50 research outputs found
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
Substitution tilings with statistical circular symmetry
Two new series of substitution tilings are introduced in which the tiles
appear in infinitely many orientations. It is shown that several properties of
the well-known pinwheel tiling do also hold for these new examples, and, in
fact, for all substitution tilings showing tiles in infinitely many
orientations.Comment: 15 pages, 5 figure
Symmetries of Monocoronal Tilings
The vertex corona of a vertex of some tiling is the vertex together with the
adjacent tiles. A tiling where all vertex coronae are congruent is called
monocoronal. We provide a classification of monocoronal tilings in the
Euclidean plane and derive a list of all possible symmetry groups of
monocoronal tilings. In particular, any monocoronal tiling with respect to
direct congruence is crystallographic, whereas any monocoronal tiling with
respect to congruence (reflections allowed) is either crystallographic or it
has a one-dimensional translation group. Furthermore, bounds on the number of
the dimensions of the translation group of monocoronal tilings in higher
dimensional Euclidean space are obtained.Comment: 26 pages, 66 figure
Weighted cut-and-project sets in bounded distance to a lattice
Recent results of Grepstad and Lev are used to show that weighted
cut-and-project sets with one-dimensional physical space and one-dimensional
internal space are bounded distance equivalent to some lattice if the weight
function is continuous on the internal space, and if is either
piecewise linear, or twice differentiable with bounded curvature.Comment: 11 pages, 1 figur