The death of Alan Hugh Schoen on 26 July 2023 at age 98 marks the passing of an extraordinary scientist. Schoen is celebrated for his 1968–70 discovery of the gyroid, an infinite periodic minimal “soap film” surface (IPMS). Its intrinsic beauty—carving space into a pair of enantiomeric labyrinths—is more than a geometric jewel. For many years it was ignored, considered at best a curio. Today the gyroid is recognized as a ubiquitous natural form whose complex geometry has been identified in materials with characteristic length scales ranging from angstroms to microns, and it is of relevance to condensed matter physics, materials science, and biology.

The story of Schoen’s discovery is fascinating. The gyroid remained unseen by 19th-century mathematicians, whose studies of minimal surfaces were fueled by the then-nascent field of complex analysis. In particular two of the pioneers of complex analysis, Bernhard Riemann and Hermann Schwarz, discovered the infinite and three-periodic minimal surfaces later christened the P(rimitive) and D(iamond) surfaces. Schoen, a solid-state physicist who was more conversant with crystallography and structural chemistry than with differential geometry, developed a fascination with IPMS in the late 1960s. He communicated and discussed his own results, derived largely from physical models, with a roll call of leading mathematicians responsible for related advances in modern geometry and topology, including Donald Coxeter, Thomas Banchoff, Robert Osserman, Blaine Lawson, Fred Almgren, Lipman Bers, Johannes Nitsche, and Stefan Hildebrandt. Some lent him a sympathetic ear, but little substantial collaboration was forthcoming. Indeed, the existence of the gyroid was only established to the satisfaction of the pure geometry community more than 25 years later.^{[1]} Schoen’s discovery relied far less on pure mathematics than it did on expert model-building and intuition, fueled by his background in condensed-matter physics and innate curiosity for all things geometric. In hindsight, the gyroid is a triumph of “experimental mathematics,” now an accepted route to mathematical discovery.

Schoen was born on 11 December 1924 to Charles John Schoen and Elizabeth Olga Dietz in Mt Vernon, New York, where he was educated at the local public school with his brothers Donald and Homer and his sister Alice. He went on to undergraduate studies at Yale, as a part of the US Navy’s V12 officer training program, then the US Navy School of Oriental Languages at the University of Colorado, in preparation for the postwar reconstruction of Japan. He spent a few years in Japan with the US forces as a civilian employee. His personal connection with Japan was lifelong: Decades later he married Reiko Takasawa, an accomplished classical pianist, with whom he shared 31 years of marriage.

Returning to the US, Schoen joined the University of Illinois at Urbana-Champaign in 1953 for an experimental PhD. Under the supervision of David Lazarus, he used radioactive tracers to measure atomic diffusion in single crystals of alloys. He found the experimental work unrewarding, drawn rather to theoretical aspects of diffusion, with particular interest in the theory of correlated random walks. That led Schoen to model diffusion of isotopes induced by vacancy hopping within a crystal. He published both experimental measurements^{[2]} and exact calculations for various lattice geometries.^{[3]} That work initiated his longstanding interest in crystalline networks, which ultimately led to his discovery of the gyroid.

While working at the private aerospace company TRW Systems in Los Angeles from 1964–67, Schoen explored collapsible, lightweight building modules in preparation for the expected rapid establishment of bases on the Moon. The designs were based on polyhedral packings and his already extensive knowledge of crystalline networks, collected in *The Third Dimension in Chemistry* (A. F. Wells). There he learnt of a theoretical trivalent network, the Laves net, known to Wells as (10,3)-a. That net has been rediscovered again and again since the 1930s, most recently occasioning a comment by Schoen.^{[4]} It deserves to be better known, since, like the tetravalent diamond and hexavalent simple cubic nets, it has just one vertex and one edge, modulo net symmetries.

