Convex Polygons for Aperiodic Tiling

TL;DR

This paper proves that any convex polygon capable of creating an edge-to-edge monohedral tiling must also be able to produce a periodic tiling, highlighting a fundamental link between convex polygon tilings and periodicity.

Contribution

It establishes a new theoretical result connecting convex polygons in edge-to-edge tilings with the necessity of periodicity in such tilings.

Findings

Convex polygons in edge-to-edge tilings can generate periodic tilings.

Non-periodic edge-to-edge tilings with convex polygons are impossible.

The result constrains the types of convex polygons suitable for aperiodic tilings.

Abstract

If all tiles in a tiling are congruent, the tiling is called monohedral. Tiling by convex polygons is called edge-to-edge if any two convex polygons are either disjoint or share one vertex or one entire edge in common. In this paper, we prove that a convex polygon that can generate an edge-to-edge monohedral tiling must be able to generate a periodic tiling.