The growth pattern of an invasive cell-to-cell propagation (called the successive coronas) on the square grid is a tilted square. On the triangular and hexagonal grids, it is an hexagon. It is remarkable that, on the aperiodic structure of Penrose tilings, this cell-to-cell diffusion process tends to a regular decagon (at the limit). In this article we generalize this result to any regular multigrid dual tiling, by defining the characteristic polygon of a multigrid and its dual tiling. Exploiting this elegant duality allows to fully understand why such surprising phenomena, of seeing highly regular polygonal shapes emerge from aperiodic underlying structures, happen.

翻译：侵入性细胞间传播（称为连续冠）在正方形网格上的生长模式是一个倾斜的正方形。在三角形和六边形网格上，它是一个六边形。值得注意的是，在彭罗斯铺砌的非周期结构上，这种细胞间扩散过程趋向于一个正十边形（在极限情况下）。在本文中，我们通过定义多重网格及其对偶铺砌的特征多边形，将这一结果推广到任意正则多重网格对偶铺砌。利用这种优雅的对偶性，可以充分理解为何这种从非周期底层结构中涌现出高度规则多边形形状的惊人现象会发生。