An aperiodic tile with edge-to-edge orientational matching rules
James J. Walton, Michael F. Whittaker
TL;DR
This paper introduces a single hexagonal tile with novel edge-to-edge matching rules that enforce non-periodic tilings, providing a new approach to aperiodicity with rigorous mathematical properties.
Contribution
The authors present a new aperiodic tile with unique edge matching rules, including a novel charge-based rule, and prove its aperiodicity and dynamical properties.
Findings
Tile enforces non-periodic tilings with simple rules
Hull of tilings is uniquely ergodic and minimal
Tilings have pure point spectrum and model set structure
Abstract
We present a single, connected tile which can tile the plane but only non-periodically. 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 tilings. 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. We show that the hull of all tilings satisfying our rules is uniquely ergodic and that almost all tilings in the hull belong to a…
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