Spherical Foams in Flat Space

Regular tesselations of space are characterized through their Schlafli symbols {p,q,r}, where each cell has regular p-gonal sides, q meeting at each vertex, and r meeting on each edge. Regular tesselations with symbols {p,3,3} all satisfy Plateau's laws for equilibrium foams. For general p, however, these regular tesselations do not embed in Euclidean space, but require a uniform background curvature. We study a class of regular foams on S^3 which, through conformal, stereographic projection to R^3 define irregular cells consistent with Plateau's laws. We analytically characterize a broad classes of bulk foam bubbles, and extend and explain recent observations on foam structure and shape distribution. Our approach also allows us to comment on foam stability by identifying a weak local maximum of A^(3/2)/V at the maximally symmetric tetrahedral bubble that participates in T2 rearrangements.Comment: 4 pages, 4 included figures, RevTe

Fitting Voronoi Diagrams to Planar Tesselations

Given a tesselation of the plane, defined by a planar straight-line graph $G$, we want to find a minimal set $S$ of points in the plane, such that the Voronoi diagram associated with $S$ "fits" \ $G$. This is the Generalized Inverse Voronoi Problem (GIVP), defined in \cite{Trin07} and rediscovered recently in \cite{Baner12}. Here we give an algorithm that solves this problem with a number of points that is linear in the size of $G$, assuming that the smallest angle in $G$ is constant.Comment: 14 pages, 8 figures, 1 table. Presented at IWOCA 2013 (Int. Workshop on Combinatorial Algorithms), Rouen, France, July 201

Adaptive Binning of X-ray data with Weighted Voronoi Tesselations

We present a technique to adaptively bin sparse X-ray data using weighted Voronoi tesselations (WVTs). WVT binning is a generalisation of Cappellari & Copin's (2001) Voronoi binning algorithm, developed for integral field spectroscopy. WVT binning is applicable to many types of data and creates unbiased binning structures with compact bins that do not lead the eye. We apply the algorithm to simulated data, as well as several X-ray data sets, to create adaptively binned intensity images, hardness ratio maps and temperature maps with constant signal-to-noise ratio per bin. We also illustrate the separation of diffuse gas emission from contributions of unresolved point sources in elliptical galaxies. We compare the performance of WVT binning with other adaptive binning and adaptive smoothing techniques. We find that the CIAO tool csmooth creates serious artefacts and advise against its use to interpret diffuse X-ray emission.Comment: 14 pages; submitted to MNRAS; code freely available at http://www.phy.ohiou.edu/~diehl/WVT/index.html with user manual, examples and high-resolution version of this pape

Magnetic flux pinning in superconductors with hyperbolic-tesselation arrays of pinning sites

We study magnetic flux interacting with arrays of pinning sites (APS) placed on vertices of hyperbolic tesselations (HT). We show that, due to the gradient in the density of pinning sites, HT APS are capable of trapping vortices for a broad range of applied magnetic fluxes. Thus, the penetration of magnetic field in HT APS is essentially different from the usual scenario predicted by the Bean model. We demonstrate that, due to the enhanced asymmetry of the surface barrier for vortex entry and exit, this HT APS could be used as a "capacitor" to store magnetic flux.Comment: 7 pages, 5 figure

Hard Discs on the Hyperbolic Plane

We examine a simple hard disc fluid with no long range interactions on the two dimensional space of constant negative Gaussian curvature, the hyperbolic plane. This geometry provides a natural mechanism by which global crystalline order is frustrated, allowing us to construct a tractable model of disordered monodisperse hard discs. We extend free area theory and the virial expansion to this regime, deriving the equation of state for the system, and compare its predictions with simulation near an isostatic packing in the curved space.Comment: 4 pages, 3 figures, included, final versio

A presentation for the baseleaf preserving mapping class group of the punctured solenoid

We give a presentation for the baseleaf preserving mapping class group $Mod(\S)$ of the punctured solenoid $\S$. The generators for our presentation were introduced previously, and several relations among them were derived. In addition, we show that $Mod(\S)$ has no non-trivial central elements. Our main tool is a new complex of triangulations of the disk upon which $Mod(\S)$ acts.Comment: 14 pages, 1 figur

Simple equation of state for hard disks on the hyperbolic plane

A simple equation of state for hard disks on the hyperbolic plane is proposed. It yields the exact second virial coefficient and contains a pole at the highest possible packing. A comparison with another very recent theoretical proposal and simulation data is presented.Comment: 3 pages, 1 figur