49 research outputs found
Pinwheel patterns and powder diffraction
Pinwheel patterns and their higher dimensional generalisations display
continuous circular or spherical symmetries in spite of being perfectly
ordered. The same symmetries show up in the corresponding diffraction images.
Interestingly, they also arise from amorphous systems, and also from regular
crystals when investigated by powder diffraction. We present first steps and
results towards a general frame to investigate such systems, with emphasis on
statistical properties that are helpful to understand and compare the
diffraction images. We concentrate on properties that are accessible via an
alternative substitution rule for the pinwheel tiling, based on two different
prototiles. Due to striking similarities, we compare our results with the toy
model for the powder diffraction of the square lattice.Comment: 7 pages, 4 figure
MLD Relations of Pisot Substitution Tilings
We consider 1-dimensional, unimodular Pisot substitution tilings with three
intervals, and discuss conditions under which pairs of such tilings are locally
isomorhphic (LI), or mutually locally derivable (MDL). For this purpose, we
regard the substitutions as homomorphisms of the underlying free group with
three generators. Then, if two substitutions are conjugated by an inner
automorphism of the free group, the two tilings are LI, and a conjugating outer
automorphism between two substitutions can often be used to prove that the two
tilings are MLD. We present several examples illustrating the different
phenomena that can occur in this context. In particular, we show how two
substitution tilings can be MLD even if their substitution matrices are not
equal, but only conjugate in . We also illustrate how the (in
our case fractal) windows of MLD tilings can be reconstructed from each other,
and discuss how the conjugating group automorphism affects the substitution
generating the window boundaries.Comment: Presented at Aperiodic'09 (Liverpool
SCD Patterns Have Singular Diffraction
Among the many families of nonperiodic tilings known so far, SCD tilings are
still a bit mysterious. Here, we determine the diffraction spectra of point
sets derived from SCD tilings and show that they have no absolutely continuous
part, that they have a uniformly discrete pure point part on the z-axis, and
that they are otherwise supported on a set of concentric cylinder surfaces
around this axis. For SCD tilings with additional properties, more detailed
results are given.Comment: 11 pages, 2 figures; Accepted for Journal of Mathematical Physic
On the Frequency Module of the Hull of a Primitive Substitution Tiling
Understanding the properties of tilings is of increasing relevance to the study of aperiodic tilings and tiling spaces. This work considers the statistical properties of the hull of a primitive substitution tiling, where the hull is the family of all substitution tilings with respect to the substitution. A method is presented on how to arrive at the frequency module of the hull of a primitive substitution tiling (the minimal -module, where is the set of integers) containing the absolute frequency of each of its patches. The method involves deriving the tiling\u27s edge types and vertex stars; in the process, a new substitution is introduced on a reconstructed set of prototiles
Primitive substitution tilings with rotational symmetries
This work introduces the idea of symmetry order, which describes the rotational symmetry types of tilings in the hull of a given substitution. Definitions are given of the substitutions σ6 and σ7 which give rise to aperiodic primitive substitution tilings with dense tile orientations and which are invariant under six- and sevenfold rotations, respectively; the derivation of the symmetry orders of their hulls is also presented
PV cohomology of pinwheel tilings, their integer group of coinvariants and gap-labelling
In this paper, we first remind how we can see the "hull" of the pinwheel
tiling as an inverse limit of simplicial complexes (Anderson and Putnam) and we
then adapt the PV cohomology introduced in a paper of Bellissard and Savinien
to define it for pinwheel tilings. We then prove that this cohomology is
isomorphic to the integer \v{C}ech cohomology of the quotient of the hull by
which let us prove that the top integer \v{C}ech cohomology of the hull
is in fact the integer group of coinvariants on some transversal of the hull.
The gap-labelling for pinwheel tilings is then proved and we end this article
by an explicit computation of this gap-labelling, showing that \mu^t
\big(C(\Xi,\ZZ) \big) = \dfrac{1}{264} \ZZ [\dfrac{1}{5}].Comment: Problems of compilation by arxiv for figures on p.6 and p.7. I have
only changed these figure