Boundary Effects in Biological Planar Networks: Pentagons Dominate Marginal Cells

The topological and geometrical features at the boundary zone of planar polygonal networks remain poorly understood. Based on observations and mathematical proofs, we propose that marginal cells in Pyropia haitanensis thalli, a two-dimensional (2D) biological polygonal network, have an average edge number of exactly five. We demonstrate that this number is maintained by specific division patterns. Furthermore, we reveal significant limitations of Lewis law and Aboav-Weaire law by comparing the topological and geometrical parameters of marginal cells and inner cells. We find strong boundary effects that are manifested in the distinct distributions of interior angles and edge lengths in marginal cells. Similar to inner cells, cell division tend to occur in marginal cells with large sizes. Our findings suggest that inner cells should be strictly defined based on their positional relationship to the marginal cells.