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first published online 30 May 2012, doi: 10.1098/rsfs.2012.002322012Interface FocusAlan H. SchoenReflections concerning triply-periodic minimal surfaces

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DISCUSSION

Reflections concerning triply-periodicminimal surfaces

Alan H. Schoen*

Carbondale, IL, USA

In recent decades, there has been an explosion in the number and variety of embedded triply-periodic minimal surfaces (TPMS) identified by mathematicians and materials scientists. Onlythe rare examples of low genus, however, are commonly invoked as shape templates in scientificapplications. Exact analytic solutions are now known for many of the low genus examples. Themore complex surfaces are readily defined with numerical tools such as S URFACE EVOLVER

software or the LandauGinzburg model. Even though table-top versions of several TPMShave been placed within easy reach by rapid prototyping methods, the inherent complexityof many of these surfaces makes it challenging to grasp their structure. The problem of dis-tinguishing TPMS, which is now acute because of the proliferation of examples, has beenaddressed by Lord & Mackay (Lord & Mackay 2003 Curr. Sci. 85, 346362).

Keywords: triply-periodic; minimal; surfaces

In recent decades, there has been an explosion in thenumber and variety of embedded triply-periodic mini-

mal surfaces (TPMS) identified by mathematiciansand materials scientists. Only the rare examples of lowgenus, however, are commonly invoked as shapetemplates in scientific applications. Exact analytic sol-utions are now known for many of the low genusexamples. The more complex surfaces are readily definedwith numerical tools such as Ken Brakkes SURFACEEVOLVER software [1] or Holyst and Gozdzs Landau Ginzburg model [2]. Even though table-top versions ofseveral TPMS have been placed within easy reach byrapid prototyping methods, the inherent complexity ofmany of these surfaces makes it challenging to grasptheir structure. The problem of distinguishing TPMS,

which is now acute because of the proliferation ofexamples, has been addressed by Eric Lord and AlanMackay [3].

I describe here some highlights of a journey thatculminated unexpectedly in 1966 in a search for newexamples of TPMS, exactly 100 years after Schwarzwrote his monumental treatise on TPMS [4] (a moredetailed account of my search can be found in [5,6]and at http://www.schoengeometry.com/e_tpms.html). The helicoid and the catenoid were the onlyminimal surfaces I had encountered before 1966.I knew only the bare rudiments of differential geometryand little more of complex analysis. In those days, soft

matter, mesoscopic, block copolymer, MCM-48, photo-nic crystals, etc., were not yet household words. My

principal research interests were random and correlatedrandom walks on lattices, atomic diffusion in crystals,the Mossbauer effect, and the combinatorial and geo-metrical connections between triply-periodic graphsand polyhedra. Some of these topics became fused inmy mind, leading me along a tortuous path to TPMS.

The first step on this journey was my discoveryin 1956 of a method for distinguishing between thevacancy and interstitial mechanisms for atomic diffu-sion in crystalline solids (http://www.schoen geometry.com/e_tpms.html) [79]. In 1951, in a mathematicalanalysis of self-diffusion by the vacancy mechanism, inwhich the elementary step of a diffusing atom is its

exchange of position with a vacancy (vacant latticesite), Conyers Hering found that the correlation betweenthe directions of consecutive steps of the atomreduces the self-diffusion coefficient by the fractionalamount 12f, where f 1 kcosulAv=1 kcos ulAv;kcos ulAv is the average value of the cosine of the angle ubetween consecutive jumps of the diffusing atom (cf.table 1); f is known as the BardeenHering correlationfactor [9].

I was startled to discover that for self-diffusion bythe vacancy mechanism, correlation reduces both thediffusion coefficient and the isotope effect by exactlythe same fractional amount. This meant that isotope

effect measurements could provide the first experimen-tal evidence to support the prevailing notion that self-diffusion in close-packed crystals of cubic symmetryoccurs by the vacancy mechanism. By contrast, one

*[email protected]

One contribution of 18 to a Theme Issue Geometry of interfaces:topological complexity in biology and materials.

Interface Focus (2012) 2, 658668

doi:10.1098/rsfs.2012.0023

Published online 30 May 2012

Received 26 April 2012Accepted 26 April 2012 658 This journal is q 2012 The Royal Society

http://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlmailto:[email protected]://rsfs.royalsocietypublishing.org/http://rsfs.royalsocietypublishing.org/http://rsfs.royalsocietypublishing.org/http://crossmark.crossref.org/dialog/?doi=10.1098/rsfs.2012.0023&domain=pdf&date_stamp=2012-05-30mailto:[email protected]://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.html


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would expect that small foreign atoms in dilute sol-ution in silicon or germanium would occupy thecomparatively large interstitial spaces (cf. figure 1)and diffuse by strictly random walk, as they hopfrom one interstitial site to another. Because the direc-tions of consecutive steps of an atom in such a case areuncorrelated, f 1. These expectations were con-firmed: in self-diffusion isotope effect experiments insingle crystals of palladium, Peterson obtained resultsconsistent with those of the vacancy mechanism [10],while measurements of the isotope effect for lithium

diffusion in silicon by Pell [11] implied diffusion byrandom walk, which is consistent only with interstitialdiffusion.

