Symmetry and other geometric constraints of surface networks in nature and science (Q675345)

Three basic equations for topological constraints upon inhomogeneous surface networks of solids are derived from the Euler equation and other identities, which lead to some insight into the essential issues of this area. In particular, a symmetry between vertices and polygons of a general surface network is shown to exist, and variations in a surface network can simple be described by a kind of reciprocal exchange between vertices and polygons. This finding establishes a quantitative basis for the description of granular and biological materials in terms of microstructures. It will also be seen that classical models correspond to a very special case of constraints. Theoretical results are in agreement with experimental data for networks that arise on a surface, such as fracture, biological cells, metallurgical gains, bubbles, leaf-vein networks and the coat pattern of a giraffe.