Abstract
The Taylor–Socolar tilings are regular hexagonal tilings of the plane but are distinguished in being comprised of hexagons of two colors in an aperiodic way. We place the Taylor–Socolar tilings into an algebraic setting, which allows one to see them directly as model sets and to understand the corresponding tiling hull along with its generic and singular parts. Although the tilings were originally obtained by matching rules and by substitution, our approach sets the tilings into the framework of a cut and project scheme and studies how the tilings relate to the corresponding internal space. The centers of the entire set of tiles of one tiling form a lattice Q in the plane. If XQ denotes the set of all Taylor–Socolar tilings with centers on Q, then XQ forms a natural hull under the standard local topology of hulls and is a dynamical system for the action of Q.The Q-adic completion Q of Q is a natural factor of XQ and the natural mapping XQ → Q is bijective except at a dense set of points of measure 0 in /Q. We show that XQ consists of three LI classes under translation. Two of these LI classes are very small, namely countable Q-orbits in XQ. The other is a minimal dynamical system, which maps surjectively to /Q and which is variously 2 : 1, 6 : 1, and 12 : 1 at the singular points. We further develop the formula of what determines the parity of the tiles of a tiling in terms of the coordinates of its tile centers. Finally we show that the hull of the parity tilings can be identified with the hull XQ; more precisely the two hulls are mutually locally derivable.
Highlights
This paper concerns the aperiodic hexagonal mono-tilings created by Joan Taylor
We learned about these tilings from the unpublished paper of Joan Taylor [1], the extended paper of Socolar and Taylor [2], and a talk given by Uwe Grimm at the KIAS conference on aperiodic order in September, 2010 [3]
The vertices of level ≥ 2 triangles: we look at the six tiles surrounding the corner tiles of level ≥ 2 triangles in a Taylor–Socolar tiling
Summary
This paper concerns the aperiodic hexagonal mono-tilings created by Joan Taylor. We learned about these tilings from the unpublished (but available online) paper of Joan Taylor [1], the extended paper of Socolar and Taylor [2], and a talk given by Uwe Grimm at the KIAS conference on aperiodic order in September, 2010 [3]. Because of the generic conditions, these levels are all bounded in any finite patch and one needs only to look a finite distance out from the patch in order to pick up all the appropriate centroids and triangle edges to decide on the coloring and shifting within the patch. We refer the reader to the paper for details, but the point is that in creating a tiling following the rules a triangle pattern emerges from the stripes of the hexagons This triangularization can be viewed as the edge-shifting of a triangulation T conforming to our edge shifting rule. The point p is a centroid and it either has infinite level, in which case it has no orientation and we go to Proposition 3.3, or it has a finite level in which two of the three w-lines through it have forced color and finite level, which is a contradiction
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