Sphere Packings
نویسنده
چکیده
This paper is a continuation of the first two parts of this series ([I],[II]). It relies on the formulation of the Kepler conjecture in [F]. The terminology and notation of this paper are consistent with these earlier papers, and we refer to results from them by prefixing the relevant section numbers with I, II, or F. Around each vertex is a modification of the Voronoi cell, called the V -cell and a collection of quarters and quasi-regular tetrahedra. These objects constitute the decomposition star at the vertex. A decomposition star may be decomposed into standard clusters. By definition, a standard cluster is the part of the given decomposition star that lies over a given standard region on the unit sphere.
منابع مشابه
Density Doubling, Double-circulants, and New Sphere Packings
New nonlattice sphere packings in dimensions 20, 22, and 44–47 that are denser than the best previously known sphere packings were recently discovered. We extend these results, showing that the density of many sphere packings in dimensions just below a power of 2 can be doubled using orthogonal binary codes. This produces new dense sphere packings in Rn for n = 25, 26, . . . , 31 and 55, 56, . ...
متن کاملPhase diagram and structural diversity of the densest binary sphere packings.
The densest binary sphere packings have historically been very difficult to determine. The only rigorously known packings in the α-x plane of sphere radius ratio α and relative concentration x are at the Kepler limit α=1, where packings are monodisperse. Utilizing an implementation of the Torquato-Jiao sphere-packing algorithm [S. Torquato and Y. Jiao, Phys. Rev. E 82, 061302 (2010)], we presen...
متن کاملSphere Packings and Error-correcting Codes
Error-correcting codes are used in several constructions for packings of equal spheres in ^-dimensional Euclidean spaces E. These include a systematic derivation of many of the best sphere packings known, and construction of new packings in dimensions 9-15, 36, 40, 48, 60, and 2 for m g 6. Most of the new packings are nonlattice packings. These new packings increase the previously greatest know...
متن کاملCharacterization of maximally random jammed sphere packings. II. Correlation functions and density fluctuations.
In the first paper of this series, we introduced Voronoi correlation functions to characterize the structure of maximally random jammed (MRJ) sphere packings across length scales. In the present paper, we determine a variety of different correlation functions that arise in rigorous expressions for the effective physical properties of MRJ sphere packings and compare them to the corresponding sta...
متن کاملDeriving Finite Sphere Packings
Sphere packing problems have a rich history in both mathematics and physics; yet, relatively few analytical analyses of sphere packings exist, and answers to seemingly simple questions are unknown. Here, we present an analytical method for deriving all packings of n spheres in R3 satisfying minimal rigidity constraints (≥ 3 contacts per sphere and ≥ 3n− 6 total contacts). We derive such packing...
متن کاملDensest local sphere-packing diversity. II. Application to three dimensions.
The densest local packings of N three-dimensional identical nonoverlapping spheres within a radius R(min)(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. B. Hopkins, F. H. Stillinger, and S. Torquato, Phys. Rev. E 81, 041305 (2010)], we described our method for finding the putative densest packings of N spher...
متن کامل