The computational techniques used before 1960 tocalculate correlation factors yielded only approximatevalues. Even though these were accurate enough forcomparison with experiment, I decided to derive exactvalues, using a direct combinatorial approach. In 1960,with the help of a key clue suggested to me by thetheoretical physicist J. Kanamori (1960, personal com-munication), R. W. Lowen Jr and I obtained solutionsfor f expressed as elliptic integrals [12], yielding exactvalues for six two- and three-dimensional structures, plus

a six-figure numericalvalueforthe simple cubic (sc) lattice.Our results are summarized in table 1. I do not recallwhether we succeeded in obtaining analytic expressionsfor the face-centred cubic (fcc) lattice. The approximate

fcc value listed there, which Bob Lowen and I were ableto confirm, was computed by Hering [9].

The smaller the value of f, the easier it is to obtainaccurate experimental values of the isotope effect. Theentries in table 1 suggested to me that for self-diffusionby the vacancy mechanism in a monatomic cubic crystalwith Z 3, fwould probably be less than 0.5. (The datain the table are consistent with the approximationkcos ulAv ffi 1=Z 1; for both two- and three-dimen-sional structures.) Although I was fascinated in 1958 byWells stereoscopic images [13] of an intertwined pair of

enantiomorphic Laves graphs [1417], for each of whichZ 3, I was disappointed to learn that Wells consideredit highly unlikely that there exists a monatomic crystal inwhich the atomic positions correspond to the vertices ofjust one Laves graph [18]. In 1960, I constructed amodel of an intertwined pair of Laves graphs, and sixyears later it became my guide when I conjectured theexistence of a TPMSthe gyroidthat separates thetwo graphs (figure 2).

While pondering the classical concepts of dual mapsand reciprocal polyhedra described by Coxeter [19,20], Iattempted to develop a systematic general procedure forconstructing a kind of dual relation for pairs of triply-

periodic graphs whose edges coincide with hypotheticaldiffusion pathwaysone graph for self-diffusion and theother for interstitial diffusion. I wondered what restric-tions would be required on the properties of a graph in

Table 1. Calculated values of the BardeenHering correlation factor f [10] for four two-dimensional and four three-dimensional

structures. Z is the coordination number of the graph.

structure Z 2kcos ulAv f

linear chain 2 1 0honeycomb layer 3 1/2 1/3square layer 4 12 2/p 0.466942triangle layer 6 5/6 2

p3/p 0.566057

diamond 4 1/3 1/2simple cubic 6 0.209841 0.653120body-centred cubic 8 12 G4(14)8p

32 8p/G4(14) 0.727194

face-centred cubic 12 0.123. . . 0.781. . .

Figure 1. Large interstitial sites (red) in the diamond crystal structure (green) (stereo image).

Discussion. Triply-periodic minimal surfaces A. H. Schoen 659

Interface Focus (2012)

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order for such a dual recipe to be effective. For example,

would the graph have to be symmetrici.e. both edge-transitive and vertex-transitive? I searched for crystalsof cubic symmetry in which there is onlyone kind of inter-stitial site of atomic proportions, because I assumed thatinterstitial diffusion would approximate random walkmost closely in such crystals.

In the spring of 1966, at the suggestion of KonradWachsmann [21], architecture chairman at the Universityof Southern California, I paid a call on Peter Pearce[22,23], a Los Angeles architect/designer who was study-ing polyhedral packings and triply-periodic networks(graphs). Peters studio was filled with ball-and-stickmodels of crystal structures, two of which I found

especially intriguing. One of them modelled the diamondcrystal structure and the other the body-centred cubic lat-tice. I will call these two the diamondgraph and the bccgraph. A single interstitial cavity in each of them wasoccupied by what Peter called a saddle polyhedronanobject whose faces are skew polygons congruent to thesmallest edge circuit in the graph (cf. figures 3b and 4b).Peter had been inspired by a museum exhibit, designedby his former mentor Charles Eames [24] and the mathe-matician Ray Redheffer [25], in which a regular skewquadrangular boundary frame was periodically immersedin a beaker of soapy water, spanning a minimal surfaceeach time it emerged from the beaker.

When I saw Peters two saddle polyhedra, I recog-nized immediately that at least for the graphs sc, bcc,Laves and diamondand probably also for othergraphsit was true that:

a point at the centre of the saddle polyhedron P1that is interstitial with respect to the graph G1 is avertex of a second graph G2, and

a point at the centre of the saddle polyhedron P2that is interstitial with respect to the graph G2 is avertex of the original graph G1.

The edges ofG1 protrude through the faces ofP2, and

the edges ofG2 protrude through the faces ofP1. I con-cluded that saddle polyhedra might serve as the basisfor a three-dimensional dual relation analogous to theconventional duality of planar graphs. I called G1 a

nodal graph and G2 an interstitial graph, but a saddlepolyhedron that is interstitial in one of the graphs isnodal in its dual and vice versa. Figures 3a,b and 4a,b

illustrate this duality for the bcc and I-WP graphs. Afew graphs, such as sc, diamond and Laves, are self-dual, but the graphs of most dual pairs, such as the bccgraph (cf. figure 3a) and the WP graph (cf. figure 4a),are dissimilar. Two dual Laves graphs are enantio-morphic because the Laves graph is chiral. The nodalpolyhedron for the Laves graph is shown in figure 5.

