An aperiodic tile with edge-to-edge orientational matching rules

Abstract

We present a single, connected tile which can tile the plane but only nonperiodically. The tile is hexagonal with edge markings, which impose simple rules as to how adjacent tiles are allowed to meet across edges. The first of these rules is a standard matching rule, that certain decorations match across edges. The second condition is a new type of matching rule, which allows tiles to meet only when certain decorations in a particular orientation are given the opposite charge. This forces the tiles to form a hierarchy of triangles, following a central idea of the Socolar-Taylor tiling. However, the new edge-to-edge orientational matching rule forces this structure in a very different way, which allows for a surprisingly simple proof of aperiodicity.