## Abstract

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. Only the rare examples of low genus, however, are commonly invoked as shape templates in scientific applications. Exact analytic solutions are now known for many of the low genus examples. The more complex surfaces are readily defined with numerical tools such as Surface Evolver software or the Landau–Ginzburg model. Even though table-top versions of several TPMS have been placed within easy reach by rapid prototyping methods, the inherent complexity of many of these surfaces makes it challenging to grasp their structure. The problem of distinguishing TPMS, which is now acute because of the proliferation of examples, has been addressed by Lord & Mackay (Lord & Mackay 2003 *Curr. Sci*. **85**, 346–362).

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. Only the rare examples of low genus, however, are commonly invoked as shape templates in scientific applications. Exact analytic solutions are now known for many of the low genus examples. The more complex surfaces are readily defined with numerical tools such as Ken Brakke's Surface Evolver software [1] or Holyst and Góźdź's Landau–Ginzburg model [2]. Even though table-top versions of several TPMS have been placed within easy reach by rapid prototyping methods, the inherent complexity of many of these surfaces makes it challenging to grasp their structure. The problem of distinguishing TPMS, which is now acute because of the proliferation of examples, has been addressed by Eric Lord and Alan Mackay [3].

I describe here some highlights of a journey that culminated unexpectedly in 1966 in a search for new examples of TPMS, exactly 100 years after Schwarz wrote his monumental treatise on TPMS [4] (a more detailed 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 only minimal surfaces I had encountered before 1966. I knew only the bare rudiments of differential geometry and little more of complex analysis. In those days, soft matter, mesoscopic, block copolymer, MCM-48, photonic crystals, etc., were not yet household words. My principal research interests were random and correlated random walks on lattices, atomic diffusion in crystals, the Mössbauer effect, and the combinatorial and geometrical connections between triply-periodic graphs and polyhedra. Some of these topics became fused in my mind, leading me along a tortuous path to TPMS.

The first step on this journey was my discovery in 1956 of a method for distinguishing between the ‘vacancy’ and ‘interstitial’ mechanisms for atomic diffusion in crystalline solids (http://www.schoengeometry.com/e_tpms.html) [7–9]. In 1951, in a mathematical analysis of self-diffusion by the vacancy mechanism, in which the elementary step of a diffusing atom is its exchange of position with a vacancy (vacant lattice site), Conyers Hering found that the correlation between the directions of consecutive steps of the atom reduces the self-diffusion coefficient by the fractional amount 1 − *f*, where ; is the average value of the cosine of the angle *θ* between consecutive jumps of the diffusing atom (cf*.* table 1); *f* is known as the Bardeen–Hering correlation factor [9].

I was startled to discover that for self-diffusion by the vacancy mechanism, correlation reduces both the diffusion coefficient and the isotope effect by exactly the same fractional amount. This meant that isotope effect measurements could provide the first experimental evidence to support the prevailing notion that self-diffusion in close-packed crystals of cubic symmetry occurs by the vacancy mechanism. By contrast, one would expect that small foreign atoms in dilute solution in silicon or germanium would occupy the comparatively large interstitial spaces (cf. figure 1) and diffuse by strictly random walk, as they hop from one interstitial site to another. Because the directions of consecutive steps of an atom in such a case are uncorrelated, *f* = 1. These expectations were confirmed: in self-diffusion isotope effect experiments in single crystals of palladium, Peterson obtained results consistent with those of the vacancy mechanism [10], while measurements of the isotope effect for lithium diffusion in silicon by Pell [11] implied diffusion by random walk, which is consistent only with interstitial diffusion.

The computational techniques used before 1960 to calculate correlation factors yielded only approximate values. Even though these were accurate enough for comparison with experiment, I decided to derive *exact* values, using a direct combinatorial approach. In 1960, with the help of a key clue suggested to me by the theoretical physicist J. Kanamori (1960, personal communication), R. W. Lowen Jr and I obtained solutions for *f* expressed as elliptic integrals [12], yielding exact values for six two- and three-dimensional structures, plus a six-figure numerical value for the simple cubic (*sc*) lattice. Our results are summarized in table 1. I do not recall whether we succeeded in obtaining analytic expressions for the face-centred cubic (*fcc*) lattice. The approximate *fcc* value listed there, which Bob Lowen and I were able to confirm, was computed by Hering [9].

