Related papers: Packing Squares in a Torus
Let $S$ be a set of $n$ points in the unit square $[0,1]^2$, one of which is the origin. We construct $n$ pairwise interior-disjoint axis-aligned empty rectangles such that the lower left corner of each rectangle is a point in $S$, and the…
This work investigates dense packings of congruent hard infinitesimally--thin circular arcs in the two-dimensional Euclidean space. It focuses on those denotable as major whose subtended angle $\theta \in \left ( \pi, 2\pi \right ]$.…
Dense packings have served as useful models of the structure of liquid, glassy and crystal states of matter, granular media, heterogeneous materials, and biological systems. Probing the symmetries and other mathematical properties of the…
We construct stable configurations of n overlapping discs of radius r in a unit square, with r = O(1/n). By a result of Diaconis, Lebeau, and Michel, this result is best possible, up to a constant factor. A consequence is that the…
Given a finite family of squares in the plane, the packing problem asks for the maximum number $\nu$ of pairwise disjoint squares among them, while the hitting problem for the minimum number $\tau$ of points hitting all of them. Clearly,…
We study the problem of discrete geometric packing. Here, given weighted regions (say in the plane) and points (with capacities), one has to pick a maximum weight subset of the regions such that no point is covered more than its capacity.…
The packing of hard spheres (HS) of diameter $\sigma$ in a cylinder has been used to model experimental systems, such as fullerenes in nanotubes and colloidal wire assembly. Finding the densest packings of HS under this type of confinement,…
Dense polyhedron packings are useful models of a variety of condensed matter and biological systems and have intrigued scientists mathematicians for centuries. Recently, organizing principles for the types of structures associated with the…
We provide a tight result for a fundamental problem arising from packing squares into a circular container: The critical density of packing squares into a disk is $\delta=\frac{8}{5\pi}\approx 0.509$. This implies that any set of (not…
We consider the problem of packing congruent circles with the maximum radius in a unit square as a mathematical optimization problem. Due to the presence of non-overlapping constraints, this problem is a notoriously difficult nonconvex…
We construct the densest known two-dimensional packings of superdisks in the plane whose shapes are defined by |x^(2p) + y^(2p)| <= 1, which contains both convex-shaped particles (p > 0.5, with the circular-disk case p = 1) and…
Dense packings of hard particles have important applications in many fields, including condensed matter physics, discrete geometry and cell biology. In this paper, we employ a stochastic search implementation of the Torquato-Jiao…
The densest local packings of N three-dimensional identical nonoverlapping spheres within a radius Rmin(N) of a fixed central sphere of the same size are obtained for selected values of N up to N = 1054. In the predecessor to this paper…
A toy model of particles packings is presented, which consists in arranging hexagons on a triangular lattice according to local stability rules. The number of stable packings is analytically computed and found to grow exponentially with the…
We study the Hard Core Model on the graphs ${\rm {\bf \scriptstyle G}}$ obtained from Archimedean tilings i.e. configurations in $\scriptstyle \{0,1\}^{{\rm {\bf G}}}$ with the nearest neighbor 1's forbidden. Our particular aim in choosing…
In this paper we formulate the problem of packing unequal rectangles/squares into a fixed size circular container as a mixed-integer nonlinear program. Here we pack rectangles so as to maximise some objective (e.g. maximise the number of…
An ellipsoid, the simplest non-spherical shape, has been extensively used as models for elongated building blocks for a wide spectrum of molecular, colloidal and granular systems. Yet the densest packing of congruent hard ellipsoids, which…
We present our exhaustive exploration of the densest ternary sphere packings (DTSPs) for 45 radius ratios and 237 kinds of compositions, which is a packing problem of three kinds of hard spheres with different radii, under periodic boundary…
The present work surveys problems in $n$-dimensional space with $n$ large. Early development in the study of packing and covering in high dimensions was motivated by the geometry of numbers. Subsequent results, such as the discovery of the…
We consider sequential random packing of cubes $z+[0,1]^n$ with $z\in \frac{1}{N}\ZZ^n$ into the cube $[0,2]^n$ and the torus $\QuotS{\RR^n}{2\ZZ^n}$ as $N\to\infty$. In the cube case $[0,2]^n$ as $N\to\infty$ the random cube packings thus…