A two-vertex theorem for normal tilings

TL;DR

This paper establishes a lower bound on the average number of non-smooth vertices in periodic 2D normal tilings, introduces a new proof for monohedral cases, and explores related predictions and 3D constructions.

Contribution

It proves a new lower bound for non-smooth vertices in periodic 2D tilings and provides a different proof for monohedral tilings, extending understanding of tiling vertex properties.

Findings

Periodic tilings have at least two non-smooth vertices on average.

A new proof technique is introduced for monohedral tilings.

A 3D monohedral tiling with zero non-smooth vertices is constructed.

Abstract

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