The smaller the value of *f*, the easier it is to obtain accurate experimental values of the isotope effect. The entries in table 1 suggested to me that for self-diffusion by the vacancy mechanism in a monatomic cubic crystal with *Z* = 3, *f* would probably be less than 0.5. (The data in the table are consistent with the approximation for both two- and three-dimensional structures.) Although I was fascinated in 1958 by Wells’ stereoscopic images [13] of an intertwined pair of enantiomorphic Laves graphs [14–17], for each of which *Z* = 3, I was disappointed to learn that Wells considered it highly unlikely that there exists a monatomic crystal in which the atomic positions correspond to the vertices of just one Laves graph [18]. In 1960, I constructed a model of an intertwined pair of Laves graphs, and six years later it became my guide when I conjectured the existence of a TPMS—the gyroid—that separates the two graphs (figure 2).

While pondering the classical concepts of dual maps and reciprocal polyhedra described by Coxeter [19,20], I attempted to develop a systematic general procedure for constructing a kind of dual relation for pairs of triply-periodic graphs whose edges coincide with hypothetical diffusion pathways—one graph for self-diffusion and the other for interstitial diffusion. I wondered what restrictions would be required on the properties of a graph in order for such a dual recipe to be effective. For example, would the graph have to be *symmetric*—i.e. both edge-transitive and vertex-transitive? I searched for crystals of cubic symmetry in which there is only one kind of interstitial site of atomic proportions, because I assumed that interstitial diffusion would approximate random walk most closely in such crystals.

In the spring of 1966, at the suggestion of Konrad Wachsmann [21], architecture chairman at the University of Southern California, I paid a call on Peter Pearce [22,23], a Los Angeles architect/designer who was studying polyhedral packings and triply-periodic networks (graphs). Peter's studio was filled with ball-and-stick models of crystal structures, two of which I found especially intriguing. One of them modelled the diamond crystal structure and the other the body-centred cubic lattice. I will call these two the *diamond* graph and the *bcc* graph. A single interstitial cavity in each of them was occupied by what Peter called a *saddle polyhedron*—an object whose faces are skew polygons congruent to the smallest edge circuit in the graph (cf*.* figures 3*b* and 4*b*). Peter had been inspired by a museum exhibit, designed by his former mentor Charles Eames [24] and the mathematician Ray Redheffer [25], in which a regular skew quadrangular boundary frame was periodically immersed in a beaker of soapy water, spanning a minimal surface each time it emerged from the beaker.

When I saw Peter's two saddle polyhedra, I recognized immediately that at least for the graphs *sc*, *bcc*, Laves and diamond—and probably also for other graphs—it was true that:

— a point at the centre of the saddle polyhedron 𝒫

_{1}that is interstitial with respect to the graph 𝒢_{1}is a vertex of a second graph 𝒢_{2}, and— a point at the centre of the saddle polyhedron 𝒫

_{2}that is interstitial with respect to the graph 𝒢_{2}is a vertex of the original graph 𝒢_{1}.

The edges of 𝒢_{1} protrude through the faces of 𝒫_{2}, and the edges of 𝒢_{2} protrude through the faces of 𝒫_{1}. I concluded that saddle polyhedra might serve as the basis for a three-dimensional dual relation analogous to the conventional duality of planar graphs. I called 𝒢_{1} a *nodal* graph and 𝒢_{2} an *interstitial* graph, but a saddle polyhedron that is interstitial in one of the graphs is nodal in its dual and *vice versa*. Figures 3*a*,*b* and 4*a*,*b* illustrate this duality for the *bcc* and *I-WP* graphs. A few graphs, such as *sc*, *diamond* and *Laves*, are self-dual, but the graphs of most dual pairs, such as the *bcc* graph (cf*.* figure 3*a*) and the *WP* graph (cf*.* figure 4*a*), are dissimilar. Two dual *Laves* graphs are enantiomorphic because the *Laves* graph is chiral. The nodal polyhedron for the Laves graph is shown in figure 5.