The concept of a “dual” net had dogged Schoen since his diffusion studies, when he reasoned that vacancies hop within spaces within the atomic structure, effectively traversing adjacent sites of a complementary or dual network. While well-defined from 2D nets, its construction for 3D nets was uncertain. He reasoned that the 3D dual was well-behaved for simpler atomic structures. For example, the diamond net was assumed to be self-dual, with each edge of the dual threading the smallest rings (skew hexagons) of the original net. Similarly, the simple-cubic net was self-dual. The same algorithm applied to the chiral Laves net gave a similar result: It too was self-dual, with the proviso that the dual pair are chiral enantiomers. (It is only in recent years that a more rigorous algorithm for construction of a dual to an arbitrary crystalline net has been advanced, resting on advances in 3D tiling theory and the concept of a “natural” tiling.^{[5]}) His dual construction relied on a second innovation, namely the notion of “interstitial” and “nodal” polyhedra, analogous to the plane-faced Voronoi cells centred on a net vertex, defining the volumes closest to that vertex. In contrast to Voronoi cells, Schoen’s polyhedra allowed nonplanar faces, bounded by skew polygons. Those curved forms were pioneered by the Californian architect Paul Pearce, whose extensive collection made a deep impression on Schoen. Pearce’s “saddle polyhedra” contained saddle-shaped faces, namely minimal surfaces spanning the skew polygons formed by their edges.^{[6]}

In 1967 Schoen relocated to Cambridge, Massachusetts, lured by an irresistible offer. A chance meeting with Lester van Atta, associate director of NASA’s Electronics Research Center, so impressed van Atta that Schoen was hired to head up a new lab at Cambridge, to be known as the Office of Geometrical Applications. The job description was, to put it mildly, relaxed: Schoen was instructed only to “follow [his] nose”^{[7]}. Until the forced closure of the Electronic Research Center in 1970 (rumored to be “prompted by [President Nixon’s] wish to damage the presidential aspirations of the senior senator from Massachusetts, Teddy Kennedy”^{[8]}), Schoen was able to think and work unhindered. The results were spectacular. Schoen merged the twin concepts of dual graphs and minimal surfaces to arrive at precisely Schwarz’ P and D IPMS in a brilliant heuristic. Imagine tubifying all edges of a simple cubic net and its dual, or a diamond net and its dual, forming a pair of labyrinths whose skeletons are a dual pair of nets, enclosed by convoluted walls. Next, inflate both convoluted crystalline labyrinths at the same rate. Eventually, the advancing walls bounding one labyrinth meet the advancing walls of its dual; further growth of one can only occur at the expense of its dual. The balanced intermediate describes a minimal surface, which dissects space into a pair of highly convoluted dual labyrinths, with equal volumes. Those surfaces are the P and D IPMS respectively, whose (self-)dual pairs of labyrinths have primitive cubic and face-centred cubic symmetry respectively. Schoen reasoned that applying the same heuristic to the almost self-dual pair of Laves graphs would give a new IPMS. That claim was reported in 1968, when he announced a third fundamental IPMS, intermediate to the P and D, which he later called the gyroid.^{[9]}

For the next two years, he discovered over a dozen further IPMS based on the concept of dual labyrinths, aided by the construction of accurate models of saddle polyhedra produced by injection molding, pioneering computer graphics experiments, and careful analysis of the differential geometry of minimal surfaces. While definite proof of the existence of these IPMS evaded Schoen at the time, virtually all of those surfaces were later shown to exist to the satisfaction of mathematicians, via analytic and numerical methods.^{[10]}

The main publication of Schoen’s work remains his 1970 NASA Technical Note "Infinite periodic minimal surfaces without self-intersections," which he later described as “sketchy writing” done in “terrible haste,” as he had been “writing against a deadline at top speed.”^{[11]} That may have been so, but its content was groundbreaking and of lasting impact. The report included explicit data cleverly extracted from his knowledge of the intrinsic equivalence of the P, D, and G IPMS via the Bonnet transformation of minimal surfaces. Those data were sufficient to compute the gyroid explicitly, as was done in the early 1980s, by which time the importance of that structure was becoming apparent in liquid crystalline mesophases,^{[12]} following the recognition of IPMS as likely structures of then-mysterious bicontinuous mesophases by the chemical engineer L. E. Scriven.^{[13]} In fact, Schoen later reported that he had recognized the possible physical relevance of IPMS during his time at NASA, inspired by IPMS-like structures reported within plant cells. That suspicion has been amply confirmed in more recent times.

