AN APERIODIC TILE WITH EDGE-TO-EDGE ORIENTATIONAL MATCHING RULES

We present a single, connected tile which can the plane but only non-periodically. The is hexagonal with edge markings, impose simple rules as to how adjacent tiles are allowed meet across edges. first of these standard matching rule, that certain decorations match second condition new type allows when in particular orientation given opposite charge. This forces form hierarchy triangles, following central idea Socolar--Taylor tilings. However, edge-to-edge orientational rule this structure very different way, for surprisingly proof aperiodicity. show hull all tilings satisfying our uniquely ergodic and almost belong minimal core generated by substitution. Identifying charge-flips each other, shown have pure point dynamical spectrum regular model set structure.