains at least three edges — that it is impossible to construct a locally-centered deficient symmetric graph (LCDSG) on the vertices. I have no idea why I failed to ask myself in those days whether there exists aLCDSGon the vertices of thebody-centered cubic lattice. The only example of aLCDSGthat I examined in 1966 was a graph of degree six I callFCC_{6}(I) (cf.Fgs. E2.45 - E2.55b). Its vertices are those of theface-centered cubic lattice. I used the nameFCCfor the familiar symmetric graph of degree twelve on the vertices of thefcclattice, in which every pair of nearest neighbor vertices is joined by an edge. I derived the deficient graphFCC_{6}(I) by removing a symmetrical set of six out-of-plane edges from each vertex, leaving behind a flat six-edge cluster that occurs in each of the four possible [111] orientations. AlthoughFCC_{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 tosymmetricgraphs only.As it happened, not only did the rule

notfail in the case ofFCC_{6}(I) — it yielded an unexpectedly interesting pair of saddle polyhedra. I call theinterstitialpolyhedron — shown in Fig. E2.51 — theDoubly Expanded Tetrahedron('DET'), and the correspondingnodalpolyhedron — shown in Fig. E2.55a — thePinwheel Polyhedron. The Doubly Expanded Tetrahedron is the first example I had encountered of a space-filling saddle polyhedron in which the faces are of two kinds — hexagons and quadrangles. As illustrated in Figs. E2.53 and E2.54, Schwarz'sPandDsurfaces 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 1970In the spring of 1967, the physicist Lester C. Van Atta, who was Associate Director of the NASA Electronics Research Center ('ERC') in Cambridge, Massachusetts, came to Los Angeles for a few days to visit his physicist son Bill, a friend and colleague of mine who happened to be a whiz at solving combinatorial puzzles. A year earlier, stimulated by a Martin Gardner column in Scientific American that described Piet Hein's SOMA puzzle, I had become hooked on investigating the possible symmetries of complementary half-cube packings by the eight solid tetrominoes (

cf.Fig. E1.13). After Bill lent his father a set of these puzzle pieces, his father told Bill that he 'wanted to meet the guy who had cost [him] a night's sleep'.

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

I wanted to immerse myself immediately in the study of TPMS. However, I believed that I should first make a better organized attack on my embryonic duality rule ('partitioning algorithm'). I knew it was unlikely that the rule would be applicable to every possible triply-periodic graph, but I had no idea how to characterize those graphs for which it worked and those for which it didn't. Inspired by Polya's rules for problem solving, I continued to emphasize

symmetricgraphs — 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

interstitialdiffusion andsubstitutionaldiffusion, 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 principalinterstitialsites of a crystalline solid.

WARNING: Continue at your own risk.

The text in the next few paragraphs has not yet been edited. Much of it duplicates other portions of text and will eventually be integrated into the main narrative.I was able to confirm my hunch that the duality rule was valid for many pairs of triply periodic graphs — one of the pair being called

substitutionaland the otherinterstitial. In every case I tested, bothnodalandinterstitialpolyhedra exhibited the following properties:(a) the

number of facesof the polyhedron is equal to thenumber of edgesof the associated periodic graph;

(b) thesymmetryof the polyhedron is identical to thesymmetryof the associated vertex of the periodic graph.It didn't matter which graph was called

substitutionaland which was calledinterstitial. 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 —substitutionaland the silicon atomic sitesinterstitial!)I call the saddle polyhedron in Fig. E2.25 the

interstitial polyhedronfor thebccgraph 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 thenodal polyhedronfor another triply periodic graph, which I namedWP. 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

nodalpolyhedron andinterstitialpolyhedron for the diamond graph. That means that in a space-filling array of these saddle polyhedra, each polyhedron can either (a) enclose at its center a vertex of the diamond graph or (b) occupy a single interstitial region bounded by the edges and vertices of the diamond graph.For some periodic graphs, either the nodal polyhedron or interstitial polyhedron (or both) may turn out to be a convex polyhedron with plane faces. For the simple cubic graph of degree six, for example, both polyhedra are cubes.