During the weeks following my first meeting withPeter Pearce, I tested my dual recipe on a variety ofother graphs, not only symmetric ones, obtaining unam-biguous results in every case. In some cases, the saddlepolyhedra degenerated into convex polyhedra. In the6-valent sc graph and the 12-valent fcc graph, for

example, both the interstitial and nodal polyhedra pro-duced by the recipe are the Voronoi polyhedra of thevertices of their respective graphs. For the fcc graph,there are two shapes of interstitial polyhedrathe

Figure 2. Two dual enanatiomorphic Laves graphs.

(a)

(b)

Figure 3. (a; stereo image) The dual graphs bcc(bluevertices andedges) and WP (orange vertices and edges); (b; stereo image)Tetragonal tetrahedron, interstitial polyhedron of the bccgraph.

660 Discussion. Triply-periodic minimal surfaces A. H. Schoen




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regular octahedron and the regular tetrahedron. Thedual graph in this case is not k-regular, because it hasboth degree 4 and degree 8 vertices. Nevertheless, thenodal polyhedron produced by the recipe for fcc is therhombic dodecahedron, as one might expect. In spite ofall these favourable indications, I recognized that therecipe, which is described in [6], is essentially an ad hocheuristic. Expecting it to fail eventually, I remained onthe lookout for the failure case.

In recent years, several authors [26,27] have appliedDelaney Dress tiling theory to formalize the conceptof saddle polyhedra, treating a greatly expanded set ofgraphs, with results equivalent to those sketched here.The treatment of the subject has become simplified, inpart because these authors have devised improvedconventions for naming both graphs and saddle polyhedra.

Peter Pearce fabricated each face of his saddle poly-hedra by pushing a metal tool in the shape of the faceboundary into a heated sheet of transparent vinyl,stretching it into an approximation of a minimal sur-face. I constructed my own plastic models of a varietyof saddle polyhedra, using my childrens toy vacuum-

forming machine and tools made in my tiny garageshop. I made each mould for vacuum-forming face poly-gons as a solid cast of polyester resin, first stretching arubber membrane across the boundary of the face tooland then pouring resin onto the membrane and allow-ing it to harden.

My first encounter with TPMS occurred soon after Imet Peter Pearce. While I was thinking about how toname saddle polyhedra, I was startled to discover twospectacular surfaces (cf. figure 6a,c) I had never seenbefore. Let me call the regular skew hexagonal face ofthe expanded octahedron shown in figure 4b F

p/2

because it has 908 face angles. Adjacent faces of this

saddle polyhedron are related by mirror reflection inthe plane that contain a shared pair of consecutiveedges. But if adjacent faces are related instead by ahalf-turn about a common edge, their union defines a

smooth surface spanned by a 10-sided skew polygon.No matter how many additional replicas of F

p/2 areattached in this fashion, the emerging triply-periodiclabyrinthine surface remains free of self-intersections.Applying the same procedure to the regular skew hexa-gon F

p/3, which has 608 face angles, yields a second

triply-periodic surface. I named the Fp/2 surface Dand the Fp/3 surface P. The lattice for D is fcc and

the lattice for P is sc. Eight Fp/2 hexagons define a lat-

tice fundamental domain for Dand four Fp/3 hexagons

define one for P.In June 1966, I telephoned the minimal surface

expert Hans Nitsche for information about the surfacesI was calling D and P. I described them as optimallysmoothed versions of the three infinite regular skewpolyhedra Coxeter and Petrie discovered as schoolboysin the 1920s [28]. Hans informed me that D and P arethe two adjoint minimal surfaces investigated in 1866by Schwarz [4], who proved that they are described by

conjugate harmonic functions in Weierstrass integrals.He explained that the smoothness at the junctionbetween their hexagonal or quadrangular faces is a con-sequence of Schwarzs reflection principle [4,29]. A fewweeks later, Norman Johnson [30] visited me at myhome, where we discussed the Coxeter maps f6,4j4g,f4,6j4g and f6,6j3g [24] and their relevance to these sur-faces. The symbol f6,4j4g, for example, describes atiling by regular six-gons, with four incident on eachvertex, and holes that are regular four-gons. Now Ibegan to study Schwarz [4], Eisenhart [31], Hilbert &Cohn-Vossen [32] and a few other authors.

While I was learning more about the mathematics of

minimal surfaces, I conducted a variety of wire-frameexperiments with soap films, starting with the catenoidand the helicoid. I was mildly curious about the shapeof the curve around the waist of the square catenoid(cf. figure 7). It was clear from soap film experimentswith two square rings at different separation distancesthat this curve is not a circle (cf. the circular waist ofthe true catenoid), but to prove that it is not, it isnecessary to invoke Bjorlings theorem [33]: if two mini-mal surfaces contain a curve C at all correspondingpoints of which the surface tangent planes are thesame, then the surfaces are the same. Early in 1968,my colleague Jim Wixson and I computed the shape

of this square catenoid waist curve, using Schwarzsequations for the P surface [4]. As I anticipated, itbulges slightly outward in the neighbourhood ofthe four points closest to the corners of the squares.

