Related papers: Hyperball packings in hyperbolic $3$-space
A hidden truncation hyperbolic (HTH) distribution is introduced and finite mixtures thereof are applied for clustering. A stochastic representation of the HTH distribution is given and a density is derived. A hierarchical representation is…
The density of a code is the fraction of the coding space covered by packing balls centered around the codewords. This paper investigates the density of codes in the complex Stiefel and Grassmann manifolds equipped with the chordal…
We give upper bounds for the density of unit ball packings relative to their outer parallel domains and discuss their connection to contact numbers. Also, packings of soft balls are introduced and upper bounds are given for the fraction of…
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…
An earlier paper describes a program to prove the Kepler conjecture on sphere packings. This paper carries out the second step of that program. A sphere packing leads to a decomposition of $R^3$ into polyhedra. The polyhedra are divided…
Questions surrounding the spatial disposition of particles in various condensed-matter systems continue to pose many theoretical challenges. This paper explores the geometric availability of amorphous many-particle configurations that…
When a thin sheet is crushed into a small three-dimensional volume, it invariably forms a structure with a low volume fraction but high resistance to further compression. Being a far-from-equilibrium process, forced crumpling is not…
The aim of the study presented here was the analysis of packings generated according to random sequential adsorption protocol consisting of identical Platonic and Archimedean solids. The computer simulations performed showed, that the…
We improve the previously best known upper bounds on the sizes of $\theta$-spherical codes for every $\theta<\theta^*\approx 62.997^{\circ}$ at least by a factor of $0.4325$, in sufficiently high dimensions. Furthermore, for sphere packing…
In this paper we get an explicit lower bound for the radius of a Bergman ball contained in the Dirichlet fundamental polyhedron of a torsion-free discrete group $G\subset PU(n,1)$ acting on complex hyperbolic space. Consequently the volume…
A polyhedron in a three-dimensional hyperbolic space is said to be generalized if finite, ideal and truncated vertices are admitted. In virtue of Belletti's theorem (2021) the exact upper bound for volumes of generalized hyperbolic…
Our main result is that for all sufficiently large $x_0>0$, the set of commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with fixed invariant trace field $k$ and systole bounded below by $x_0$ has density one within the…
Let M be a symplectic-toric manifold of dimension at least four. This paper investigates the so called symplectic ball packing problem in the toral equivariant setting. We show that the set of toric symplectic ball packings of M admits the…
Advancements in the synthesis of faceted nanoparticles and colloids have spurred interest in the phase behavior of polyhedral shapes. Regular tetrahedra have attracted particular attention because they prefer local symmetries that are…
All hard, convex shapes are conjectured by Ulam to pack more densely than spheres, which have a maximum packing fraction of {\phi} = {\pi}/\sqrt18 ~ 0.7405. For many shapes, simple lattice packings easily surpass this packing fraction. For…
We obtain an asymptotic formula for the number of circles of curvature at most T in any given bounded Apollonian circle packing. For an integral packing, we obtain the upper bounds for the number of circles with prime curvature as well as…
We extend to the context of hyperbolic 3-manifolds with geodesic boundary Thurston's approach to hyperbolization by means of geometric triangulations. In particular, we introduce moduli for (partially) truncated hyperbolic tetrahedra, and…
This paper gives an exposition of the authors' harmonic deformation theory for 3-dimensional hyperbolic cone-manifolds. We discuss topological applications to hyperbolic Dehn surgery as well as recent applications to Kleinian group theory.…
The problem of finding the asymptotic behavior of the maximal density of sphere packings in high Euclidean dimensions is one of the most fascinating and challenging problems in discrete geometry. One century ago, Minkowski obtained a…
In this paper, we study the rigidity of hyperbolic polyhedral 3-manifolds and the volume optimization program of angle structures. We first study the rigidity of decorated 1-3 type hyperbolic polyhedral metrics on 3-manifolds which are…