In the simplest cases, the graph is

unary,i.e.,all the vertices of the graph are equivalent. But the duality rule works smoothly without requiring anyad hocadjustments even for many non-unary graphs. (I plan eventually to post a picture or two of the polyhedra for such a graph.)As explained below, in early 1968 I searched for — and eventually found — a periodic graph for which the duality rule failed, and that failure led to the discovery of the pseudo-gyroid, which is composed of hexagons with perfectly helical boundary curves.

Recall that

FCC_{6}(I) is a locally-centered deficient graph of degree six on the vertices of thefcclattice (cf.Fig. E2.48). In February, 1968, I realized that I had never searched for the obviousbcc.counterpart toFCC_{6}(I):a locally-centered deficient graph on the vertices of the bcc. lattice. (It's easy to prove that no locally-centered deficient graph on the vertices of thesclattice exists.) Once I started looking, it didn't take me long to discoverBCC, a symmetric graph of degree six on the vertices of the_{6}bcc.lattice. A portion of this graph is shown in Fig. E1.20d.BCCproved to be the long-sought counterexample to my 'duality rule'. Its edges are those of the_{6}infinite regular warped polyhedron('IRWP') that I callM_{4}(cf.Fig. E1.16a and E1.20a to E1.20f).In the case of

BCC, the breakdown in the duality rule occurs at the very first step — the construction of the_{6}interstitial polyhedron. Instead of thefiniteinterstitial polyhedron the duality rule was intended to generate, aninfiniteone —M_{4}— appeared. I shed no tears over this failure of the duality rule, because the two-labyrinth character ofM_{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 ofM_{4}were enantiomorphic Laves graphs. This was potentially exciting, because it suggested thatM_{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 Self-Intersections 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 skewquadranglesspanned by minimal surfaces. I was looking for a way to 'smooth out the wrinkles' inM_{4}. The hexagons inM_{6}are thedualsof the quadrangles inM_{4}. InM_{4}, the bump across the edge shared by adjacent faces is 60°. InM_{6}, it is only ~44.4°. I hoped that making the bump smaller by ~15.6° than the bump inM_{4}would enableM_{6}to look at least a little more like a continuous minimal surface thanM_{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 ofThis was a wild and woolly guess, with no theoretical justification whatsoever, but the new physical model I constructed (M_{6}by helical arcs, the ~44.4° bump between adjacent faces might shrink to nearly zero.cf.Fig. E1.17) was encouraging. The edge shape in the authentic gyroid just happens to differ so slightly from the shape of a helical arc that it is impossible to detect the difference betwen the two by eye.The glaringly obvious hints that I had missed from the outset stemmed from the identical combinatorial structure of each of the three regular tessellations of the three surfaces —

P,D, and the pseudo-gyroid surface in Fig. E1.17. All three of these surfaces can be constructed of surface patches that correspond to faces of any of the three Coxeter-Petrie infinite regular skew polyhedra, with tangent planesidentically orientedat corresponding vertices. Since I was already familiar with the details of how curves transform and how the tangent plane remains invariant at each point of the surface in the Bonnet bending of the catenoid into the helicoid, it should have occurred to me (but didn't!) that the almost perfectlycircularlines of curvature in the coordinate planes of Schwarz'sPsurface would be transformed by Bonnet bending into almost perfectlyhelicallines of curvature before they finally became the linear asymptotics in Schwarz'sDsurface 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

