New Tetrahedrally Close-Packed Structures
John M. Sullivan · 2010
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