It is sobering to note the underwhelming response to Schoen’s preliminary announcement of the gyroid. Although formally unproven, the claim was profound and deserving of more attention at the time by geometers, since the gyroid, like Pearce’s saddle polyhedra, were in many senses bona fide regular infinite polyhedra, akin to the five finite Platonic polyhedra. An important review of the geometry of polyhedra by Branko Grünbaum in 1977 noted Schoen’s American Mathematical Society abstract but found the description incomprehensible.^{[14]} However obscure they may have been when they first appeared, those publications eventually cemented Schoen’s legacy and mark the birthday of a most remarkable geometry, the gyroid.

Today Schoen’s gyroid and its two cousins (Schwarz’s P and D surfaces) are key models for the so-called bicontinuous phases in soft materials, found in lyotropic liquid crystals (e.g. lipid-water mixtures). The widespread existence of those phases confirms that supramolecular self-assembled materials can spontaneously form complex labyrinthine patterns, well described by hyperbolic (saddle-shaped) interfaces tracing out IPMS, in addition to simpler spherical, cylindrical, or flat interfaces characteristic of other phases of those materials. The origin of hyperbolic interfaces can be understood by simple geometric reasoning.^{[15]} When the constituent molecules such as lipids with hydrophobic and hydrophilic ends assemble into sheets, the wedgelike shape of these two-part molecules determines the curvature of the resulting sheet, much like bimetallic strips bend in response to temperature. Geometry dictates that a sheet formed by a molecular bilayer must be warped, forming a saddle to accommodate the bulkier ends within the interior of the molecular sandwich. The bilayer therefore ideally warps to adopt equal (negative) Gaussian curvature everywhere if the wedge-shaped molecules within the bilayer are uniform in shape. Moreover, to accommodate equally packed back-to-back monolayers, the bilayer must be equally convex and concave everywhere, with zero mean curvature, so it forms a minimal surface. Indeed, it is now certain that the three bicontinuous mesophases found in amphiphile-water mixtures are well modelled by the three simplest IPMS, the P, D, and gyroid surfaces at the molecular scale.^{[16]}^{,}^{[17]}

Molecular packing goes some way to explaining that geometry in chemistry and nature. But a deep conundrum remains. Why complex crystalline IPMS, rather than molten hyperbolic shapes? In contrast to atomic scale crystals, the answer is most likely geometric rather than physical. It turns out that the geometric constraints imposed by molecular packing cannot be realized in our space. All minimal surfaces, except for the flat plane, have varying Gaussian curvature. On the other side of the ledger, the pseudosphere, a surface with constant negative Gaussian curvature, cannot be embedded in 3D Euclidean space without unphysical singularities (a famous theorem due to Hilbert). Chemically monodisperse hyperbolic bilayers are therefore necessarily frustrated. Along with the P and D surfaces, Schoen’s gyroid is likely the least frustrated solution to the twin demands of equal Gaussian curvature and zero mean curvature, with very small variations of Gaussian curvature among singularity-free hyperbolic surfaces in three-space—smaller than other minimal surfaces, including simpler surfaces which are infinite but not three-periodic. In other words, 2D local geometric constraints acting within the bilayer membrane impose global 3D crystallinity at the supramolecular scale. The relative abundance of the gyroid phase compared with the related Schwarz IPMS may be due to a further optimal feature of the gyroid: the most uniform channel dimensions within its dual labyrinths.^{[18]}^{,}^{[19]}^{,}^{[20]}

Typically, these bicontinuous phases have lattice parameters of the order of 10 nm, beyond which they melt. The potential to tune those dimensions within the complex 3D topology makes these self-assembled bicontinuous mesostructures attractive molecular sponges for controlled diffusion of both membrane- and water-bound molecules within the intertwined aqueous domains and the convoluted membrane, both of which form extended connected components, enabling diffusion. Today these liquid crystalline materials are used in the lab and industry as hosts for slow-release of pharmaceuticals and matrices for so-called “in cubo” crystallisation of membrane proteins.^{[21]} Their macroscopic spatial connectivity also gives these phases their gel-like stiffness. When turned into a solid sheet or network structure, the gyroid has been shown to possess excellent mechanical stiffness,^{[22]} which, to a degree, can be rationalized by its high degree uniformity with regards to domain sizes^{[23]}; recently, this has led to its use as an "infill" geometry when 3D-printing solid shapes.