(a)

(b)

Figure 4. (a; stereo image) WP graph; (b; stereo image)Expanded octahedron, interstitial polyhedron of the WPgraph.

Figure 5. nodal polyhedron of the Lavesgraph. Its mirror imageis the interstitial polyhedron of the graph (stereo image).





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(In September 1968, I made use of Schwarzs analyticexpression for the length of this curve to derive theBonnet angle [4] of the gyroid.)

In the summer of 1966, I began calling the inter-twined pair of labyrinth graphs in a TPMS skeletalgraphs. I believed that there must exist other examplesof TPMS besides Schwarzs D, P, H and CLP sur-

faces [4], but I did not undertake a very systematicsearch. In July, I began to suspect that there exists aTPMS I named L (for Laves), with two intertwinedlabyrinths that are essentially swollen versions ofenantiomorphic Laves graphs [14 17]. I later changedits name to gyroid, or G, which is what I will call ithere. I believed in the existence of G partly becauseits skeletal graphslike the sc and diamond skeletalgraphs of P and Dwould be symmetric, i.e. bothvertex-transitive and edge-transitive. Even now Iknow of no other examples of dual pairs of symmetricgraphs on a cubic lattice. The genus of G would bethree, as it is for D and P, and the space lattice of G

would be bcc, implying that there is a TPMS of genusthree for each of the three cubic lattices (the spacelattices of D and P are fcc and sc, respectively).

Schwarz demonstrated how to derive the Weierstrassintegrals that define the coordinates of sufficientlysimple symmetrical minimal surfaces bounded byeither straight lines or plane geodesics (or both) [4].Because it was impossible for G to contain eitherstraight lines or plane geodesics, I had no idea how toconstruct it. It didnt occur to me that the key to thegyroid problem was Ossian Bonnets associate surfacetransformation. Bonnet proved in 1853 [34] that everysimply-connected minimal surface S can be bent in

such a way that (a) the orientation of the tangentplane at every point is unchanged, (b) the Gaussiancurvature at every point is unchanged, and (c) themean curvature at every point remains zero.

D and P are examples of adjoint minimal surfaces.Plane geodesics in one surface of an adjoint pair S1and S2 correspond to straight lines, orthogonal tothat plane, in the other surface. If a point Ocommon to S1 and S2 is fixed, and r1 and r2 are corre-sponding points of S1 and S2, then r*(u), the image ofr1 and

r2 under bending, is given by

r

*(u) r

1 cos ur2 sin u. Hence the points r1, r2, and r*(u) lie on anellipse centered at O. For S1 and S2, u 0 and p/2,respectively. Every surface produced by Bonnet bend-ing is called associate to S1 and S2 and isparameterized by the Bonnet angle u. If S1 D,S2 P, and the Bonnet angle uG cot

21 (K0[1/2]/K[1/2])ffi38.0147748, the resulting intersection-freeassociate surface has all of the properties I had antici-pated for G. Figure 8 shows ellipses through sets ofcorresponding points on D, P, and G. These ellipseshave the same eccentricity.

In May, 1968, I made an incomplete survey of the

regular and uniform tilings on D, G, and P bystraight-edged skew polygons. (In unpublished work,Norman Johnson later filled in the gaps in my inven-tory.) Because I was beginning to receive not-so-subtlepressure from NASA headquarters to do somethinguseful, I decided to apply my analysis of these tilingsto the design of expandable spaceframes, includingone based on the Laves graph. I spent the next twomonths developing these designs, writing NASApatent applications, andwith the assistance ofCharles Strauss and Bob Davismaking computer-animated movies of what I called the collapse trans-formation for several examples of triply-periodic

graphs, including the Laves graph [6].Let us consider the kinematics of the collapse trans-

formation applied to the infinite Laves graph. At firstwe regard the graph as embedded in Schwarzs D sur-face, with adjacent vertices of the graph at the centresof adjacent skew hexagonal faces of D, as infigure 9a. The initial directions of the vertex displace-ments are along perpendiculars to D, adjacent verticesmoving oppositely with respect to the two sides of D.But it is convenient instead to describe all of thevertex trajectories relative to the position of a singlefixed vertex at the origin O. Now the initial directionsof the vertex displacements are as shown in figure 9b.

The trajectory of every vertex V is an ellipse centred atthe origin. The three vertices nearest the origina,b, and c in figure 10rotate on circulararcs in orthog-onal coordinate planes. Figure 11 shows the circular

(a) (b) (c)

Figure 6. The Coxeter-Petrie f6,4j4gmap [28] on the surfaces (a) D, (b) G and (c) P.

Figure 7. The linear asymptotics and plane geodesics in thesquare catenoid of Schwarzs P surface (stereo image).





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trajectories of vertices a, b, and c offigure 10 as well as theelliptical trajectories of three vertices shown in figure 9that are farther from the origin than a, b and c are.

