Degenerate polygonal tilings in simple animal tissues.

The salient feature of one-cell-thick epithelia is their en face view, which reveals the polygonal cross section of the close-packed prismatic cells. The physical mechanisms that shape these tissues were hitherto explored using theories based on cell proliferation, which were either entirely topological or included certain morphogenetic forces. But mitosis itself may not be instrumental in molding the tissue. We show that the structure of simple epithelia can be explained by an equilibrium model where energy-degenerate polygons in an entropy-maximizing tiling are described by a single geometric parameter encoding their inflation. The two types of tilings found numerically--ordered and disordered--closely reproduce the patterns observed in Drosophila, Hydra, and Xenopus and they generalize earlier theoretical results. Free of a specific cell self-energy, cell-cell interaction, and cell division kinetics, our model provides an insight into the universality of living and inanimate two-dimensional cellular structures.