Homochiral inflation for the aperiodic monotile Tile(1,1)

TL;DR

This paper introduces a homochiral inflation method for the Tile(1,1) monotile, enabling quasiperiodic tilings with fixed chirality, explicit orientation coding, and a clear separation of symmetry elements, advancing understanding of aperiodic tilings.

Contribution

The paper presents a novel homochiral inflation construction for Tile(1,1), fixing chirality at all steps and explicitly coding orientations, which improves the analysis of its quasiperiodic tilings.

Findings

Tilings are decomposed into two clusters with six orientations each.

Positions of metatiles are determined by translations along three directions.

Orientation distribution at each inflation step is explicitly computed.

Abstract

The recently discovered chiral monotile Tile(1,1) is tiling the plane in a quasiperiodic fashion by taking twelve different orientations when applying $2\pi/12$ rotation. An homochiral inflation construction of such a quasiperiodic tiling is proposed where the chirality of the monotile is completely fixed at all inflation steps, avoiding to exchange its chirality between two successive steps. Doing so, the twelve possible orientations of the monotile are explicitly coded and the key difference between odd and even orientations is taken into account. The tiling is decomposed using only two different clusters, $\Gamma$ and $\Omega$, each of them taking six possible orientations. This gives a total set of twelve metatiles, which assembly can be mapped onto a triangular lattice. This approach allows to properly separate rotation and translation symmetry elements relating monotiles together. As all possible orientations of the two clusters are already incorporated in the twelve metatiles, positions of adjacent metatiles are given by translations which are along three equivalent directions ($2\pi/3$ rotation) as evidenced by junction lines. Finally, thanks to the homochiral inflation, the orientation distribution of the monotile at each inflation step is computed.