During the weeks following my first meeting with Peter Pearce, I tested my dual recipe on a variety of other graphs, not only symmetric ones, obtaining unambiguous results in every case. In some cases, the saddle polyhedra ‘degenerated’ into convex polyhedra. In the 6-valent *sc* graph and the 12-valent *fcc* graph, for example, both the interstitial and nodal polyhedra produced by the recipe are the Voronoi polyhedra of the vertices of their respective graphs. For the *fcc* graph, there are *two* shapes of interstitial polyhedra—the regular octahedron and the regular tetrahedron. The dual graph in this case is not ‘*k*-regular’, because it has both degree 4 and degree 8 vertices. Nevertheless, the nodal polyhedron produced by the recipe for *fcc* is the rhombic dodecahedron, as one might expect. In spite of all these favourable indications, I recognized that the recipe, which is described in [6], is essentially an *ad hoc* heuristic. Expecting it to fail eventually, I remained on the lookout for the ‘failure’ case.

In recent years, several authors [26,27] have applied Delaney–Dress tiling theory to formalize the concept of saddle polyhedra, treating a greatly expanded set of graphs, with results equivalent to those sketched here. The treatment of the subject has become simplified, in part because these authors have devised improved conventions for naming both graphs and saddle polyhedra.

Peter Pearce fabricated each face of his saddle polyhedra by pushing a metal tool in the shape of the face boundary into a heated sheet of transparent vinyl, stretching it into an approximation of a minimal surface. I constructed my own plastic models of a variety of saddle polyhedra, using my children's toy vacuum-forming machine and tools made in my tiny garage shop. I made each mould for vacuum-forming face polygons as a solid cast of polyester resin, first stretching a rubber membrane across the boundary of the face tool and then pouring resin onto the membrane and allowing it to harden.

My first encounter with TPMS occurred soon after I met Peter Pearce. While I was thinking about how to name saddle polyhedra, I was startled to discover two spectacular surfaces (cf*.* figure 6*a*,*c*) I had never seen before. Let me call the regular skew hexagonal face of the *expanded octahedron* shown in figure 4*b Φ_{π}*

_{/2}because it has 90° face angles. Adjacent faces of this saddle polyhedron are related by mirror reflection in the plane that contain a shared pair of consecutive edges. But if adjacent faces are related instead by a half-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

*Φ*

_{π}

_{/2}are attached in this fashion, the emerging triply-periodic labyrinthine surface remains free of self-intersections. Applying the same procedure to the regular skew hexagon

*Φ*

_{π}

_{/3}, which has 60° face angles, yields a second triply-periodic surface. I named the

*Φ*

_{π}

_{/2}surface

*D*and the

*Φ*

_{π}

_{/3}surface

*P*. The lattice for

*D*is

*fcc*and the lattice for

*P*is

*sc*. Eight

*Φ*

_{π}

_{/2}hexagons define a lattice fundamental domain for

*D*and four

*Φ*

_{π}

_{/3}hexagons define one for

*P*.

In June 1966, I telephoned the minimal surface expert Hans Nitsche for information about the surfaces I was calling *D* and *P*. I described them as optimally smoothed versions of the three infinite regular skew polyhedra Coxeter and Petrie discovered as schoolboys in the 1920s [28]. Hans informed me that *D* and *P* are the two *adjoint* minimal surfaces investigated in 1866 by Schwarz [4], who proved that they are described by conjugate harmonic functions in Weierstrass integrals. He explained that the smoothness at the junction between their hexagonal or quadrangular faces is a consequence of *Schwarz's reflection principle* [4,29]. A few weeks later, Norman Johnson [30] visited me at my home, where we discussed the Coxeter maps {6,4|4}, {4,6|4} and {6,6|3} [24] and their relevance to these surfaces. The symbol {6,4|4}, for example, describes a tiling by regular *six*-gons, with *four* incident on each vertex, and ‘holes’ that are regular *four*-gons. Now I began 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-frame experiments with soap films, starting with the catenoid and the helicoid. I was mildly curious about the shape of the curve around the waist of the ‘square catenoid’ (cf*.* figure 7). It was clear from soap film experiments with two square rings at different separation distances that this curve is not a circle (cf*.* the circular waist of the true catenoid), but to prove that it is not, it is necessary to invoke Björling's theorem [33]: if two minimal surfaces contain a curve *C* at all corresponding points of which the surface tangent planes are the same, 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 Schwarz's equations for the *P* surface [4]. As I anticipated, it bulges slightly outward in the neighbourhood of the four points closest to the corners of the squares. (In September 1968, I made use of Schwarz's analytic expression for the length of this curve to derive the *Bonnet angle* [4] of the gyroid.)

