Related papers: Gap Distributions in Circle Packings
We have studied the packing of congruent disks on a spherical cap, for caps of different size and number of disks, $N$. This problem has been considered before only in the limit cases of circle packing inside a circle and on a sphere…
It has been recently pointed out that local volume fluctuations in granular packings follow remarkably well a shifted and rescaled Gamma distribution named the kGamma distribution [T. Aste, T. Di Matteo, Phys. Rev. E 77 (2008) 021309]. In…
Traditional clustering identifies groups of objects that share certain qualities. Tangles do the converse: they identify groups of qualities that often occur together. They can thereby identify and discover 'types': of behaviour, views,…
We review a recently proposed theory of random packings. We describe the volume fluctuations in jammed matter through a volume function, amenable to analytical and numerical calculations. We combine an extended statistical mechanics…
We investigate equal spheres packings generated from several experiments and from a large number of different numerical simulations. The structural organization of these disordered packings is studied in terms of the network of common…
In this article, we attempt to study the possible link between the dynamics of a circle map and the caustics of its iterations. The attention is on a geometrically defined off-center reflections, which, coincidentally, is also a…
This paper studies the asymptotic behaviors of the pairwise angles among n randomly and uniformly distributed unit vectors in R^p as the number of points n -> infinity, while the dimension p is either fixed or growing with n. For both…
The paper reports a generalized expression for filling the congruent circles (of radius r) in a circle (of radius R). First, a generalized expression for the biggest circle (r) inscribed in the nth part of the bigger circle (R) was…
We provide a general framework for constructing probability distributions on Riemannian manifolds, taking advantage of area-preserving maps and isometries. Control over distributions' properties, such as parameters, symmetry and modality…
We present an approach for the study and design of reflectors with rotational or translational symmetry that redirect light from a point source into any desired radiant intensity distribution. This method is based on a simple conformal map…
A typical computational geometry problem begins: Consider a set P of n points in R^d. However, many applications today work with input that is not precisely known, for example when the data is sensed and has some known error model. What if…
Apollonian gaskets are formed by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We experimentally study the pair correlation, electrostatic energy, and nearest neighbor spacing of…
The subalgebra of the tautological ring of the moduli of curves of compact type generated by the kappa classes is studied. Relations, constructed via the virtual geometry of the moduli of stable maps, are used to prove universality results…
We consider circles on a translation surface $X$, consisting of points joined to a common center point by a geodesic of length $R$. We show that as $R \to \infty$ these circles distribute to a measure on $X$ which is equivalent to the area.…
A better understanding of the formation of mass structures in the universe can be obtained by determining the amount and distribution of dark and luminous matter in spiral galaxies. To investigate such matters a sample of 12 galaxies, most…
It is shown here how prior estimates on the local shape of the universe can be used to reduce, to a small region, the full parameter space for the search of circles in the sky. This is the first step towards the development of efficient…
In this paper, we introduce discrete approximate circle bundles, a class of objects designed to serve as the data science analog of circle bundles from algebraic topology. We show that, under appropriate conditions, one can meaningfully and…
In this work we propose a heuristic algorithm for the layout optimization for disks installed in a rotating circular container. This is a unequal circle packing problem with additional balance constraints. It proved to be an NP-hard…
In this paper, we consider the problem of manifold approximation with affine subspaces. Our objective is to discover a set of low dimensional affine subspaces that represents manifold data accurately while preserving the manifold's…
We propose a new approach to the value distribution theory of entire holomorphic curves. We define a ``packing density'' of an entire holomorphic curve, and show that it has various non-trivial properties. We prove a ``gap theorem'' for…