SYMMETRY AND OTHER GEOMETRIC CONSTRAINTS OF SURFACE NETWORKS IN NATURE AND SCIENCE

Three basic equations for topological constraints upon inhomogeneous s urface networks of solids are derived from the Euler equation and othe r identities which lead to some insight into the essential issues of t his area. In particular, a symmetry between vertices and polygons of a general surface network is shown to exist, and variations in a surfac e network can simply be described as a kind of reciprocal exchange bet ween vertices and polygons. The number of three-ordered or three-fold 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-edge d polygons to the total number of polygons. When the minimum-edged pol ygon has five edges, the number of three-ordered vertices is automatic ally greater than two-thirds of the total number of vertices. The domi nant occupation of three-ordered vertices can still retain under certa in conditions after appearance of three-edged and/or four-edged polygo ns. The critical distribution of polygons for the maintenance of this kind of domination is determined. The gap between the critical distrib ution and natural or artificial surface networks allows the geometric structure of a network to be changed greatly without loss of the domin ation. This finding establishes a quantitative basis for the descripti on of granular and biological materials in terms of microstructures. I t will also be seen that classical models correspond to a very special case of constraints. Theoretical results are in agreement with experi mental data for networks that arise in surfaces, such as fracture, bio logical cells, metallurgical grains, bubbles, leaf-vein networks and t he coat pattern of a giraffe.