In the summer of 1966, I began calling the intertwined pair of labyrinth graphs in a TPMS ‘skeletal graphs’. I believed that there must exist other examples of TPMS besides Schwarz's *D*, *P, H* and *CLP* surfaces [4], but I did not undertake a very systematic search. In July, I began to suspect that there exists a TPMS I named *L* (for Laves), with two intertwined labyrinths that are essentially swollen versions of enantiomorphic Laves graphs [14–17]. I later changed its name to *gyroid*, or *G*, which is what I will call it here. I believed in the existence of *G* partly because its skeletal graphs—like the *sc* and *diamond* skeletal graphs of *P* and *D*—would be *symmetric*, i.e. both vertex-transitive and edge-transitive. Even now I know of no other examples of dual pairs of symmetric graphs on a cubic lattice. The genus of *G* would be three, as it is for *D* and *P*, and the space lattice of *G* would be *bcc*, implying that there is a TPMS of genus three for each of the three cubic lattices (the space lattices of *D* and *P* are *fcc* and *sc*, respectively).

Schwarz demonstrated how to derive the Weierstrass integrals that define the coordinates of sufficiently simple symmetrical minimal surfaces bounded by either straight lines or plane geodesics (or both) [4]. Because it was impossible for *G* to contain either straight lines or plane geodesics, I had no idea how to construct it. It didn't occur to me that the key to the gyroid problem was Ossian Bonnet's *associate surface transformation*. Bonnet proved in 1853 [34] that every simply-connected minimal surface *S* can be *bent* in such a way that (a) the orientation of the tangent plane at every point is unchanged, (b) the Gaussian curvature at every point is unchanged, and (c) the mean curvature at every point remains zero.

*D* and *P* are examples of *adjoint* minimal surfaces. Plane geodesics in one surface of an adjoint pair *S*_{1} and *S*_{2} correspond to straight lines, orthogonal to that plane, in the other surface. If a point O common to *S*_{1} and *S*_{2} is fixed, and *r*_{1} and *r*_{2} are corresponding points of *S*_{1} and *S*_{2}, then ** r***(

*θ*), the image of

*r*_{1}and

*r*_{2}under bending, is given by

***(**

*r**θ*) =

*r*_{1}cos

*θ*+

*r*_{2}sin

*θ*. Hence the points

*r*_{1},

*r*_{2}, and

***(**

*r**θ*) lie on an ellipse centered at O. For

*S*

_{1}and

*S*

_{2},

*θ*= 0 and

*π*/2, respectively. Every surface produced by Bonnet bending is called

*associate*to

*S*

_{1}and

*S*

_{2}and is parameterized by the

*Bonnet angle*

*θ*. If

*S*

_{1}=

*D*,

*S*

_{2}=

*P*, and the Bonnet angle

*θ*

_{G}= cot

^{−1}(

*K*′[1/2]/

*K*[1/2])≅38.014774°, the resulting intersection-free associate surface has all of the properties I had anticipated for

*G*. Figure 8 shows ellipses through sets of corresponding points on

*D*,

*P*, and

*G*. These ellipses have the same eccentricity.