Every vertex in the graph belongs to one of fourclasses1, 2, 3, or 4according to whether it is relatedby a translational symmetry of the graph to the vertex

a, b, c, or O. In figure 10, the points A, B, C, and O,with coordinates rA, rB, rC, and (0,0,0), respectively,lie at the corners of the regular tetrahedron ABCO.Every vertex of the graph is mapped onto one of

the four vertices of ABCO. If a vertex is in class 1,with initial position r1, its collapse trajectory is rr1 cos u rA sin u. The collapse trajectories of verticesin class 2 and 3, with initial positions r2 and r3, arer r2 cos u rB sin uand r r3 cos u rC sin u, respect-ively. The trajectory of a vertex in class 4, with initial

position r4, is along the line through O, r r4 cos u.For a vertex Vwith initial position (x,y,z), the major

radius of its trajectory ellipse is equal to j(x,y,z)j. IfV isin class 1, 2, or 3, the minor radius of the ellipse is equal

P1

G1

G2

G3

G4

G6

G5

O

D1

D2

D3

D6

D4

D5

P6

P5

P2

P3

P4

P1

G1

G2

G3

G4

G6

G5

O

D1

D2

D3

D6

D4

D5

P6

P5

P2

P3 P

4

Figure 8. (stereo image) Ellipses, all of the same eccentricity, through corresponding points on D, Gand P. There is a fixed pointO at the centre of the hexagonal patches.

da a

O O

b bc c

d

d

(a)

a

O

bc

d a

O

bc

(b)

Figure 9. (a; stereo image) Directions of initial displacements of vertices of the Laves graph embedded in Schwarzs D surface.(b; stereo image) Directions of initial displacements of vertices of the Laves graph embedded in Schwarzs D surface, with onevertex (yellow) fixed at the origin.





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to the edge length of the graph, and if Vis in class 4, theminor radius is equal to zero. Consequently, the ellipseeccentricities are not all equal, unlike the eccentricitiesof the elliptical trajectories in the associate surfacetrnsformation. The one-to-one mapping of points inthe associate surface transformation is unrelated tothe mapping of vertices in the collapse transformation,which is many-to-one.

In June 1966, in the course of my campaign to testthe robustness of the dual graph recipe, I defined a sym-metric graph on a given set of vertices as defectiveif notall pairs of nearest-neighbor vertices are joined by anedge [6]. Aside from an uninteresting 3-valent graph

on the vertices of the sc lattice, the only example of adefective symmetric graph I investigated on thatoccasion was FCC6, a 6-valent graph on the verticesof the fcc lattice. The duality recipe yielded the

interstitial and nodal polyhedra shown in figures 12and 13 [6] (http://www.schoengeometry.com/e_tpms.html). If an infinite set of the interstitial polyhedrafor this graph is arranged so that every adjacent pairof polyhedra share quadrangular faces, the hexagonalfaces define the P surface, and if every adjacent pairof polyhedra share hexagonal faces, the quadrangularfaces define the D surface.

In July 1967, at the invitation of the physicist Lester

Van Atta, I joined the staff of the NASA ElectronicsResearch Center (ERC) in Cambridge, MA, where hewas Associate Director. My responsibilities were notvery clearly defined. Van Atta told me just to followmy nose. But to give my position some bureaucraticheft, he created for me the Office of Geometrical Appli-cations. I understood, of course, that I was expectedeventually to produce something NASA might regardas useful to its mission. Exploiting the vagueness ofmy job description, however, I resumed what I hadbarely begun a year earlierexploring the connectionsamong polyhedra, triply-periodic graphs, and minimalsurfaces. Van Atta provided generous support for my

research and never tried to influence my choices ofwhat to work on.

On 14 February 1968, it occurred to me that when Iconstructed the FCC6 graph on the vertices of the fcclattice a year earlier, I forgot to devise a defective sym-metric graph on the vertices of the bcc lattice. I quicklydiscovered BCC6, a symmetric graph of degree six,which turned out to provide the long anticipated coun-terexample to the duality recipe [6]. The breakdown inthe recipe occurred at the first step. When each quad-rangular interstice of BCC6 is spanned by a minimalsurface, the resulting interstitial polyhedron is an infi-nite saddle polyhedron that I call M4 (cf. figure 14).

Its skeletal graphs are enantiomorphic Laves graphs.Weeks later I designed an improvised nodal polyhedron[5,6] (http://www.schoengeometry.com/e_tpms.html)for BCC6 (cf. figure 15).

d d

a a

B B

C, DC, D

O O

A A

c c

bb

Figure 10. Collapse trajectories of vertices a !A, b !B, c !C, d!D in the Laves graph for rotation angle u[ [0, p/2]. The ver-

tices a, b and c move on circularpaths. The vertex d, for example, moves on an ellipse of eccentricity ffiffiffiffiffiffiffiffi2=3

p(stereo image).

(a)

(b)

Figure 11. (a,b) Circular and elliptical collapse trajectories ofLaves graph vertices for rotation angles u[ [0, p/2].



