Symmetry and other geometric constraints of surface networks in nature and science

Abstract

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 simply be described as a kind of reciprocal exchange between vertices and polygons. The number of three-ordered or threefold vertices, as well as many-edged polygons and many-ordered vertices, is controlled by the ratio of the number of three-edged and/or four-edged polygons to the total number of polygons. When the minimum-edged polygon has five edges, the number of three-ordered vertices is automatically greater than two-thirds of the total number of vertices. The dominant occupation of three-ordered vertices can still retain under certain conditions after appearance of three-edged and/or four-edged polygons. The critical distribution of polygons for the maintenance of this kind of domination is determined. The gap between the critical distribution and natural or artificial surface networks allows the geometric structure of a network to be changed greatly without loss of the domination. 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 in surfaces, such as fracture, biological cells, metallurgical grains, bubbles, leaf-vein networks and the coat pattern of a giraffe.