Related papers: New dense superball packings in three dimensions
Unraveling the complexities of random packing in three dimensions has long puzzled physicists. While both experiments and simulations consistently show a maximum density of 64 percent for tightly packed random spheres, we still lack an…
A recent paper on the large-scale structure of the Universe presented evidence for a rectangular three-dimensional lattice of galaxy superclusters and voids, with lattice spacing ~120 Mpc and called for some ``hitherto unknown process'' to…
Motivated by the search for best lattice sphere packings in Euclidean spaces of large dimensions we study randomly generated perfect lattices in moderately large dimensions (up to d=19 included). Perfect lattices are relevant in the…
In this paper we consider the ball and horoball packings belonging to $3$-dimensional Coxeter tilings that are derived by simply truncated orthoschemes with parallel faces. The goal of this paper to determine the optimal ball and horoball…
We determine the optimal horoball packing densities for Koszul-type Coxeter simplex tilings in hyperbolic $3$-space. Using a parametrization of horoballs by the Busemann function and the symmetry of the tilings, we obtain families of…
3 families of 4-dimensional lattices $L_k, M_k, M_k / 2 \subset \mathbb{R}^2$ are defined. Each lattice is defined by 2 quadratic extensions and has a \emph{finite} number of unit vectors, but the number of unit vectors in each of the 3…
We prove that the Leech lattice is the unique densest lattice in R^24. The proof combines human reasoning with computer verification of the properties of certain explicit polynomials. We furthermore prove that no sphere packing in R^24 can…
It is shown how in 3+3 dimensions, it is possible to have a superparticle Lagrangian that has manifest supersymmetry both on the world line and in the target space.
We call a periodic ball packing in d-dimensional Euclidean space periodically (strictly) jammed with respect to a period lattice if there are no nontrivial motions of the balls that preserve the period (that maintain some period with…
We show that in any $d$-dimensional real normed space, unit balls can be packed with density at least \[\frac{(1-o(1))d\log d}{2^{d+1}},\] improving a result of Schmidt from 1958 by a logarithmic factor and generalizing the recent result of…
We consider three-dimensional clusters of identical bubbles packed around a central bubble and calculate their energy and optimal shape. We obtain the surface area and bubble pressures to improve on existing growth laws for…
In this paper using the concept of the extended Hamming code we give a construction for dense packing of points at distance at least one in such unit cubes which dimension are a power of two.
We prove explicit stability estimates for the sphere packing problem in dimensions 8 and 24, showing that, in the lattice case, if a lattice is $\sim \varepsilon$ close to satisfying the optimal density, then it is, in a suitable sense,…
We develop an analogue for sphere packing of the linear programming bounds for error-correcting codes, and use it to prove upper bounds for the density of sphere packings, which are the best bounds known at least for dimensions 4 through…
In this paper, we study the problem of hyperball (hypersphere) packings in $n$-dimensional hyperbolic space ($n \ge 4$). We prove that to each $n$-dimensional congruent saturated hyperball packing, there is an algorithm to obtain a…
Superballs represent a class of particles whose shapes are defined by ${|x|}^{2p}+{|y|}^{2p}+{|z|}^{2p} \le R^{2p}$, with $p\in(0,\infty)$ being the "deformation parameter". $0<p<0.5$ represents a family of hexapodlike (concave…
Let $\Delta$ be the optimal packing density of $\mathbb R^n$ by unit balls. We show the optimal packing density using two sizes of balls approaches $\Delta + (1 - \Delta) \Delta$ as the ratio of the radii tends to infinity. More generally,…
In many areas of research it is interesting how lattices can be filled with particles that have no nearest neighbors, or they are in limited quantities. Examples may be found in statistical physics, chemistry, materials science, discrete…
The lattice size of a lattice polytope is a geometric invariant which was formally introduced in the context of simplification of the defining equation of an algebraic curve, but appeared implicitly earlier in geometric combinatorics.…
Dense random packings of hard particles are useful models of granular media and are closely related to the structure of nonequilibrium low-temperature amorphous phases of matter. Most work has been done for random jammed packings of…