Gyroid structures are also firmly established in phase diagrams of block copolymeric melts. Here too, the local and global geometric homogeneity of the gyroid is responsible, optimising competing chain entropy of constituent blocks and interfacial tensions between different blocks. These mesostructured copolymeric materials afford robust templates for synthesis of solid inorganic bicontinuous materials, giving mesoporous metals^{[24]}^{,}^{[25]} and silicates.^{[26]}

The gyroid, an unknown geometric form just 50 years ago, is today an important structure for materials scientists, chemists, physicists and, more recently, biologists. It transpires that nature has formed gyroids in many living systems for millions of years, remaining unidentified until recently. Its form has been found within many organelles in vivo, including within the retinal cells of tree shrews, chloroplasts in algae, mitochondria of amoeba, and many others across all kingdoms of life.^{[27]} Those living soft materials are complex self-assemblies of proteins, lipids, and water.

Nature also grows solid gyroidlike matrices in vivo, likely templated by soft membranes, identified to date within wing scales of numerous green butterflies^{[28]}^{,}^{[29]}^{,}^{[30]} as well as on beetle cuticles and bird feathers.^{[31]} Its literally most brilliant function is found in some green butterflies whose wing scales are scaffolded by a chitin matrix based on the single gyroid with one solid and one hollow channel. The structure has the right length scale to act as photonic crystals causing green reflections.^{[32]} Inspired by the optical effect in butterflies and by the observation that the single gyroid is a chiral structure,^{[33]} gyroid photonics has emerged as a subfield within its own right, with explorations of the gyroid and related patterns as dielectric photonic crystals and conducting metamaterials.^{[34]}

The formation mechanism for these very swollen hard and soft biomaterials—whose lattice parameters are roughly twice those of the largest synthetic assemblies—remains an open question for both biologists and physicists. Does biology harness the gyroid for specific biological functions, directed by evolutionary pressures? While there are indications, such as light-triggered structural transitions of the bicontinuous membranes in plant chloroplast cells^{[35]} or starvation effects in amoeba,^{[36]} the purposes for which nature uses the gyroid and its relatives remains a mystery. We have little doubt that geometry will play a crucial part in that future puzzle.

Clearly, the occurrence of the gyroid in both synthetic and biological materials with characteristic lengths spanning at least three orders of magnitude hints at the geometric nature of its various formation mechanisms. Indeed, the gyroid has been shown to emerge in entropic simulations of hard pear-shaped particles with no prescribed length scale, suggesting a common origin.

We have focussed here on the most influential aspects of Schoen’s work to the international research community, ignoring his many other activities. He remained a much-loved mentor at Carbondale SIU, with whom he was affiliated for many years up to his death. Schoen also explored numerous other geometric issues, from polyhedra to tessellations, and—in line with his self-described “addiction to recreational mathematics”—mathematical puzzle toys.^{[37]}

We were privileged to meet Alan and his wife Reiko at a conference in Primošten, Croatia, in 2011. Although he was in his mid 80s, his extraordinary interest and enthusiasm for all things scientific remained undimmed. Surely everyone at that conference knew of his work, yet none of us had met him personally until then. All of us—mathematicians, physicists, biologists and material scientists—were dazzled by the presence of this gracious, loquacious, and curious gentleman, brimming with stories of his adventures in scientific research, seemingly from a lost world. There, Schoen presented a magnificent lecture, full of generous attribution to the work of others, which is now accessible to all.^{[38]}

Schoen is survived by his wife Reiko and by his children Cathy, Andrew, and Alison.

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#### References

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^{[6]} P. Pearce, *Structure in nature is a strategy for design*, The MIT Press, Cambridge, Massachusetts, and London, England (1978).

^{[7]} A. H. Schoen, https://schoengeometry.com/e-tpms.html.

^{[8]} A. H. Schoen, https://schoengeometry.com/e-tpms.html, §42.

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