In May, 1968, I made an incomplete survey of the regular and uniform tilings on *D*, *G*, and *P* by straight-edged skew polygons. (In unpublished work, Norman Johnson later filled in the gaps in my inventory.) Because I was beginning to receive not-so-subtle pressure from NASA headquarters to do something ‘useful’, I decided to apply my analysis of these tilings to the design of expandable spaceframes, including one based on the Laves graph. I spent the next two months developing these designs, writing NASA patent applications, and—with the assistance of Charles Strauss and Bob Davis—making computer-animated movies of what I called the ‘collapse transformation’ for several examples of triply-periodic graphs, including the Laves graph [6].

Let us consider the kinematics of the collapse transformation applied to the infinite Laves graph. At first we regard the graph as embedded in Schwarz's *D* surface, with adjacent vertices of the graph at the centres of adjacent skew hexagonal faces of *D*, as in figure 9*a*. The initial directions of the vertex displacements are along perpendiculars to *D*, adjacent vertices moving oppositely with respect to the two sides of *D*. But it is convenient instead to describe all of the vertex trajectories *relative to the position of a single fixed vertex* at the origin O. Now the initial directions of the vertex displacements are as shown in figure 9*b*. The trajectory of every vertex *V* is an ellipse centred at the origin. The three vertices nearest the origin—a,b, and c in figure 10—rotate on *circular* arcs in orthogonal coordinate planes. Figure 11 shows the circular trajectories of vertices a, b, and c of figure 10 as well as the elliptical trajectories of three vertices shown in figure 9 that are farther from the origin than a, b and c are.

Every vertex in the graph belongs to one of four classes—1, 2, 3, or 4—according to whether it is related by 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 *r*_{A}, *r*_{B}, *r*_{C}, 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 *r*_{1}, its collapse trajectory is ** r** =

*r*_{1}cos

*θ*+

*r*_{A}sin

*θ*. The collapse trajectories of vertices in class 2 and 3, with initial positions

*r*_{2}and

*r*_{3}, are

**=**

*r*

*r*_{2}cos

*θ*+

*r*_{B}sin

*θ*and

**=**

*r*

*r*_{3}cos

*θ*+

*r*_{C}sin

*θ*, respectively. The trajectory of a vertex in class 4, with initial position

*r*_{4}, is along the line through O,

**=**

*r*

*r*_{4}cos

*θ*.

For a vertex *V* with initial position (*x*,*y*,*z*), the major radius of its trajectory ellipse is equal to |(*x*,*y*,*z*)|. If *V* is in class 1, 2, or 3, the minor radius of the ellipse is equal to the edge length of the graph, and if *V* is in class 4, the minor radius is equal to zero. Consequently, the ellipse eccentricities are not all equal, unlike the eccentricities of the elliptical trajectories in the associate surface trnsformation. The one-to-one mapping of points in the associate surface transformation is unrelated to the mapping of vertices in the collapse transformation, which is many-to-one.

In June 1966, in the course of my campaign to test the robustness of the dual graph recipe, I defined a symmetric graph on a given set of vertices as *defective* if not all pairs of nearest-neighbor vertices are joined by an edge [6]. Aside from an uninteresting 3-valent graph on the vertices of the *sc* lattice, the only example of a defective symmetric graph I investigated on that occasion was *FCC*_{6}, a 6-valent graph on the vertices of the *fcc* lattice. The duality recipe yielded the interstitial and nodal polyhedra shown in figures 12 and 13 [6] (http://www.schoengeometry.com/e_tpms.html). If an infinite set of the interstitial polyhedra for this graph is arranged so that every adjacent pair of polyhedra share quadrangular faces, the hexagonal faces define the *P* surface, and if every adjacent pair of polyhedra share hexagonal faces, the quadrangular faces define the *D* surface.

In July 1967, at the invitation of the physicist Lester Van Atta, I joined the staff of the NASA Electronics Research Center (ERC) in Cambridge, MA, where he was Associate Director. My responsibilities were not very clearly defined. Van Atta told me just to ‘follow my nose’. But to give my position some bureaucratic heft, he created for me the Office of Geometrical Applications. I understood, of course, that I was expected eventually to produce something NASA might regard as useful to its mission. Exploiting the vagueness of my job description, however, I resumed what I had barely begun a year earlier—exploring the connections among polyhedra, triply-periodic graphs, and minimal surfaces. Van Atta provided generous support for my research and never tried to influence my choices of what to work on.