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It seemed plausible to me that M4 could somehow betransformed into the gyroid TPMS I had imagined in1966. If the dihedral angle wat an edge shared by twocongruent skew polygons is equal to p, as in the caseof the skew hexagons in Schwarzs D surface (cf.figure 6a), the angle deficit d p2 w is zero, but if

w,p, d is positive. I judged that d would be smallerin M6the dual of M4than in M4, and a calculation

confirmed that it is: for M4, d p/3, and for M6, dp2 cos21(5/7) ffi 44.4158. On 1 April 1968, after con-structing a physical model of M6 (cf. figure 16), Idecided to replace the infinite regular skew helical poly-gons that wind around the outside of the tunnelsparallel to the cube axes of M6 by helices, believingthat M6 might thereby be transformed into a TPMS.The straight edges of each hexagonal face of M6 wouldthen become helical arcs, each of one-quarter pitch, in asequence of alternating handedness. When I constructeda plastic model of this modified surface (G in figure 6b),

I found that it did indeed appear to resemble a TPMS,but I had no idea how to prove that it is one. I telephonedBob Osserman and explained my predicament: I had ahexagonal surface patch, bounded by helical arcs, thatlooked like a plausible candidate for the unit patch ofan embedded TPMS, but I had no idea how to solvethe equations for the patch. After I sent him a model ofG, he asked his PhD student Blaine Lawson to investi-gate. I introduced myself to Blaine by telephone, toldhim everything I knew about the surface and decided towait for himto prove that G is an embedded TPMS.

At the end of August, 1968, I returned to Cambridgefrom an AMS summer meeting in Madison, where I had

given a talk about the gyroid. When I submitted myabstract [5] for this talk a few months earlier, I naivelyassumed that Blaine would surely complete his proof byAugust. When I telephoned him early in September to

ask whether he had finished the proof, he replied thathe was too busy finishing his dissertation and would beunable to devote any more time to my problem. Ibegged him not to give up, saying that I felt mysteriouslyconfident that the gyroid is a minimal surface and thatthe proof must be right around the corneror words

to that effect (even though I hadnt the slightest ideahow to carry it out!).Then I abruptly changed the subject and described

what had been my principal concern for the previousseveral monthsinvestigating the values of the skew-ness of the faces, vertex figures, and holes of theCoxeter maps {6,4|4}, {4,6|4}, and {6,6|3} on Gand com-paring them to their values on Dand P. I explained howthis analysis [5] had led me to design expandable space-frames based on the geometry of collapsing graphs[35]. I had not previously even mentioned this subjectto Blaine. I tried, clumsily, to explain what I called anamusing coincidence: both the vertices in the collap-

sing graph transformation and also points onSchwarzs D and P surfaces subjected to the associatesurface transformation move on elliptical trajectories,although the ellipses in the two transformations arenot related.

Suddenlyin what was to become a EurekamomentBlaine asked me if I was saying that thegyroid is associate to Schwarzs D and P surfaces. Ihad not said that, because this quite obvious idea hadnot crossed my mind. But I immediately became greatlyexcited, because I recognized that Blaine had pointed tothe solution of the problem! For two months I had beenblindly applying truncations and other algebraically

complicated operations to the three Coxeter maps onD, P, and G, without stopping to think about whythese maps match the combinatorial structure of allthree surfaces, why the tangent planes at corresponding

Figure 15. BCC6 pinwheel polyhedron, improvised nodal poly-hedron of BCC6 graph (stereo image).

Figure 14. M4, an infinite warped polyhedron (stereo image).

Figure 12. Doubly expanded tetrahedron, interstitial poly-hedron of the FCC6 graph (stereo image).

Figure 13. FCC6 pinwheel polyhedron, nodal polyhedron ofthe FCC6 graph (stereo image).





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points of these surfaces are parallel, why the surfaceshave the same genus, etc. Now I understood for thefirst time that the spiral curves in each lattice funda-mental domain of G are simply intermediate curves inthe morphing of the round waists of P into the straightlines of D. It was at last clear that these spiral curves

could not be the perfect helices I had naively imaginedthem to be, since their pre-imagesthe round waists ofPare not quite perfect circles. I soon used computer-animated stereoscopic images to support theseconclusions and also to demonstrate that G is the onlyembedded surface associate to D and P. Eighteenyears later, Karsten Grosse-Brauckmann and MeinhardWohlgemuth published a rigorous proof that the gyroidis embedded [36].

I immediately proposed to Blaine that we publish ajoint paper announcing the gyroid, but he declined,explaining that he had done no more than misunderstandwhat I said about the ellipses of the collapse transform-

ation and the ellipses of the associate surfacetransformation. When I insisted that it was impossibleto know how long it would have taken me to discoverthe facts by myself if he had not asked that crucialquestion, Blaine reluctantly agreed to co-publish.

Two days later, when Dr Van Atta returned to ERCfrom an out-of-town trip, I told him the exciting news.To my surprise, he became angry and insisted that I tele-phone Blaine, explain that I had made a serious mistakeand publish alone. Reluctantly, I acceded to his demand.

In the spring of 1967, I was puzzled by a reference onp. 271 of Hilbert and Cohn-Vossens Geometry and theImagination [32] to a TPMS with the symmetry of

the diamond structure, discovered by E. R. Neovius,Schwarzs doctoral student. In 1969, when at last I exam-ined Neoviuss dissertation, I was startled to see adrawing of a lattice fundamental domain of an embeddedTPMS of genus nine [37] in which the set of embeddedstraight lines is exactly the same as the set in SchwarzsP surface!When I inspected my model of the Dsurface,I confirmed that it too has an embedded complementof genus nineteenthat I named C(D) (http://www.schoengeometry.com/e_tpms.html) [38]. It is shown infigure 17. In 1970, I conjectured from soap film exper-iments inside transparent polyhedral cells that both Dand P have unbounded arithmetic sequences of comp-

lements of increasing genus and that complements existfor other TPMS as well [6]. Ken Brakkes experimentswith his SURFACE EVOLVER [1,39], beginning in 1999,provide qualified support for these conjectures.

