Tiling Problem: Convex Pentagons for Edge-to-Edge Monohedral Tiling and Convex Polygons for Aperiodic Tiling

TL;DR

This paper classifies convex pentagons capable of edge-to-edge monohedral tilings into eight types and proves that no convex polygon alone can serve as an aperiodic prototile without additional matching rules.

Contribution

It provides a complete classification of convex pentagons for edge-to-edge tilings and establishes the impossibility of convex polygons as aperiodic prototiles without extra conditions.

Findings

Convex pentagons for edge-to-edge tilings are exactly eight types.

No convex polygon can be an aperiodic prototile without matching conditions.

The classification aids in understanding tiling possibilities with convex polygons.

Abstract

We show that convex pentagons that can generate edge-to-edge monohedral tilings of the plane can be classified into exactly eight types. Using these results, it is also proved that no single convex polygon can be an aperiodic prototile without matching conditions other than "edge-to-edge."