Computer modelling of three-dimensional cellular pattern growth

The time evolution of three-dimensional cellular patterns is discussed on the basis of the oertex model which consists of a set of equations for the motion of vertices obtained from the full curvature-driven equation of motion of cell boundaries (faces) by a reduction in the degrees of freedom. Some difficulty in the earlier vertex models, originating from the fact that faces are non-coplanar, is now side-stepped by introducing virtual vertices as supplementary degrees of freedom. Equations of motion are then readily derived assuming that the whole process is purely dissipative. By solving these equations numerically, scaling properties concerning cellular pattern growth are obtained. In addition, geometrical features such as the three-dimensional versions of the Aboav-Weaire law and the Lewis law are verified with fair accuracy. We emphasize that our vertex model has the potentiality for efficient improvements of indefinite degree by introducing arbi- trary numbers of virtual vertices.