  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 $\bar v^{\star}=0$.