possibilityof some 'practical' offshoots. Since I had no idea how to obtain an analytic solution for a minimal surface patch bounded by six helical arcs, I decided to give up trying to prove that the pseudo-gyroid is a minimal surface. Instead I sent a physical model of the pseudo-gyroid to Bob Osserman, who passed it along to his PhD student Blaine Lawson at Stanford. Blaine agreed to think about the problem, but he warned me that his dissertation would be keeping him extremely busy.I delved more deeply into the analysis of the 'continuous transformation on vertices and edges' mentioned in the abstract of Fig. E2.10. I attempted to identify every possible example of non-self-intersecting 'infinite regular warped polyhedra' (and 'infinite

quasi-regularwarped polyhedra'), whose faces are regular skew polygons. (cf.Figs. E1.35b and E1.35c.). At the same time, I analyzed the geometry of what I called the 'graph collapse' transformation, which is diagrammed for the 2-dimensional square graph in Fig. E2.68c. To the exclusion of almost everything else, I concentrated for several weeks on the engineering requirements for the application of this transformation to the design of expandable space frames. Finally I wrote a patent application with the help of two NASA patent attorneys who for two weeks flew up to Cambridge every morning from Washington. Here is a synopsis of the expandable space-frame patent, which was issued in 1975, and here is the complete text of the NASA patent, from which an illustration is shown in Fig. E2.68a. I estimated that with realistically designed struts, a value of 80:1 was feasible for the ratio of the expanded to collapsed volume of the space frame.With the help of Charles Strauss, Randy Lundberg, Bob Davis, Ken Paciulan, and Jay Epstein, I made an animated film of the collapse of the Laves graph and of three other symmetric graphs. The portion of the video '1969 'Part 4' that shows examples of the graph collapse transformation begins at 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. 86-88 of

Infinite Periodic Minimal Surfaces Without Self-Intersections. In the fully collapsed state, the vertices and edges of the [infinite] Laves graph are mapped onto the four vertices and six edges, respectively, of a singleregular 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) isfixed, the collapse trajectories of all the other vertices are ellipses centered on that vertex. For every vertexV, the major radius of the ellipse is equal to the initial distance ofVfrom O. The minor radius of the ellipse is equal to the edge length of the graph ifVis related to vertex a, b, or c by a translation that is a symmetry of the Laves graph —i.e.,ifVis red, green, or blue. The minor radius is equal to zero ifVis related to vertex O by a translation that is a symmetry of the Laves graph —i.e.,ifVis yellow. Collapse onto tetrahedron AOBC occurs twice in each period of the transformation: at the two moments when either one-quarter or three-quarters of each elliptical trajectory has been traversed.Each of the vertices a, b, c rotates on a

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

Fig. E2.68a

A hinged joint in the expandable space frameThe collapse of the Laves graph is readily depicted by regarding the graph as initially embedded in the

Dsurface (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 theDsurface are labeled A and B, with motion in the direction from A to B defined aspositiveand motion in the direction from B to A defined asnegative, 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 theDsurface

The arrows indicate the initial directions of the curvilinear displacements of the vertices.

Green diplacements are calledpositive, and

red diplacements are callednegative.

Fig. E2.68b.1

The yellow vertex at O is nowfixed.

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

Thecirculartrajectories of vertices a, b, and c

and theellipticaltrajectory of vertex d, in Fig. E2.68b.1

One-fourth of a complete trajectory period is shown here for these four vertices.

Perhaps the easiest way to illustrate the collapse is to depict

thecirculartrajectories, in orthogonal coordinate planes,

of the three vertices that are nearest neighbors ofvertexanyV

.if V is regarded as fixed.The next four images illustrate these trajectories for

V= a yellow, red, green, or blue vertex.

