718 research outputs found
Local symmetry dynamics in one-dimensional aperiodic lattices
A unifying description of lattice potentials generated by aperiodic
one-dimensional sequences is proposed in terms of their local reflection or
parity symmetry properties. We demonstrate that the ranges and axes of local
reflection symmetry possess characteristic distributional and dynamical
properties which can be determined for every aperiodic binary lattice. A
striking aspect of such a property is given by the return maps of sequential
spacings of local symmetry axes, which typically traverse few-point symmetry
orbits. This local symmetry dynamics allows for a classification of inherently
different aperiodic lattices according to fundamental symmetry principles.
Illustrating the local symmetry distributional and dynamical properties for
several representative binary lattices, we further show that the renormalized
axis spacing sequences follow precisely the particular type of underlying
aperiodic order. Our analysis thus reveals that the long-range order of
aperiodic lattices is characterized in a compellingly simple way by its local
symmetry dynamics.Comment: 15 pages, 12 figure
Saturated simplicial complexes
AbstractAmong shellable complexes a certain class has maximal modular homology, and these are the so-called saturated complexes. We extend the notion of saturation to arbitrary pure complexes and give a survey of their properties. It is shown that saturated complexes can be characterized via the p-rank of incidence matrices and via the structure of links. We show that rank-selected subcomplexes of saturated complexes are also saturated, and that order complexes of geometric lattices are saturated
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
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