A two-vertex theorem for normal tilings

Gábor Domokos, Ákos G. Horváth, Krisztina Regős

Abstract: We regard a smooth, 𝑑=2-dimensional manifold 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 𝑣¯⋆ and we prove that if M is periodic then 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 𝑣¯⋆=0.