Fig. E2.68b.3

Rotation of R, G, and B vertices around Y vertex

stereo view

Fig. E2.68b.4

Rotation of G, B, and Y vertices around R vertex

stereo view

Fig. E2.68b.5

Rotation of B, Y, and R vertices around G vertex

stereo view

Fig. E2.68b.6

Rotation of Y, R, and G vertices around B vertex

stereo view

The stereo mages in Fig. E2.68c illustrate the application of the graph collapse transformation to the

square graph, which is initially embedded in a minimal surface — the plane. In this 2-dimensional example, all the edges of the graph coalesce into a single vertical edge. The images below are parametrized by the value ofθ, the angle of rotation of each edge of the graph out of the horizontal plane. The discussion below applies to a 16-vertex square portion of the infinite graph, but in principle the analysis applies to any finite [connected] piece of the graph, however large.In Fig. E2.68c.0, the vertex at A is held fixed. The horizontal positions of the vertex at A and the seven vertices (C, F, H, I, K, N, P) to which downward-pointing red arrows are attached are maintained at a constant level. (Ignore the red arrows. I shouldn't have included them in this image, which I plan to replace.) The vertices to which green arrows are attached are displaced upward, but they are constrained to move sideways because the [identical] links that connect the vertices are inextensible. The net result is that the links all rotate,

moving on elliptical trajectories centered on A(cf.Fig. E2.68c.9). The ellipses for vertices F, K, and P degenerate into straight lines, and those for vertices B and E are circles. Collapse of all the links onto a single vertical edge occurs atθ=90° andθ=270° (cf.Figs. E2.68c.4 and Figs. E2.68c.5). (I'll soon make an animated sequence showing a full period of the motion.)

After I sketched the notes shown in Fig. E.2.70, I phoned Blaine Lawson, who was already an expert on minimal surfaces. He was then approaching the last stages of his PhD dissertation research at Stanford under Bob Osserman. I asked him if he thought it was plausible that a hybrid derived from these two straight-edged polygons would define an embedded surface. Blaine replied that it was not an unreasonable idea, because the

Fig. E2.68c.0

Square grid graph before the start

of the collapse transformation

θ=0°

Fig. E2.68c.1

θ=18°

Fig. E2.68c.2

θ=52.2°

Fig. E2.68c.3

θ=81°

Fig. E2.68c.4

θ=90°

Fig. E2.68c.5

θ=279°

Fig. E2.68c.6

θ=307.8°

Fig. E2.68c.7

θ=345°

Fig. E2.68c.8

θ=360°

Fig. E2.68c.9

Some additional history

Inserting handles into a TPMS

(See also Figs. E2.83 to E2.85.)It occurred to me one day in the spring of 1969 that for some pairs

S_{1}and

S_{2}of moderately low-genus embedded TPMS for which the respective

adjoint surfacesS_{1}† andS_{2}† have elementary patches that are bounded by

simply-connected straight-edged polygonsP_{1}andP_{2}, there must exist an embeddedhybridTPMSS_{h}whose adjoint surfaceS_{h}† has an elementary

patch with boundary polygon equal toa linear combinationofP_{1}andP_{2}.

(I defined a linear combination of two polygonsP_{1}andP_{2}to be a polygon

interpolatedbetweenP_{1}andP_{2}. The idea of constructing a linear combination

of two polygons occurred to me after I recalled a description oflinear combinations of convex polyhedrathat 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 ofPandC(P).

Fig. E.2.70

1969 proposal for a genus-14 embeddedhybridofP(genus 3) andC(P)(genus 9)

The two scribbled captions 'Schwarz's "' are erroneous.D"

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 quadrangleP_{2}at the lower right are

12, 23, 34, 41.

The transformation ofP_{1}intoP_{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 zero at a steady rate.The relative weights assigned to the adjoint polygons

P_{1}andP_{2}

require trial-and-error adjustment to make arcs 23 and 15coplanar.

The cognoscenti call this process 'killing periods'.

intermediate value theoremguarantees 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 ofCosta-Hoffman-Meekssurfaces. The original surface from that family is shown below, in a stereo image due to Hermann Karcher.

