A two-vertex theorem for normal tilings

Abstract

We regard a smooth, $d=2$-dimensional manifold $\mathcal{M}$ and its normal tiling $M$, the cells of which may have non-smooth or smooth vertices (at the latter, two edges meet at 180 degrees.) We denote the average number (per cell) of non-smooth vertices by ${\bar{v}}^{\star }$ and we prove that if $M$ is periodic then $v^{\star }\ge 2$ and we show the same result for the monohedral case by an entirely different argument. Our theory also makes a closely related prediction for non-periodic tilings. In 3 dimensions we show a monohedral construction with ${\bar{v}}^{\star }=0$.

11 pages, 4 figures