Rotationally Symmetric Tilings with Convex Pentagons and Hexagons

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.

8 pages, 10 figures. Tilings in $2$ dimensions, preprint to appear in Elemente der Mathematik (EMS) Vol. 71 (2016)