61 research outputs found
Structural aspects of tilings
In this paper, we study the structure of the set of tilings produced by any
given tile-set. For better understanding this structure, we address the set of
finite patterns that each tiling contains. This set of patterns can be analyzed
in two different contexts: the first one is combinatorial and the other
topological. These two approaches have independent merits and, once combined,
provide somehow surprising results. The particular case where the set of
produced tilings is countable is deeply investigated while we prove that the
uncountable case may have a completely different structure. We introduce a
pattern preorder and also make use of Cantor-Bendixson rank. Our first main
result is that a tile-set that produces only periodic tilings produces only a
finite number of them. Our second main result exhibits a tiling with exactly
one vector of periodicity in the countable case.Comment: 11 page
Aperiodic Tilings: Breaking Translational Symmetry
Classical results on aperiodic tilings are rather complicated and not widely
understood. Below, an alternative approach is discussed in hope to provide
additional intuition not apparent in classical works.Comment: 4 pages, 2 figures, minor change
1D Effectively Closed Subshifts and 2D Tilings
Michael Hochman showed that every 1D effectively closed subshift can be
simulated by a 3D subshift of finite type and asked whether the same can be
done in 2D. It turned out that the answer is positive and necessary tools were
already developed in tilings theory. We discuss two alternative approaches:
first, developed by N. Aubrun and M. Sablik, goes back to Leonid Levin; the
second one, developed by the authors, goes back to Peter Gacs.Comment: Journ\'ees Automates Cellulaires, Turku : Finland (2010
Aperiodic tilings and entropy
In this paper we present a construction of Kari-Culik aperiodic tile set -
the smallest known until now. With the help of this construction, we prove that
this tileset has positive entropy. We also explain why this result was not
expected
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
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