Four Sided Domains in Hierarchical Space Dividing Patterns

Abstract

The cracks observed in the glaze of ceramics form networks, which divide the 2D plane into domains. It is shown that, on the average, the number of sides of these domains is four. This contrasts with the usual 2D space divisions observed in Voronoi tessellation or 2D soap froths. In the latter networks, the number of sides of a domain coincides with the number of its neighbors, which, according to Euler's theorem, has to be six on average. The four sided property observed in cracks is the result of a formation process which can be understood as the successive divisions of domains with no later reorganization. It is generic for all networks having such hierarchical construction rules. We introduce a "geometrical charge," analogous to Euler's topological charge, as the difference from four of the number of sides of a domain. It is preserved during the pattern formation of the crack pattern.