14 research outputs found
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
Averaged shelling for quasicrystals
The shelling of crystals is concerned with counting the number of atoms on
spherical shells of a given radius and a fixed centre. Its straight-forward
generalization to quasicrystals, the so-called central shelling, leads to
non-universal answers. As one way to cope with this situation, we consider
shelling averages over all quasicrystal points. We express the averaged
shelling numbers in terms of the autocorrelation coefficients and give explicit
results for the usual suspects, both perfect and random.Comment: 4 pages, several figures, 2 tables; updated version with minor
corrections and improvements; to appear in the proceedings of ICQ
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
Similarity and Coincidence Isometries for Modules
The groups of (linear) similarity and coincidence isometries of certain
modules in d-dimensional Euclidean space, which naturally occur in
quasicrystallography, are considered. It is shown that the structure of the
factor group of similarity modulo coincidence isometries is the direct sum of
cyclic groups of prime power orders that divide d. In particular, if the
dimension d is a prime number p, the factor group is an elementary Abelian
p-group. This generalizes previous results obtained for lattices to situations
relevant in quasicrystallography.Comment: 14 page
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
Coincidence rotations of the root lattice
The coincidence site lattices of the root lattice are considered, and
the statistics of the corresponding coincidence rotations according to their
indices is expressed in terms of a Dirichlet series generating function. This
is possible via an embedding of into the icosian ring with its rich
arithmetic structure, which recently (arXiv:math.MG/0702448) led to the
classification of the similar sublattices of .Comment: 13 pages, 1 figur
A dimensionally continued Poisson summation formula
We generalize the standard Poisson summation formula for lattices so that it
operates on the level of theta series, allowing us to introduce noninteger
dimension parameters (using the dimensionally continued Fourier transform).
When combined with one of the proofs of the Jacobi imaginary transformation of
theta functions that does not use the Poisson summation formula, our proof of
this generalized Poisson summation formula also provides a new proof of the
standard Poisson summation formula for dimensions greater than 2 (with
appropriate hypotheses on the function being summed). In general, our methods
work to establish the (Voronoi) summation formulae associated with functions
satisfying (modular) transformations of the Jacobi imaginary type by means of a
density argument (as opposed to the usual Mellin transform approach). In
particular, we construct a family of generalized theta series from Jacobi theta
functions from which these summation formulae can be obtained. This family
contains several families of modular forms, but is significantly more general
than any of them. Our result also relaxes several of the hypotheses in the
standard statements of these summation formulae. The density result we prove
for Gaussians in the Schwartz space may be of independent interest.Comment: 12 pages, version accepted by JFAA, with various additions and
improvement
Multiple planar coincidences with N-fold symmetry
Planar coincidence site lattices and modules with N-fold symmetry are well
understood in a formulation based on cyclotomic fields, in particular for the
class number one case, where they appear as certain principal ideals in the
corresponding ring of integers. We extend this approach to multiple
coincidences, which apply to triple or multiple junctions. In particular, we
give explicit results for spectral, combinatorial and asymptotic properties in
terms of Dirichlet series generating functions.Comment: 13 pages, two figures. For previous related work see math.MG/0511147
and math.CO/0301021. Minor changes and references update
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
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Shelling of homogeneous media
A homogeneous medium is characterised by a point set in Euclidean space (for the atomic positions, say), together with some self-averaging property. Crystals and quasicrystals are homogeneous, but also many structures with disorder still are. The corresponding shelling is concerned with the number of points on shells around an arbitrary, but fixed centre. For non-periodic point sets, where the shelling depends on the chosen centre, a more adequate quantity is the averaged shelling, obtained by averaging over points of the set as centres. For homogeneous media, such an average is still well defined, at least almost surely (in the probabilistic sense). Here, we present a two-step approach for planar model sets