Related papers: On the Packing Density Bound in 3-Space
This is the sixth in a series of papers giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is the…
We construct a dense packing of regular tetrahedra, with packing density $D > >.7786157$.
This is the second in a series of papers giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is the…
Define the superball with radius $r$ and center ${\boldsymbol 0}$ in $\mathbb{R}^n$ to be the set $$ \left\{{\boldsymbol x}\in\mathbb{R}^n:\sum_{j=1}^{m}\left(x_{k_j+1}^2+x_{k_j+2}^2+\cdots+x_{k_{j+1}}^2\right)^{p/2}\leq…
In hyperbolic space density cannot be defined by a limit as we define it in Euclidean space. We describe the local density bounds for sphere packings and we discuss the different attempts to define optimal arrangements in hyperbolic space.
We improve the best known upper bound on the density of a planar measurable set A containing no two points at unit distance to 0.25442. We use a combination of Fourier analytic and linear programming methods to obtain the result. The…
We present a new lower bound on the Bowen-Radin maximal density of radius-R ball packings in the m-dimensional hyperbolic space, improving on the basic covering bound by factor \Omega(m(R+\ln m)) as m tends to infinity. This is done by…
We prove that the highest density of non-overlapping translates of a given centrally symmetric convex domain relative to its outer parallel domain of given outer radius is attained by a lattice packing in the Euclidean plane. This…
In this article we obtain linear programming bounds for the maximal sphere packing density of commutative spaces. A special case of our results solves a conjecture by Cohn and Zhao on linear programming bounds for sphere packings in…
This is the eighth and final paper in a series giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is…
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…
We study the problem of high-dimensional multiple packing in Euclidean space. Multiple packing is a natural generalization of sphere packing and is defined as follows. Let $ N>0 $ and $ L\in\mathbb{Z}_{\ge2} $. A multiple packing is a set…
We prove a lower bound on the entropy of sphere packings of $\mathbb R^d$ of density $\Theta(d \cdot 2^{-d})$. The entropy measures how plentiful such packings are, and our result is significantly stronger than the trivial lower bound that…
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,…
Based on Minkowski's work on critical lattices of 3-dimensional convex bodies we present an efficient algorithm for computing the density of a densest lattice packing of an arbitrary 3-polytope. As an application we calculate densest…
We consider circle packings in the plane with circles of sizes $1$, $r\simeq 0.834$ and $s\simeq 0.651$. These sizes are algebraic numbers which allow a compact packing, that is, a packing in which each hole is formed by three mutually…
Packings of equal disks in the plane are known to have density at most $\pi/\sqrt{12}$, although this density is never achieved in the square torus, which is what we call the plane modulo the square lattice. We find packings of disks in a…
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…
In \cite{Sz17-2} we considered hyperball packings in $3$-dimensional hyperbolic space. We developed a decomposition algorithm that for each saturated hyperball packing provides a decomposition of $\HYP$ into truncated tetrahedra. In order…
In this paper we study the problem of hyperball (hypersphere) packings in $3$-dimensional hyperbolic space. We introduce a new definition of the non-compact saturated ball packings and describe to each saturated hyperball packing, a new…