The original Costa-Hoffman-Meeks surfaceI mailed (faxed?) Blaine a copy of my Fig. E.2.70 sketch, but then I abandoned the

P—C(P) hybrid, because I realized that vacuum-forming a surface patch with a severe undercut — like the one shown at left center in Fig. E.2.70 — would be difficult or impossible. Determined to build a physical model ofsomehybrid surface, I turned instead to a topologically simpler case — the hybrid ofPandI-WPthat I callO,C-TO(cf.Figs. E2.79 to E2.81). Its genus is only 10. Then for the next forty-two years, I forgot all about theP—C(P) hybrid!This process of hybridization is equivalent to

attaching a handleto a minimal surface. During the years 1969-1975, I employed an extremely laborious experimental technique I had devised, based on a method of successive approximations, to hybridize several examples of TPMS. With this technique, a laser is used to measure the orientation of the tangent plane of a long-lasting polyoxyethylene soap filmS† at a sequence of closely spaced points near its boundary.S† is spanned by a straight-edged skew polygonP†, which is a candidate for the boundary of theadjointof the curved-edge boundary polygonPof an elementary patchSof the hybrid (the surface containing the added handles). The relative lengths of the straight edges of the first of several successive candidates forP† are just rough estimates. When the shapes of the curved edges of the corresponding version ofPare numerically derived (using a very simple computer program) from the measurements of this firstP† candidate, it is inevitably found thatPfails to close. Corrections are then applied to the relative lengths of the edges ofP†, and an improved version ofP—i.e.,one that is more nearly aclosedpolygon — is obtained. After four or five iterations of this procedure, a satisfactory approximation toPis obtained (unless, of course, there is no such surface asP).The laser measurements and the accompanying numerical calculations are the least tedious steps in this procedure. The most time-consuming step by far is fabricating an accurate physical model of

P†. By 1991, Ken Brakke's Surface Evolver had rendered this oppressively tedious method obsolete.

Many other people subsequently discovered the idea of handle attachment and applied it to a variety of minimal surfaces, not just periodic ones. (I have been publicly scolded by several mathematicians — most often by J.C.C. Nitsche — for failing to publish my work in refereed journals.

_{The laser spectrometer Hal Robinson assembled in 1968 }

_{for measuring the surface orientation of a soap film }

_{ I no longer have any photos of the similar instrument I constructed in 1975.}

Mea culpa.)In May, 2011, I discovered in my files the long-forgotten sketch shown in Fig. E.2.70 and emailed a copy to Ken Brakke. He quickly confirmed (with his Surface Evolver) that the

P—C(P) hybridisembedded. Ken's pictures of this surface, which he dubbed 'N14', are shown in Figs. E2.71, E2.72, and E2.73 and also at his Triply Periodic Minimal Surfaces web site, under the name N14.

Fig. E2.71

Ken Brakke's May 2011 Surface Evolver solution for an elementary patch of

N14, the embeddedhybridofPand C(P)

(cf.Fig. E2.70)

(image courtesy of Ken Brakke)

Fig. E2.72 E2.73

A cubic unit cell of theP—C(P) hybrid (genus 14)

The unit cells in the two images are displaced with respect to

each other by one-half of the body diagonal of an enclosing cube.

(cf.Fig. E2.71)

(images courtesy of Ken Brakke)

Fig. E2.74

1969 sketch showing how to generateO,C-TO

The 'bcc labyrinth surface' referred to at the top

of the page is the surface I later renamedI-WP.

