Cluster Model of Decagonal Tilings

A relaxed version of Gummelt's covering rules for the aperiodic decagon is considered, which produces certain random-tiling-type structures. These structures are precisely characterized, along with their relationships to various other random tiling ensembles. The relaxed covering rule has a natural realization in terms of a vertex cluster in the Penrose pentagon tiling. Using Monte Carlo simulations, it is shown that the structures obtained by maximizing the density of this cluster are the same as those produced by the corresponding covering rules. The entropy density of the covering ensemble is determined using the entropic sampling algorithm. If the model is extended by an additional coupling between neighboring clusters, perfectly ordered structures are obtained, like those produced by Gummelt's perfect covering rules.Comment: 10 pages, 20 figures, RevTeX; minor changes; to be published in Phys. Rev.

Hyperconvexity and Tight Span Theory for Diversities

The tight span, or injective envelope, is an elegant and useful construction that takes a metric space and returns the smallest hyperconvex space into which it can be embedded. The concept has stimulated a large body of theory and has applications to metric classification and data visualisation. Here we introduce a generalisation of metrics, called diversities, and demonstrate that the rich theory associated to metric tight spans and hyperconvexity extends to a seemingly richer theory of diversity tight spans and hyperconvexity.Comment: revised in response to referee comment

Space-Time Approach to Scattering from Many Body Systems

We present scattering from many body systems in a new light. In place of the usual van Hove treatment, (applicable to a wide range of scattering processes using both photons and massive particles) based on plane waves, we calculate the scattering amplitude as a space-time integral over the scattering sample for an incident wave characterized by its correlation function which results from the shaping of the wave field by the apparatus. Instrument resolution effects - seen as due to the loss of correlation caused by the path differences in the different arms of the instrument are automatically included and analytic forms of the resolution function for different instruments are obtained. The intersection of the moving correlation volumes (those regions where the correlation functions are significant) associated with the different elements of the apparatus determines the maximum correlation lengths (times) that can be observed in a sample, and hence, the momentum (energy) resolution of the measurement. This geometrical picture of moving correlation volumes derived by our technique shows how the interaction of the scatterer with the wave field shaped by the apparatus proceeds in space and time. Matching of the correlation volumes so as to maximize the intersection region yields a transparent, graphical method of instrument design. PACS: 03.65.Nk, 3.80 +r, 03.75, 61.12.BComment: Latex document with 6 fig

Some comments on the inverse problem of pure point diffraction

In a recent paper, Lenz and Moody (arXiv:1111.3617) presented a method for constructing families of real solutions to the inverse problem for a given pure point diffraction measure. Applying their technique and discussing some possible extensions, we present, in a non-technical manner, some examples of homometric structures.Comment: 6 pages, contribution to Aperiodic 201

What is a crystal?

Almost 25 years have passed since Shechtman discovered quasicrystals, and 15 years since the Commission on Aperiodic Crystals of the International Union of Crystallography put forth a provisional definition of the term crystal to mean ``any solid having an essentially discrete diffraction diagram.'' Have we learned enough about crystallinity in the last 25 years, or do we need more time to explore additional physical systems? There is much confusion and contradiction in the literature in using the term crystal. Are we ready now to propose a permanent definition for crystal to be used by all? I argue that time has come to put a sense of order in all the confusion.Comment: Submitted to Zeitschrift fuer Kristallographi

Quasicrystalline Order in Binary Dipolar Systems

Motivated by recent experimental findings, we investigate the possible occurrence and characteristics of quasicrystalline order in two-dimensional mixtures of point dipoles with two sorts of dipole moments. Despite the fact that the dipolar interaction potential does not exhibit an intrinsic length scale and cannot be tuned a priori to support the formation of quasicrystalline order, we find that configurations with long--range quasicrystallinity yield minima in the potential energy surface of the many particle system. These configurations emanate from an ideal or perturbed ideal decoration of a binary tiling by steepest descent relaxation. Ground state energy calculations of alternative ordered states and parallel tempering Monte-Carlo simulations reveal that the quasicrystalline configurations do not correspond to a thermodynamically stable state. On the other hand, steepest descent relaxations and conventional Monte-Carlo simulations suggest that they are rather robust against fluctuations. Local quasicrystalline order in the disordered equilibrium states can be strong.Comment: 10 pages, 7 figure

Direct AFM observation of individual micelles, tile decorations and tiling rules of a dodecagonal liquid quasicrystal.

We performed an atomic force microscopy study of the dendron-based dodecagonal quasicrystal, the material that had been reported in 2004 as the first soft quasicrystal. We succeeded in orienting the 12-fold axis perpendicular to the substrate, which allowed imaging of the quasiperiodic xy plane. Thus for the first time we could obtain direct real-space information not only on the arrangement of the tiles, but also on their "decorations" by the individual spherical micelles or "nanoatoms". The high-resolution patterns recorded confirm the square-triangle tiling, but the abundance of the different nodes corresponds closely to random tiling rather than to any inflation rule. The previously proposed model of three types of decorated tiles, two triangular and one square, has been confirmed, and the basic Frank-Kasper mode of alternating dense-sparse-dense-sparse layer stacking along z is confirmed too, each of the four sublayers being 2 nm thick. The consecutive dense layers are seen to be rotated by 90°, as expected. The 2 nm steps on the surface correspond to one layer of spheres, however with a dense layer always remaining on top, which implies a layer slip underneath and possibly the existence of screw dislocations

Overlapping Unit Cells in 3d Quasicrystal Structure

A 3-dimensional quasiperiodic lattice, with overlapping unit cells and periodic in one direction, is constructed using grid and projection methods pioneered by de Bruijn. Each unit cell consists of 26 points, of which 22 are the vertices of a convex polytope P, and 4 are interior points also shared with other neighboring unit cells. Using Kronecker's theorem the frequencies of all possible types of overlapping are found.Comment: LaTeX2e, 11 pages, 5 figures (8 eps files), uses iopart.class. Final versio

Pattern equivariant functions and cohomology

The cohomology of a tiling or a point pattern has originally been defined via the construction of the hull or the groupoid associated with the tiling or the pattern. Here we present a construction which is more direct and therefore easier accessible. It is based on generalizing the notion of equivariance from lattices to point patterns of finite local complexity.Comment: 8 pages including 2 figure

Absence of dissipative solutions of the schrodinger and klein-gordon equations with logarithmic

It is shown that neither the Schrödinger equation nor the Klein-Gordon one with logarithmic nonlinearities have dissipative solutions. In the case of one-dimensional space, numerical experiments with different Cauchy data, in the nonrelativistic case, lead always to final states consisting only in oscillating gaussons