Spherical foams in flat space

Abstract

Regular tessellations of three dimensional space are characterized through their Schläfli symbols {p, q, r}, where each cell has regular p-gonal sides, q sides meeting at each vertex of a cell, and r cells meeting around each edge. Regular tessellations with symbols {p, 3, 3} all satisfy Plateau's laws for equilibrium foams. For general p, however, these regular tessellations do not embed in Euclidean space, but require a uniform background curvature. We study a class of regular foams on S³ which, through conformal, stereographic projection to ³ define irregular cells consistent with Plateau's laws. We analytically characterize a broad classes of bulk foam bubbles, and extend and explain recent observations on foam structure and shape distribution.