Fig. E2.75

O,C-TO(genus 10), a hybrid ofI-WP(genus 4) andP(genus 3)

cubic unit cell

view: [100] direction

Fig. E2.76

O,C-TO

unit cell

view: [111] direction

Fig. E2.77

O,C-TO

1.5 unit cells

oblique view

Fig. E2.78 shows Ken's picture of the Manta surface, which is one of many examples of hypothetical minimal surfaces whose existence I conjectured in 1971. Some of them were inspired by experiments with soap films blown inside a kaledioscopic cell. Others were inspired by considering the structure of various highly symmetrical inorganic crystals. Manta is abalancedsurface; theP—C(P) hybrid isnon-balanced. If you compare Fig. E2.73 and Fig. E2.74, you will see that theP—C(P) hybrid has a simpler topology than Manta. Manta has [100] tunnels, while theP—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 theP—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: (Here's a useful Wikipedia article about ionic radii.)

corner sphere radius: 0.18560130

center sphere radius: 0.18560130

midedge sphere radius: 0.23553163

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 father-in-law)

that is included in Schwarz'sGesammelte Werke

The illustration shows eight triangularFlächenstückeof

Schwarz's diamond surfaceDinside a tetragonal disphenoid.

Fig. E2.82b

Free translation of the note by Kummer in Fig. E2.82a

One weekend in 1968, while I was reading p. 150 of Schwarz's

Collected Works(cf.Fig. E2.82a), it occurred to me that one might be able to model a small portion of a triply-periodic minimal surface, like the portion of Schwarz'sDsurface 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: atriangularpatch ofDand aquadrangularpatch ofC_{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 triangular-prism-shaped Coxeter cell. The nylon thread that is stretched horizontally across the interior of the cell lies on an axis of 2-fold rotational symmetry of the surface and serves to stabilize the soap film. Without the thread, the film would be in an unstable stationary state and would quickly slide away from its equilibrium position and collapse. Many of the vacuum-formed plastic models of TPMS I constructed after Hal began to work with me were made from modules whose boundary curves were derived by tracing the curved edges of soap films like this one. }My model of the tetragonal disphenoid was a flimsy one. When I arrived at NASA on the following Monday morning, I phoned Hal Robinson, the sculptor and model-maker who had recently started to work with me part-time and asked him to make a more physically rugged tetragonal disphenoid. Within a couple of days or so, he produced a lucite model with highly accurate proportions, using monofilament nylon instead of cotton threads for the internal symmetry axes. Next I asked Hal to make me a lucite model of another Coxeter cell relevant for cubic TPMS — the quadrirectangular tetrahedron, which is one-quarter of the tetragonal disphenoid. It contains only

oneinternal axis of two-fold rotational syrmmetry, nottwo. With this cell, you can toggle back and forth between a patch ofPand a patch of the Neovius surfaceC_{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

Psurface 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 four-sidedFlächenstückof C_{21}(P) as a soap film in a stationary — butunstable— state inside the quadrirectangular tetrahedron. This was a more difficult soap film experiment than toggling back and forth betweenPand C_{9}(P), which are instableequilibrium. Although the C_{21}(P) soap film corresponds to a [mathematical] stationary state, its area islargerthan that of nearby lying [non-minimal-surface] soap films, and I had to struggle to maneuver the film into its unstable equilibrium position long enough for a camera to capture it.In 1992 I made an impromptu video about minimal surfaces in which I attempted to demonstrate the art of blowing these unstable soap films inside Coxeter cells, but I had run out of glycerine that day. After many tries, I succeeded for a fleeting moment in capturing the gracefully curved

Flächenstückof 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 insideglasstetrahedra I have recently made.

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

Fig. E2.83a

Twelve elementary triangularFlächenstücke

of Neovius'sembeddedsurfaceC_{9}(P)

Fig. E2.83b

Twelve elementary triangularFlächenstücke

of Neovius'sself-intersectingsurfaceC_{9}(P)†In both

C_{9}(P) andC_{9}(P)†, the triangularFlächenstückabc

is analytically continued by reflection in its edges.

The normal vectors (red arrows) at corresponding points of the

two adjoint surfacesC_{9}(P) andC_{9}(P)† have the same directions.