While studying a 1934 paper by Stessmann [40], I con-jectured the existence of several intersection-free TPMS,bounded by plane geodesics, that contain no embeddedstraight lines. The elementary patch of each of these sur-faces, which include I-WP and F-RD, is a simply-connected surface in a stationary state inside a Coxeter

cell [6], derived from its adjoint surface. In 1989, Karcherconstructed a rigorous proof of the existence of these sur-faces and derived Weierstrass parametrizations for manyofthem[41]. Fogden and Hyde independently derived theWeierstrass parametrizations for these and otherexamples of TPMS [4244].

In 1969, I invented a technique for grafting handlesonto TPMS [6] (http://www.schoengeometry.com/e_tpms.html). It is based on the construction of anembedded surface g that is a hybrid of two knownexamples, a and b, of embedded surfaces. First, oneconstructsthe surfaceg, which is a linearcombination ofthe straight-edged polygons a and b, the adjoints ofa

and b. The surface g is then derived as the adjoint ofthe adjoint: g (g). Translational periodicity for gand the absence of self-intersections in g are achievedby properly adjusting relative weights for a and b, atechnique that came to be called killing periods.

The first example of a hybrid surface for which I con-structed a physical model was the surface O,C-TO[6], anunattractive genus 10 hybrid of Pand I-WP. While pre-paring this article, I found in my files a long-forgotten1969 sketch of a genus 14 hybrid of P and NeoviussC(P) [6,37], the first example of a conjectured hybrid sur-face. Not long after I circulated that sketch amongmathematicians in 1969, grafting handles onto minimal

surfacesand not just those of the periodic varietybecame rather fashionable. Many attractive surfaceswere devised by handle grafting. Richard Schoen (towhom I am not closely related!) proved a famous No-Go theorem that asserts the impossibility of grafting ahorizontal handle inside a catenoid that is coaxial witha vertical line (cf. Karchers interactive applet [45],which beautifully illustrates Schoens theorem).

The PC(P) hybrid I sketched in 1969 finally sprangto life recently when I sent a copy of the original hybridproposal to Ken Brakke. Ken used his SURFACE EVOLVERprogram [1] to derive a lattice fundamental domain(cf. figure 18), and he named the surface N14.

In 1968, with the collaboration of my colleague, thesculptor/model-maker Harald Robinson, I developed atechnique that uses a laser to measure the orientationof the surface normal at points near the boundary of a

Figure 17. One-fourth of a lattice fundamental domain of theminimal surface C(D) (genus 19).

Figure 16. M6, dual of M4 viewed from near [100] axis (stereoimage).



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long-lasting polyoxyethylene film spanned by a straight-edged skew polygon. I used this data to derive theapproximate shape of the plane geodesic boundarycurves of a surface patch of the adjoint TPMS, whichhas no embedded straight lines. In many of theseexamples, it was necessary to kill periods in order to

determine the relative lengths of the edges of the patch.Fortunately, the SURFACE EVOLVER program [1] hasmade this tedious experimental procedure obsolete. In1999, Ken Brakke and I began a collaboration to validateexamples of these TPMS, whose existence I hadconjectured between 1969 and 1972 (http://www.schoengeometry.com/e_tpms.html).

In my last eighteen months at NASA/ERC, Ibecame aware of several possible scientific applicationsof TPMS. A literature search in 1966 revealed thatthe structure of the prolamellar body for some etiolatedgreen plants [46,47] invites comparison with SchwarzsP surface. In 1969, I conferred twice with the Harvard

biologists Lawrence Bogorad and Christopher Wood-cock, who are experts in this field. When I telephonedthe biophysicist Donald Caspar [48] in 1969 to inquirewhether he knew of any chemical compounds withspace group Ia3d (the space group of the gyroid), heinstantly referred me to Polymorphism of lipids, a1966 paper by Luzzati and Spegt [49] on the structureof a high-temperature phase of divalent cation soaps.That was the first hint I had of the existence of crystal-line matter that incorporates the geometry of anenantiomorphic pair of Laves graphs. The first signifi-cant report of the gyroid in liquid crystals was anarticle by Stephen Hyde et al. [50]. Additional

recommended readings are in [51 53].In lectures between 1969 and 1975, I described sev-

eral potential scientific applications of TPMS, but Inever published anything about them. In 1969,Arthur Drexler, Director of Design at the N.Y.Museum of Modern Art, commissioned me to build alarge sculpture of the gyroid for a 1970 exhibition atMOMA. Dr. Van Atta obtained funds for the projectfrom NASA Headquarters, but President Nixon closedNASA/ERC just a few months later, and the projectwas terminated very soon after Jim Wixson and Ibegan work on the CAD/CAM phase of the project.In 1971, I was asked by the NSF College Science Curri-

culum Improvement Program to design a packagedcourse about TPMS, graphs, and polyhedra, but afterNSF approved my preliminary proposal, fundingfor the entire CSCIP was cancelled by an influential

U.S. Senator. In September 1969, thanks to the kind-ness of Robert Osserman and Lipman Bers, Iparticipated in an international conference in Tbilisi,Georgia, USSR on Optimal Control Theory, PartialDifferential Equations and Minimal Surfaces (http://www.schoengeometry.com/e_tpms.html). In 1972 and1974, I made two black-and-white videos on the subject

of TPMS. I edited these videos in 1999, and I expect tomake them available soon.

