A short history of an elusive yet ubiquitous structure in chemistry, materials, and mathematics.

Herein we describe some properties and the occurrences of a beautiful geometric figure that is ubiquitous in chemistry and materials science, however, it is not as well-known as it should be. We call attention to the need for mathematicians to pay more attention to the richly structured natural world, and for materials scientists to learn a little more about mathematics. Our account is informal and eschews any pretence of mathematical rigor, but does start with some necessary mathematics. Regular figures such as the five regular Platonic polyhedra are an enduring part of human culture and have been known and celebrated for thousands of years. Herein we consider them as the five regular tilings on the surface of a sphere (a two-dimensional surface of positive curvature). A flag of a tiling of a two-dimensional surface consists of a combination of a coincident tile, edge, and vertex. A generally accepted definition of regularity is flag transitivity, which means that all flags are related by symmetries of the tiling (i.e. there is just one kind of flag). In addition to the five Platonic solids, there are three regular tilings of the plane (a surface of zero curvature), and these are the familiar coverings of the plane by triangles, squares, or hexagons tiled edge-to-edge. The corresponding regular tilings of three-dimensional space are also wellknown. Flags are now a polyhedron (tile) with a coincident face, edge, and vertex, and the regular tilings of the three-sphere are the six nonstellated regular polytopes of four dimensional space. We remark that four dimensions is the richest space in this regard; higher dimensions have only three regular polytopes (and of course three dimensions has five). However, in flat threedimensional (Euclidean) space, the space of our day-to-day experience, there is disappointingly only one regular tiling—the familiar space filling by cubes sharing faces (face-to-face). The classic reference to these figures is Coxeter!s Regular Polytopes, in which he remarks on the tilings of threedimensional Euclidean space: “For the development of a general theory, it is an unhappy accident that only one honeycomb [tiling] is regular...”. Unhappy indeed, because, perhaps as a consequence, the rich world of periodic graphs, which are the underlying topology of crystal structures, has been largely neglected by mathematicians. The graph associated with (carried by) the regular tiling by cubes is the set of edges and vertices. It is notably the structure of a form of elemental polonium, and chemists often refer to it as the a-Po net. Recently a system of symbols for nets has been developed and this net has the symbol pcu. Our review is concerned with another such periodic graph, and an associated surface.