A two-vertex theorem for normal tilings

Abstract

AbstractWe regard a smooth, $$d=2$$ d = 2 -dimensional manifold $$\mathcal {M}$$ 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 $$\bar{v}^{\star }$$ v ¯ ⋆ and we prove that if M is periodic then $$\bar{v}^{\star } \ge 2$$ v ¯ ⋆ ≥ 2 . 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 $$\bar{v}^{\star }=0$$ v ¯ ⋆ = 0 .