This article is dedicated to the memory of Alexander F. JumboWells, without whose inspiration I would almost certainlynot have embarked on my journey. I cherish my memories ofa day spent with Wells during a 1968 visit to his homein Storrs, Connecticut. Jim Tanaka has written an oralhistory of Wells life and work, which can be found at http://schoengeometry.com/a_f_wells_oral_history.pdf. I am gratefulto an anonymous referee for suggesting improvements of thisarticle. Note: All of the stereoscopic image pairs shown aboveare arranged for cross-eyed viewing. If they are viewed with astereoscope, the left and right images should be exchanged.

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Figure 18. Views of Ken Brakkes SURFACE EVOLVER solutionfor two lattice fundamental domains of N14, a genus 14hybrid of P and C(P) (images courtesy of Ken Brakke).



http://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://dx.doi.org/10.1098/rsta.1996.0095http://dx.doi.org/10.1103/PhysRevLett.76.2726http://materials.iisc.ernet.in/~lord/webfiles/tpmbs.pdfhttp://materials.iisc.ernet.in/~lord/webfiles/tpmbs.pdfhttp://materials.iisc.ernet.in/~lord/webfiles/tpmbs.pdfhttp://materials.iisc.ernet.in/~lord/webfiles/tpmbs.pdfhttp://materials.iisc.ernet.in/~lord/webfiles/tpmbs.pdfhttp://materials.iisc.ernet.in/~lord/webfiles/tpmbs.pdfhttp://materials.iisc.ernet.in/~lord/webfiles/tpmbs.pdfhttp://materials.iisc.ernet.in/~lord/webfiles/tpmbs.pdfhttp://materials.iisc.ernet.in/~lord/webfiles/tpmbs.pdfhttp://dx.doi.org/10.1103/PhysRevLett.1.138http://dx.doi.org/10.1103/PhysRevLett.1.138http://dx.doi.org/10.1103/PhysRevLett.1.138http://dx.doi.org/10.1080/14786435908238264http://dx.doi.org/10.1080/14786435908238264http://dx.doi.org/10.1080/14786435908238264http://dx.doi.org/10.1103/PhysRev.136.A568http://dx.doi.org/10.1103/PhysRev.136.A568http://dx.doi.org/10.1103/PhysRev.136.A568http://dx.doi.org/10.1103/PhysRev.119.1014http://dx.doi.org/10.1103/PhysRev.119.1014http://dx.doi.org/10.1103/PhysRev.119.1014http://dx.doi.org/10.4153/CJM-1955-003-7http://dx.doi.org/10.4153/CJM-1955-003-7http://dx.doi.org/10.4153/CJM-1955-003-7http://dx.doi.org/10.4153/CJM-1955-003-7http://rsfs.royalsocietypublishing.org/http://rsfs.royalsocietypublishing.org/http://rsfs.royalsocietypublishing.org/http://dx.doi.org/10.4153/CJM-1955-003-7http://dx.doi.org/10.4153/CJM-1955-003-7http://dx.doi.org/10.1103/PhysRev.119.1014http://dx.doi.org/10.1103/PhysRev.119.1014http://dx.doi.org/10.1103/PhysRev.136.A568http://dx.doi.org/10.1103/PhysRev.136.A568http://dx.doi.org/10.1080/14786435908238264http://dx.doi.org/10.1080/14786435908238264http://dx.doi.org/10.1103/PhysRevLett.1.138http://materials.iisc.ernet.in/~lord/webfiles/tpmbs.pdfhttp://materials.iisc.ernet.in/~lord/webfiles/tpmbs.pdfhttp://dx.doi.org/10.1103/PhysRevLett.76.2726http://dx.doi.org/10.1098/rsta.1996.0095http://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.htmlhttp://www.schoengeometry.com/e_tpms.html


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51 Hyde, S. T., OKeeffe, M. & Proserpio, D. M. 2008 A shorthistory of an elusive yet ubiquitous structure in chemistry,material, and mathematics. Angew. Chem. Int. Ed. 47,79968000. (doi:10.1002/anie.200801519)

52 Hyde, S. T., Andersson, S., Ericsson, B. & Larsson, K. 1984A cubic structure consisting of a lipid bilayer forming an infi-nite periodic minimal surface of the gyroid type in theglycerolmonooleate water system. Zeit. Kristallogr. 168,213219. (doi:10.1524/zkri.1984.168.1-4.213)

53 Dierkes, U., Hildebrandt, S. & Sauvigny, F. Second edition2010, Minimal Surfaces. Springer, 708 p., 140 illus. Seehttp://www.springer.com/mathematics/book/978-3-642-11697-1.



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