On 14 February 1968, it occurred to me that when I constructed the *FCC*_{6} graph on the vertices of the *fcc* lattice a year earlier, I forgot to devise a defective symmetric graph on the vertices of the *bcc* lattice. I quickly discovered *BCC*_{6}, a symmetric graph of degree six, which turned out to provide the long anticipated counterexample to the duality recipe [6]. The breakdown in the recipe occurred at the first step. When each quadrangular interstice of *BCC*_{6} is spanned by a minimal surface, the resulting interstitial polyhedron is an *infinite* saddle polyhedron that I call *M*_{4} (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 *BCC*_{6} (cf*.* figure 15).

It seemed plausible to me that *M*_{4} could somehow be transformed into the gyroid TPMS I had imagined in 1966. If the dihedral angle *φ* at an edge shared by two congruent skew polygons is equal to *π*, as in the case of the skew hexagons in Schwarz's *D* surface (cf*.* figure 6*a*), the *angle deficit δ* =

*π*−

*φ*is zero, but if

*φ*<

*π*,

*δ*is positive. I judged that

*δ*would be smaller in

*M*

_{6}—the dual of

*M*

_{4}—than in

*M*

_{4}, and a calculation confirmed that it is: for

*M*

_{4},

*δ*=

*π*/3, and for

*M*

_{6},

*δ*=

*π*− cos

^{−1}(5/7) ≅ 44.415°. On 1 April 1968, after constructing a physical model of

*M*

_{6}(cf

*.*figure 16), I decided to replace the infinite regular skew

*helical polygons*that wind around the outside of the tunnels parallel to the cube axes of

*M*

_{6}by

*helices*, believing that

*M*

_{6}might thereby be transformed into a TPMS. The straight edges of each hexagonal face of

*M*

_{6}would then become helical arcs, each of one-quarter pitch, in a sequence of alternating handedness. When I constructed a plastic model of this modified surface (

*G*in figure 6

*b*), I found that it did indeed appear to resemble a TPMS, but I had no idea how to prove that it is one. I telephoned Bob Osserman and explained my predicament: I had a hexagonal surface patch, bounded by helical arcs, that looked like a plausible candidate for the unit patch of an embedded TPMS, but I had no idea how to solve the equations for the patch. After I sent him a model of

*G*, he asked his PhD student Blaine Lawson to investigate. I introduced myself to Blaine by telephone, told him everything I knew about the surface and decided to wait for

*him*to prove that

*G*is an embedded TPMS.

At the end of August, 1968, I returned to Cambridge from an AMS summer meeting in Madison, where I had given a talk about the gyroid. When I submitted my abstract [5] for this talk a few months earlier, I naively assumed that Blaine would surely complete his proof by August. When I telephoned him early in September to ask whether he had finished the proof, he replied that he was too busy finishing his dissertation and would be unable to devote any more time to my problem. I begged him not to give up, saying that I felt mysteriously confident that the gyroid is a minimal surface and that the proof must be ‘right around the corner’—or words to that effect (even though I hadn't the slightest idea how to carry it out!).

Then I abruptly changed the subject and described what had been my principal concern for the previous several months—investigating the values of the *skewness* of the faces, vertex figures, and holes of the Coxeter maps {6,4|4}, {4,6|4}, and {6,6|3} on *G* and comparing them to their values on *D* and *P*. I explained how this analysis [5] had led me to design expandable spaceframes based on the geometry of ‘collapsing graphs’ [35]. I had not previously even mentioned this subject to Blaine. I tried, clumsily, to explain what I called an ‘amusing coincidence’: both the vertices in the collapsing graph transformation and also points on Schwarz's *D* and *P* surfaces subjected to the associate surface transformation move on *elliptical trajectories*, although the ellipses in the two transformations are not related.

