New Tetrahedrally Close-Packed Structures

We consider mathematical models of foams and froths, as collections of surfaces which minimize area under volume constraints. Combinatorially, because of Plateau’s rules, a foam is dual to some triangulation of space. We examine the class of foams known as tetrahedrally close-packed (TCP) structures, which includes the one used by Weaire and Phelan in their counterexample to the Kelvin conjecture. In particular, we construct infinite families of new periodic TCP structures, all of which are convex combinations of the three basic TCP structures (A15, Z and C15). The construction can also be used to create TCP triangulations of three-manifolds other than Euclidean space; these are not such convex combinations. 1 Soap films and foams Soap films, bubble clusters, and foams and froths can be modeled mathematically as collections of surfaces which minimize their surface area subject to volume constraints. Remember that a surface in space has (at each point) two principal curvatures k1 and k2. Because there is no way to globally distinguish the two, only their symmetric functions are physically meaningful. The mean curvature is their sum H = k1 + k2 (twice the average normal curvature). A soap film spanning a wire boundary, with no pressure difference across it, will be a minimal surface (with H = 0), but in general we need to consider the effect of volume constraints in bubble clusters. With these constraints, we find that each surface has constant mean curvature H ≡ c, so it is called a CMC surface. Here the constant is the Lagrange multiplier for the volume constraint, which is exactly the pressure difference across the surface; this relation H = ∆p is known as Laplace’s law. Almgren proposed the following bubble cluster problem: enclose and separate regions in space with given volumes V1, . . . , Vk, using the least total area. He then proved that a minimizer exists and is a smooth surface almost everywhere [1] (see also [6]). It follows that the smooth pieces are CMC surfaces; these meet along certain singularities. Later, Taylor was able to show [14] that the singularities observed empirically in soap films by Plateau [10] are the only ones possible in a minimizing bubble cluster. That is, such a cluster consists of a finite union of smooth surfaces, meeting in threes at 120◦ dihedral angles along a finite number of smooth curves; these curves in turn meet at finitely many tetrahedral corners (which look like the soap film obtained when dipping a tetrahedral frame into soapy water—six sheets come together into the central singularity, along four triple curves). We will define a foam mathematically as a (locally finite) collection of CMC surfaces, meeting according to Plateau’s rules, and satisfying the cocycle condition: pressures can be assigned to each component of the complement so that the mean curvature of each interface is the pressure difference. We will call the components of the complement the cells of the foam, call the interfaces between them simply the faces, call the triple junction lines where they meet the (Plateau) borders, and call the tetrahedral singularities the corners. 2 Combinatorics of foams For many purposes, we can ignore the geometric parts of Plateau’s rules, and only pay attention to the combinatorics of the foam’s cell complex. The combinatorial rules mean that the dual cell complex to a foam is in fact a simplicial complex, that is, a triangulation of space. To construct this dual, we put a vertex in each cell of the foam. Vertices in a pair of adjacent cells are connected by an edge, which is dual to a face of the foam. Where three faces come together along a border, we span the three corresponding edges with a triangle; and where four borders come together at a corner, we fill in a tetrahedron. Combinatorially, the Plateau rules mean that a foam and its dual triangulation are like a Voronoi decomposition of space and the dual Delone triangulation. Given a set of sites in space, the Voronoi cell [7, 11] for each site is the convex polyhedron consisting of points in space closer to that site than to any other. If the sites are in general position (with no five on a common sphere) then the dual Delone complex (whose vertices are the original sites) is completely triangulated; each Delone tetrahedron has the property that no other sites are inside its circumsphere. This similarity suggests that we might look for foams as relaxations of Voronoi decompositions. For instance, we can give a modern interpretation to Kelvin’s construction [15] of his candidate for a least-area partition of space into equal volume cells as follows: Start with sites in the body-centered cubic (BCC) lattice. Their Voronoi cells are truncated octahedra, packed to fill space. If we let the films in this packing relax (until the geometric parts as well as the combinatorial parts of Plateau’s rules are satisfied) we should get a periodic foam, the one proposed by Kelvin. Now consider a triangulation of any compact three-dimensional manifold, and let V , E, F and T be the numbers of vertices, edges, triangles and tetrahedra, respectively. Let z be the average number of edges at a vertex (which is the average number of faces on a cell of the dual foam), and n be the average valence of an edge (the average degree—or number of borders—of a face in the foam). Then a multiple-counting argument shows that 4T = 2F, 3F = nE, 2E = zV. But since the Euler characteristic of any compact 3-manifold is zero, we have 0 = χ = V − E + F − T . Combining these results gives