40 research outputs found
Cyclotomic Aperiodic Substitution Tilings
The class of Cyclotomic Aperiodic Substitution Tilings (CAST) is introduced.
Its vertices are supported on the 2n-th cyclotomic field. It covers a wide
range of known aperiodic substitution tilings of the plane with finite
rotations. Substitution matrices and minimal inflation multipliers of CASTs are
discussed as well as practical use cases to identify specimen with individual
dihedral symmetry Dn or D2n, i.e. the tiling contains an infinite number of
patches of any size with dihedral symmetry Dn or D2n only by iteration of
substitution rules on a single tile.Comment: 60 pages, 31 figures. Parts of Theorem 2.1 (primitive substitution
matrices) and Theorem 2.2 (proof of aperiodicity) were revised. A reference
to [Hib15] was added, due to a prior claim regarding the generalized
Lancon-Billard tilin
Combinatorial problems of (quasi-)crystallography
Several combinatorial problems of (quasi-)crystallography are reviewed with
special emphasis on a unified approach, valid for both crystals and
quasicrystals. In particular, we consider planar sublattices, similarity
sublattices, coincidence sublattices, their module counterparts, and central
and averaged shelling. The corresponding counting functions are encapsulated in
Dirichlet series generating functions, with explicit results for the triangular
lattice and the twelvefold symmetric shield tiling. Other combinatorial
properties are briefly summarised.Comment: 12 pages, 2 PostScript figures, LaTeX using vch-book.cl
A Note on Shelling
The radial distribution function is a characteristic geometric quantity of a
point set in Euclidean space that reflects itself in the corresponding
diffraction spectrum and related objects of physical interest. The underlying
combinatorial and algebraic structure is well understood for crystals, but less
so for non-periodic arrangements such as mathematical quasicrystals or model
sets. In this note, we summarise several aspects of central versus averaged
shelling, illustrate the difference with explicit examples, and discuss the
obstacles that emerge with aperiodic order.Comment: substantially revised and extended, 15 pages, AMS LaTeX, several
figures included; see also math.MG/990715
Mathematical diffraction of aperiodic structures
Kinematic diffraction is well suited for a mathematical approach via measures, which has substantially been developed since the discovery of quasicrystals. The need for further insight emerged from the question of which distributions of matter, beyond perfect crystals, lead to pure point diffraction, hence to sharp Bragg peaks only. More recently, it has become apparent that one also has to study continuous diffraction in more detail, with a careful analysis of the different types of diffuse scattering involved. In this review, we summarise some key results, with particular emphasis on non-periodic structures. We choose an exposition on the basis of characteristic examples, while we refer to the existing literature for proofs and further details
Kinematic Diffraction from a Mathematical Viewpoint
Mathematical diffraction theory is concerned with the analysis of the
diffraction image of a given structure and the corresponding inverse problem of
structure determination. In recent years, the understanding of systems with
continuous and mixed spectra has improved considerably. Simultaneously, their
relevance has grown in practice as well. In this context, the phenomenon of
homometry shows various unexpected new facets. This is particularly so for
systems with stochastic components. After the introduction to the mathematical
tools, we briefly discuss pure point spectra, based on the Poisson summation
formula for lattice Dirac combs. This provides an elegant approach to the
diffraction formulas of infinite crystals and quasicrystals. We continue by
considering classic deterministic examples with singular or absolutely
continuous diffraction spectra. In particular, we recall an isospectral family
of structures with continuously varying entropy. We close with a summary of
more recent results on the diffraction of dynamical systems of algebraic or
stochastic origin.Comment: 30 pages, invited revie
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
Averaged coordination numbers of planar aperiodic tilings
We consider averaged shelling and coordination numbers of aperiodic tilings. Shelling numbers count the vertices on radial shells around a vertex. Coordination numbers, in turn, count the vertices on coordination shells of a vertex, defined via the graph distance given by the tiling. For the Ammann–Beenker tiling, we find that coordination shells consist of complete shelling orbits, which enables us to calculate averaged coordination numbers for rather large distances explicitly. The relation to topological invariants of tilings is briefly discussed