Suddenly—in what was to become a Eureka moment—Blaine asked me if I was saying that ‘the gyroid is associate to Schwarz's *D* and *P* surfaces’. I had not said that, because this quite obvious idea had not crossed my mind. But I immediately became greatly excited, because I recognized that Blaine had pointed to the solution of the problem! For two months I had been blindly applying truncations and other algebraically complicated operations to the three Coxeter maps on *D*, *P*, and *G*, without stopping to think about why these maps match the combinatorial structure of all three surfaces, why the tangent planes at corresponding points of these surfaces are parallel, why the surfaces have the same genus, *etc*. Now I understood for the first time that the spiral curves in each lattice fundamental domain of *G* are simply intermediate curves in the morphing of the round waists of *P* into the straight lines of *D*. It was at last clear that these spiral curves could not be the perfect helices I had naively imagined them to be, since their pre-images—the round waists of *P*—are not quite perfect circles. I soon used computer-animated stereoscopic images to support these conclusions and also to demonstrate that *G* is the *only* embedded surface associate to *D* and *P*. Eighteen years later, Karsten Grosse-Brauckmann and Meinhard Wohlgemuth published a rigorous proof that the gyroid is embedded [36].

I immediately proposed to Blaine that we publish a joint paper announcing the gyroid, but he declined, explaining that he had done no more than misunderstand what I said about the ellipses of the collapse transformation and the ellipses of the associate surface transformation. When I insisted that it was impossible to know how long it would have taken me to discover the facts by myself if he had not asked that crucial question, Blaine reluctantly agreed to co-publish.

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

In the spring of 1967, I was puzzled by a reference on p. 271 of Hilbert and Cohn-Vossen's ‘Geometry and the Imagination’ [32] to a TPMS with the symmetry of the diamond structure, discovered by E. R. Neovius, Schwarz's doctoral student. In 1969, when at last I examined Neovius's dissertation, I was startled to see a drawing of a lattice fundamental domain of an embedded TPMS of genus nine [37] *in which the set of embedded straight lines is exactly the same as the set in Schwarz's P surface!* When I inspected my model of the *D* surface, I confirmed that it too has an embedded complement—of genus nineteen—that I named C(*D*) (http://www.schoengeometry.com/e_tpms.html) [38]. It is shown in figure 17. In 1970, I conjectured from soap film experiments inside transparent polyhedral cells that both *D* and *P* have unbounded arithmetic sequences of complements of increasing genus and that complements exist for other TPMS as well [6]. Ken Brakke's experiments with his Surface Evolver [1,39], beginning in 1999, provide qualified support for these conjectures.

While studying a 1934 paper by Stessmann [40], I conjectured the existence of several intersection-free TPMS, bounded by plane geodesics, that contain no embedded straight lines. The elementary patch of each of these surfaces, 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, Karcher constructed a rigorous proof of the existence of these surfaces and derived Weierstrass parametrizations for many of them [41]. Fogden and Hyde independently derived the Weierstrass parametrizations for these and other examples of TPMS [42–44].

In 1969, I invented a technique for grafting handles onto TPMS [6] (http://www.schoengeometry.com/e_tpms.html). It is based on the construction of an embedded surface *γ* that is a *hybrid* of two known examples, *α* and *β*, of embedded surfaces. First, one constructs the surface *γ*†, which is a linear combination of the straight-edged polygons *α*† and *β*†, the adjoints of *α* and *β*. The surface *γ* is then derived as the ‘adjoint of the adjoint’: *γ* = (*γ*†)^{†}. Translational periodicity for *γ* and the absence of self-intersections in *γ* are achieved by properly adjusting relative weights for *α* and *β*, a technique that came to be called ‘killing periods’.

The first example of a hybrid surface for which I constructed a physical model was the surface *O,C-TO* [6], an unattractive genus 10 hybrid of *P* and *I-WP*. While preparing this article, I found in my files a long-forgotten 1969 sketch of a genus 14 hybrid of *P* and Neovius's C(*P*) [6,37], the first example of a conjectured hybrid surface. Not long after I circulated that sketch among mathematicians in 1969, grafting handles onto minimal surfaces—and not just those of the periodic variety—became rather fashionable. Many attractive surfaces were devised by handle grafting. Richard Schoen (to whom I am not closely related!) proved a famous ‘No-Go’ theorem that asserts the impossibility of grafting a horizontal handle inside a catenoid that is coaxial with a vertical line (cf*.* Karcher's interactive applet [45], which beautifully illustrates Schoen's theorem).

