112 research outputs found
Tiling Problems on Baumslag-Solitar groups
We exhibit a weakly aperiodic tile set for Baumslag-Solitar groups, and prove
that the domino problem is undecidable on these groups. A consequence of our
construction is the existence of an arecursive tile set on Baumslag-Solitar
groups.Comment: In Proceedings MCU 2013, arXiv:1309.104
On the noncommutative geometry of tilings
This is a chapter in an incoming book on aperiodic order. We review results
about the topology, the dynamics, and the combinatorics of aperiodically
ordered tilings obtained with the tools of noncommutative geometry
An aperiodic monotile
A longstanding open problem asks for an aperiodic monotile, also known as an
"einstein": a shape that admits tilings of the plane, but never periodic
tilings. We answer this problem for topological disk tiles by exhibiting a
continuum of combinatorially equivalent aperiodic polygons. We first show that
a representative example, the "hat" polykite, can form clusters called
"metatiles", for which substitution rules can be defined. Because the metatiles
admit tilings of the plane, so too does the hat. We then prove that generic
members of our continuum of polygons are aperiodic, through a new kind of
geometric incommensurability argument. Separately, we give a combinatorial,
computer-assisted proof that the hat must form hierarchical -- and hence
aperiodic -- tilings.Comment: 89 pages, 57 figures; Minor corrections, renamed "fylfot" to
"triskelion", added the name "turtle", added references, new H7/H8 rules (Fig
2.11), talk about reflection
The Domino Problem is Undecidable on Surface Groups
We show that the domino problem is undecidable on orbit graphs of non-deterministic substitutions which satisfy a technical property. As an application, we prove that the domino problem is undecidable for the fundamental group of any closed orientable surface of genus at least 2
Construction of the discrete hull for the combinatorics of a regular pentagonal tiling of the plane
The article 'A "regular" pentagonal tiling of the plane' by P. L. Bowers and
K. Stephenson defines a conformal pentagonal tiling. This is a tiling of the
plane with remarkable combinatorial and geometric properties. However, it
doesn't have finite local complexity in any usual sense, and therefore we
cannot study it with the usual tiling theory. The appeal of the tiling is that
all the tiles are conformally regular pentagons. But conformal maps are not
allowable under finite local complexity. On the other hand, the tiling can be
described completely by its combinatorial data, which rather automatically has
finite local complexity. In this paper we give a construction of the discrete
hull just from the combinatorial data. The main result of this paper is that
the discrete hull is a Cantor space
- …