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 $v^{ \star} \geq 2$ 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 $\bar v^{ \star}=0$.
Categories: math.DG