Flächenstückefor higher-genus variants of thePandDsurfaces and a variety of non-cubic TPMS. All of these experiments involved modifying the shapes of stationary-state soap films inside transparent plastic models of Coxeter cells by blowing on them.Fig. E2.84 shows some of Ken Brakke's Surface Evolver for some of these higher-genus variant surfaces. I was unable to produce the genus-15 soap film, but occasionally I succeeded with the genus-21 case.

genus=9

genus=15

genus=21

genus=27

genus=33Fig. E2.84

Ken Brakke's Surface Evolver solutions for the first few

high-genus variants of Neovius'sC_{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 ofP,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 hisSurface Evolver

software to kill periods, thereby generating each of the embedded surfaces (shown at the

right of the corresponding adjoint surface outlines).

The genus

pof the_{k}k^{th}surfaceMin the family {_{k}M} is defined as_{k}p=_{k}p_{0}+kgap(k= 0, 1, 2, ...); the values ofp_{0}and the positive integergapare characteristic of the family. These families include — but are not limited to — high-genus complements ofPandD. For thePandDfamilies, for example,p_{0}=3 andgap=6. Hence the surfaces in these two families are of genus 3, 9, 15, 21, ... .On that day in 1971, John and Bob and I blew a large variety of "finely filigreed" soap films in a variety of Coxeter cells, and we made detailed drawings of our results. Most — but not all — of these soap films included one or two nylon threads stretched along 2-fold symmetry axes of the enclosing polyhedral cell. The curved soap film boundary edges lying in face planes of the cell are 'mirror-symmetric plane lines of curvature'. Every face plane is a plane of reflection symmetry for both the assembly of cells and the soap films in their interiors. The soap films meet the enclosing face planes orthogonally. Each soap film is an approximate model of the

stationary stateof 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 withk=0 or 1 are in stable quilibrium. Ifk=2, the film is in a delicate state of unstable equilibrium. I found it impossible to produce films fork>2. Some skill is required to arrest a film fork=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 Evolvercomputer program. Many of these authenticated surfaces are illustrated on his Triply Periodic Minimal Surfaces web site. In this work, Ken uses Surface Evolver to 'kill periods' —i.e.,to derive the unique values for relative edge lengths that allow the surface to be embedded (cf.the brief description of this problem on pp. 45-46 of Infinite Periodic Minimal Surfaces Without Self-Intersections). In Infinite Periodic Minimal Surfaces Without Self-Intersections, I included only two examples of hybrid surfaces —C(H)(genus 7) andO,C-TO(genus 10), because at the time of writing, these were the only examples of such surfaces for which I had already constructed and photographed vacuum-formed plastic models. (In a footnote on p. 46, I mentioned a third example, of genus 5, that I calledg-g'. I soon renamed that surfaceg-W, after I confirmed that its dual skeletal graphs are related to the structures of hexagonal graphite and of wurtzite.) Figs. E3.7, E3.8, and E3.9 show three views of g-W.

E3. Triangle lattice surfacesAll of the minimal surfaces described in this section are named according to the conventions in Infinite Periodic Minimal Surfaces Without Self-Intersections.

Fig. E3.1a

Schwarz'sHsurface (genus 3)

Fig. E3.1b

A smaller piece of Schwarz'sHsurface

Fig. E3.2

C(H)(genus 7)

The [first-order] complement of Schwarz'sHsurface

unit cell

view: c-axis

Fig. E3.3

C(H)

view alongc-axis

Fig. E3.4

C(H)

view: c2 axis (intersection of horizontal and vertical mirror planes) in basal plane

Fig. E3.5

C(H)

view: line in basal plane that is below and parallel to a linear asymptotic (2-fold axis embedded in the surface)

Note the infinitely long straight tunnels with pointy oval cross-section

Fig. E3.6

C(H)

view: c-axis, silhouetted by bright summer sky backlighting

Note the infinitely long straight tunnels.

(The trigonal symmetry of the surface would be more apparent

if the image were rotated 30º in the image plane,

as in Fig. 3.3!)

