Rotationally Symmetric Tilings with Convex Pentagons and Hexagons

TL;DR

This paper explores rotationally symmetric tilings of the plane using a specific class of convex pentagons and hexagons, demonstrating their ability to generate monohedral and spiral tilings with various symmetry types.

Contribution

It introduces a new class of convex pentagons capable of creating all symmetry types of rotationally symmetric tilings, extending results to convex hexagons.

Findings

Existence of monohedral tilings with any given rotational symmetry

Tilings are also spiral with n arms for n>1

Convex hexagons can produce similar symmetric tilings

Abstract

In contrast to many known results concerning periodic tilings of the Euclidean plane with pentagons, here tilings with rotational symmetry are investigated. A certain class of convex pentagons is introduced. It can be shown that for any given symmetry type $\mathbf{C}_{n}$ or $\mathbf{D}_{n}$ there exists a monohedral tiling generated by a pentagon from this class. For $n>1$ each of these tilings is also a spiral tiling with $n$ arms. As a byproduct it follows that the same holds for convex hexagons.