A two-vertex theorem for normal tilings
Gábor Domokos, Ákos G. Horváth, Krisztina Regős
Abstract: We regard a smooth, ?=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 ?¯⋆ and we prove that if M is periodic then ?¯⋆≥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 ?¯⋆=0.
![](https://www.morph.bme.hu/wp-content/uploads/2022/06/Fig3_HTML.png)