742 research outputs found
An aperiodic tiles machine
The results we introduce in this work lead to get an algorithm which produces aperiodic sets of tiles using Voronoi diagrams. This algorithm runs in optimal worst-case time O(nlogn). Since a wide range of new examples can be obtained, it could shed some new light on non-periodic tilings. These examples are locally isomorphic and exhibit the 5-fold symmetry which appears in Penrose tilings and quasicrystals. Moreover, we outline a similar construction using Delaunay triangulations and propose some related open problems
Fixed Point and Aperiodic Tilings
An aperiodic tile set was first constructed by R.Berger while proving the
undecidability of the domino problem. It turned out that aperiodic tile sets
appear in many topics ranging from logic (the Entscheidungsproblem) to physics
(quasicrystals) We present a new construction of an aperiodic tile set that is
based on Kleene's fixed-point construction instead of geometric arguments. This
construction is similar to J. von Neumann self-reproducing automata; similar
ideas were also used by P. Gacs in the context of error-correcting
computations. The flexibility of this construction allows us to construct a
"robust" aperiodic tile set that does not have periodic (or close to periodic)
tilings even if we allow some (sparse enough) tiling errors. This property was
not known for any of the existing aperiodic tile sets.Comment: v5: technical revision (positions of figures are shifted
Quasiperiodicity and non-computability in tilings
We study tilings of the plane that combine strong properties of different
nature: combinatorial and algorithmic. We prove existence of a tile set that
accepts only quasiperiodic and non-recursive tilings. Our construction is based
on the fixed point construction; we improve this general technique and make it
enforce the property of local regularity of tilings needed for
quasiperiodicity. We prove also a stronger result: any effectively closed set
can be recursively transformed into a tile set so that the Turing degrees of
the resulted tilings consists exactly of the upper cone based on the Turing
degrees of the later.Comment: v3: the version accepted to MFCS 201
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
A Universal Semi-totalistic Cellular Automaton on Kite and Dart Penrose Tilings
In this paper we investigate certain properties of semi-totalistic cellular
automata (CA) on the well known quasi-periodic kite and dart two dimensional
tiling of the plane presented by Roger Penrose. We show that, despite the
irregularity of the underlying grid, it is possible to devise a semi-totalistic
CA capable of simulating any boolean circuit on this aperiodic tiling.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249
Local Rules for Computable Planar Tilings
Aperiodic tilings are non-periodic tilings characterized by local
constraints. They play a key role in the proof of the undecidability of the
domino problem (1964) and naturally model quasicrystals (discovered in 1982). A
central question is to characterize, among a class of non-periodic tilings, the
aperiodic ones. In this paper, we answer this question for the well-studied
class of non-periodic tilings obtained by digitizing irrational vector spaces.
Namely, we prove that such tilings are aperiodic if and only if the digitized
vector spaces are computable.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249
- …