1,038 research outputs found
Combinatorial substitutions and sofic tilings
A combinatorial substitution is a map over tilings which allows to define
sets of tilings with a strong hierarchical structure. In this paper, we show
that such sets of tilings are sofic, that is, can be enforced by finitely many
local constraints. This extends some similar previous results (Mozes'90,
Goodman-Strauss'98) in a much shorter presentation.Comment: 17 pages, 16 figures. In proceedings of JAC 201
Self-dual tilings with respect to star-duality
The concept of star-duality is described for self-similar cut-and-project
tilings in arbitrary dimensions. This generalises Thurston's concept of a
Galois-dual tiling. The dual tilings of the Penrose tilings as well as the
Ammann-Beenker tilings are calculated. Conditions for a tiling to be self-dual
are obtained.Comment: 15 pages, 6 figure
Algorithm for determining pure pointedness of self-affine tilings
Overlap coincidence in a self-affine tiling in is equivalent to pure
point dynamical spectrum of the tiling dynamical system. We interpret the
overlap coincidence in the setting of substitution Delone set in and
find an efficient algorithm to check the pure point dynamical spectrum. This
algorithm is easy to implement into a computer program. We give the program and
apply it to several examples. In the course the proof of the algorithm, we show
a variant of the conjecture of Urba\'nski (Solomyak \cite{Solomyak:08}) on the
Hausdorff dimension of the boundaries of fractal tiles.Comment: 21 pages, 3 figure
Geometric realizations of two dimensional substitutive tilings
We define 2-dimensional topological substitutions. A tiling of the Euclidean
plane, or of the hyperbolic plane, is substitutive if the underlying 2-complex
can be obtained by iteration of a 2-dimensional topological substitution. We
prove that there is no primitive substitutive tiling of the hyperbolic plane
. However, we give an example of substitutive tiling of \Hyp^2
which is non-primitive.Comment: 30 pages, 13 figure
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