The *P*–C(*P*) hybrid I sketched in 1969 finally sprang to life recently when I sent a copy of the original hybrid proposal to Ken Brakke. Ken used his Surface Evolver program [1] to derive a lattice fundamental domain (cf*.* figure 18), and he named the surface *N*_{14}.

In 1968, with the collaboration of my colleague, the sculptor/model-maker Harald Robinson, I developed a technique that uses a laser to measure the orientation of the surface normal at points near the boundary of a long-lasting polyoxyethylene film spanned by a straight-edged skew polygon. I used this data to derive the approximate shape of the plane geodesic boundary curves of a surface patch of the *adjoint* TPMS, which has no embedded straight lines. In many of these examples, 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] has made this tedious experimental procedure obsolete. In 1999, Ken Brakke and I began a collaboration to validate examples of these TPMS, whose existence I had conjectured between 1969 and 1972 (http://www.schoengeometry.com/e_tpms.html).

In my last eighteen months at NASA/ERC, I became aware of several possible scientific applications of TPMS. A literature search in 1966 revealed that the structure of the prolamellar body for some etiolated green plants [46,47] invites comparison with Schwarz's *P* surface. In 1969, I conferred twice with the Harvard biologists Lawrence Bogorad and Christopher Woodcock, who are experts in this field. When I telephoned the biophysicist Donald Caspar [48] in 1969 to inquire whether he knew of any chemical compounds with space group (the space group of the gyroid), he instantly referred me to *Polymorphism of lipids*, a 1966 paper by Luzzati and Spegt [49] on the structure of a high-temperature phase of divalent cation soaps. That was the first hint I had of the existence of crystalline matter that incorporates the geometry of an enantiomorphic pair of Laves graphs. The first significant report of the gyroid in liquid crystals was an article by Stephen Hyde *et al.* [50]. Additional recommended readings are in [51–53].

In lectures between 1969 and 1975, I described several potential scientific applications of TPMS, but I never published anything about them. In 1969, Arthur Drexler, Director of Design at the N.Y. Museum of Modern Art, commissioned me to build a large sculpture of the gyroid for a 1970 exhibition at MOMA. Dr. Van Atta obtained funds for the project from NASA Headquarters, but President Nixon closed NASA/ERC just a few months later, and the project was terminated very soon after Jim Wixson and I began work on the CAD/CAM phase of the project. In 1971, I was asked by the NSF College Science Curriculum Improvement Program to design a ‘packaged course’ about TPMS, graphs, and polyhedra, but after NSF approved my preliminary proposal, funding for the entire CSCIP was cancelled by an influential U.S. Senator. In September 1969, thanks to the kindness of Robert Osserman and Lipman Bers, I participated in an international conference in Tbilisi, Georgia, USSR on ‘Optimal Control Theory, Partial Differential Equations and Minimal Surfaces’ (http://www.schoengeometry.com/e_tpms.html). In 1972 and 1974, I made two black-and-white videos on the subject of TPMS. I edited these videos in 1999, and I expect to make them available soon.

## Acknowledgments

This article is dedicated to the memory of Alexander F. ‘Jumbo’ Wells, without whose inspiration I would almost certainly not have embarked on my journey. I cherish my memories of a day spent with Wells during a 1968 visit to his home in Storrs, Connecticut. Jim Tanaka has written an oral history of Wells' life and work, which can be found at http://schoengeometry.com/a_f_wells_oral_history.pdf. I am grateful to an anonymous referee for suggesting improvements of this article. Note: All of the stereoscopic image pairs shown above are arranged for cross-eyed viewing. If they are viewed with a stereoscope, the left and right images should be exchanged.

## Footnotes

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

- Received April 26, 2012.
- Accepted April 26, 2012.

- This journal is © 2012 The Royal Society