Fig. E3.7

g-W("graphite-wurtzite") (genus 5)

oblique view

Fig. E3.8

g-W

oblique view

Fig. E3.9

g-WandC(H)

oblique view

Fig. E3.10

H''-R(genus 5)

view: c-axis

Fig. E3.11

H''-R(genus 5)

view: c-axis, silhouetted by bright summer sky backlighting

Fig. E3.12

H''-R

oblique view

Fig. E3.13

H''-R

view: c2 axis (intersection of horizontal and vertical mirror planes) in basal plane

Fig. E3.14

H''-R

view: line in basal plane that is below and parallel to a linear asymptotic (2-fold axis embedded in the surface)

Note the infinitely long straight tunnels with pointy oval cross-section

Fig. E3.15

H'-T(genus 4)

view (stereo): c-axis

E4. Surfaces on other lattices

Fig. E4.1

S'-S''

genus 4

view: oblique

E5. A few minimal surface people from around the world

Christian Bär (right), and Hermann Karcher

Ken Brakke at Selinsgrove, Pennsylvania

Tomonari Dotera and Junichi Matsuzawa

at Carbondale, Illinois (November, 2013)

Shoichi Fujimori at Bloomington, Indiana (2008)

Wojciech Góźdź

Bathsheba Grossman at Santa Cruz

Stefan Hildebrandt at Berkeley (1979)

photo by George M. Bergman

©George M. Bergman

Source: Mathematisches Forschungsinstitut Oberwolfach gGmbH

David Hoffman

at the University of Granada Minimal Surface Conference, June 17, 2013

higher resolution image

Stephen Hyde at Canberra

Hermann Karcher (left),

David Hoffman (center), and

Manfredo Perdigão do Carmo (right) at Granada (1991)

photo by Dirk Ferus

©Dirk Ferus

Source: Mathematisches Forschungsinstitut Oberwolfach gGmbH

Katsuei Kenmotsu at Oberwohlfach (2009)

photo by Renate Schmid

Source: Mathematisches Forschungsinstitut Oberwohlfach gGmbH

Rafael López Camino (right) and me

at the University of Granada, June 11, 2013

higher resolution image

Blaine Lawson (left) and Bill Meeks (right) at Rio (1980)

photo by Dirk Ferus

©Dirk Ferus

Source: Mathematisches Forschungsinstitut Oberwolfach gGmbH

Bill Meeks

at the University of Granada Minimal Surface Conference, June 17, 2013

higher resolution image

Johannes C. C. Nitsche (1925-2006)

photo by Ludwig Danzer

©Ludwig Danzer

Source: Mathematisches Forschungsinstitut Oberwolfach gGmbH

Robert Osserman (1926-2011) at Berkeley in 1979

photo by George M. Bergman

©Mathematisches Forschungsinstitut Oberwolfach gGmbH

Raymond Redheffer (1921-2005)

A light moment during a break at the University of Granada

Minimal Surface Conference, June 17, 2013

higher resolution image

Magdalena Rodriguez, Matthias Weber, and Bill Meeks

at the University of Granada Minimal Surface Conference, June 17, 2013

higher resolution image

Antonio Ros Mulero, me, and my wife Reiko Takasawa

at the University of Granada Minimal Surface Conference, June 17, 2013

high resolution image

Harold Rosenberg and Magdalena Rodriguez

at the University of Granada Minimal Surface Conference, June 17, 2013

high resolution image

Gerd Schröder-Turk at Erlangen-Nürnberg

In 2015, Gerd joined the faculty of the maths

department of Murdoch University in Perth, Australia.

Isaac Van Houten at Carbondale (2008)

photo by the author

Matthias Weber at Oberwohlfach (2009)

photo by Renate Schmid

©Mathematisches Forschungsinstitut Oberwohlfach gGmbH

Adam Weyhaupt (right) at Edwardsville

with two of his students — Darren Garbuz and Caroline Coggeshall

photo by the author

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