7 research outputs found
Taylor-Socolar hexagonal tilings as model sets
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
in the plane. If denotes the set of all Taylor-Socolar tilings with
centers on then forms a natural hull under the standard local
topology of hulls and is a dynamical system for the action of . The -adic
completion of is a natural factor of and the natural
mapping is bijective except at a dense set of
points of measure 0 in . We show that consists of three LI
classes under translation. Two of these LI classes are very small, namely
countable -orbits in . The other is a minimal dynamical system which
maps surjectively to and which is variously , , and
at the singular points.
We further develop the formula of Socolar and Taylor (2011) that determines
the parity of the tiles of a tiling in terms of the co-ordinates of its tile
centers. Finally we show that the hull of the parity tilings can be identified
with the hull ; more precisely the two hulls are mutually locally
derivable.Comment: 45 pages, 33 figure
Planar aperiodic tile sets: from Wang tiles to the Hat and Spectre monotiles
A brief history of planar aperiodic tile sets is presented, starting from the
Domino Problem proposed by Hao Wang in 1961. We provide highlights that led to
the discovery of the Taylor--Socolar aperiodic monotile in 2010 and the Hat and
Spectre aperiodic monotiles in 2023. The Spectre tile is an amazingly simple
monotile; a single tile whose translated and rotated copies tile the plane but
only in a way that lacks any translational periodicity. We showcase this
breakthrough discovery through the 60 years that aperiodic tile sets have
been considered.Comment: Expositor
An aperiodic monotile that forces nonperiodicity through dendrites
We introduce a new type of aperiodic hexagonal monotile; a prototile that admits infinitely many tilings of the plane, but any such tiling lacks any translational symmetry. Adding a copy of our monotile to a patch of tiles must satisfy two rules that apply only to adjacent tiles. The first is inspired by the Socolar--Taylor monotile, but can be realised by shape alone. The second is a local growth rule; a direct isometry of our monotile can be added to any patch of tiles provided that a tree on the monotile connects continuously with a tree on one of its neighbouring tiles. This condition forces tilings to grow along dendrites, which ultimately results in nonperiodic tilings. Our local growth rule initiates a new method to produce tilings of the plane
An aperiodic tile with edge-to-edge orientational matching rules
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 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 minimal core of tilings generated by substitution. Identifying tilings which are charge-flips of each other, these tilings are shown to have pure point dynamical spectrum and a regular model set structure