698 research outputs found
A guide to Penrose tilings
The aim of this book is to provide an elementary introduction, complete with
detailed proofs, to the celebrated tilings of the plane discovered by Sir Roger
Penrose in the `70s. The book covers many aspects of Penrose tilings, including
the study of the space parameterizing Penrose tilings from the point of view of
Connes' Noncommutative Geometry.Comment: This is a preview of the following work: Francesco D'Andrea, "A Guide
to Penrose Tilings", 2023, Springer, ISBN: 978-3-031-28427-4. It is
reproduced with permission of Springe
Geometrical Penrose Tilings are characterized by their 1-atlas
Rhombus Penrose tilings are tilings of the plane by two decorated rhombi such
that the decoration match at the junction between two tiles (like in a jigsaw
puzzle). In dynamical terms, they form a tiling space of finite type. If we
remove the decorations, we get, by definition, a sofic tiling space that we
here call geometrical Penrose tilings. Here, we show how to compute the
patterns of a given size which appear in these tilings by two different method:
one based on the substitutive structure of the Penrose tilings and the other on
their definition by the cut and projection method. We use this to prove that
the geometrical Penrose tilings are characterized by a small set of patterns
called vertex-atlas, i.e., they form a tiling space of finite type. Though
considered as folk, no complete proof of this result has been published, to our
knowledge
Deformed Penrose tilings
Monte Carlo (MC) simulation of a model quasicrystal (2D Penrose rhomb tiling) shows that the kinds of local distortions that result from size-effect-like relaxations are in fact very similar to mathematical constructions called deformed model sets. Of particular interest
is the fact that these deformed model sets are pure point-diffractive, i.e. they give diffraction patterns that have sharp Bragg peaks and no diffuse scattering. Although the aforementioned MC simulations give diffraction patterns displaying some diffuse scattering, this can be attributed to the fact that the simulations include a certain amount of unavoidable randomness. Examples of simple deformed
model sets have been constructed by simple prescription and hence contain no randomness. In this case the diffraction patterns show no diffuse scattering. It is demonstrated that simple deformed model sets can be constructed, based on the 2D Penrose rhomb tiling, by using deformations which are defined in terms of the local environment of each vertex. The resulting model sets are all topologically equivalent to the Penrose tiling (same connectedness), are perfectly quasicrystalline but show an enormous variation in the Bragg peak intensities. For the examples described, which are based on nearest-neighbour environments, 12 deformation parameters may be chosen independently. If more distant neighbours are taken into account further sets of parameters may be defined. One example of these simple deformed tilings, which shows great similarity to the size-effect-distorted tiling, is discussed in detail
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
- …