Isoperimetric Inequalities and the Dodecahedral Conjecture

1997 ◽  
Vol 08 (06) ◽  
pp. 759-780 ◽  
Author(s):  
Károly Bezdek
Keyword(s):  
Unit Sphere ◽  
Sphere Packing ◽  
Isoperimetric Inequalities ◽  
Sphere Packings ◽  
Voronoi Polyhedron ◽  
Regular Dodecahedron ◽  
Nonoverlapping Unit

The dodecahedrad conjecture, posed more than 50 years ago, says that the volume of any Voronoi polyhedron of a unit sphere packing in [Formula: see text] is at least as large as the volume of a regular dodecahedron of inradius 1. In this paper we show how the dodecahedral conjecture can be obtained from the distance conjecture of 14 and 15 nonoverlapping unit spheres and from the isoperimetric conjecture of Voronoi faces of unit sphere packings.

scholarly journals Geometry and arithmetic of crystallographic sphere packings

2018 ◽  
Vol 116 (2) ◽  
pp. 436-441 ◽  
Author(s):  
Alex Kontorovich ◽  
Kei Nakamura
Keyword(s):  
Opposite Direction ◽  
Sphere Packing ◽  
Infinite Family ◽  
Reflection Group ◽  
Limit Set ◽  
Higher Dimension ◽  
Sphere Packings ◽  
Geometrically Finite

We introduce the notion of a “crystallographic sphere packing,” defined to be one whose limit set is that of a geometrically finite hyperbolic reflection group in one higher dimension. We exhibit an infinite family of conformally inequivalent crystallographic packings with all radii being reciprocals of integers. We then prove a result in the opposite direction: the “superintegral” ones exist only in finitely many “commensurability classes,” all in, at most, 20 dimensions.

scholarly journals Upper bounds for packings of spheres of several radii

2014 ◽  
Vol 2 ◽  
Author(s):  
DAVID DE LAAT ◽  
FERNANDO MÁRIO DE OLIVEIRA FILHO ◽  
FRANK VALLENTIN
Keyword(s):  
Linear Programming ◽  
Euclidean Space ◽  
Semidefinite Programming ◽  
Unit Sphere ◽  
Upper Bound ◽  
Convex Bodies ◽  
Classical Problem ◽  
Upper Bounds ◽  
Sphere Packings ◽  
Spherical Caps

AbstractWe give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds for packings of spherical caps and of convex bodies through the use of semidefinite programming. We perform explicit computations, obtaining new bounds for packings of spherical caps of two different sizes and for binary sphere packings. We also slightly improve the bounds for the classical problem of packing identical spheres.

A transition path from the zinc-blende to the NaCl type

2000 ◽  
Vol 215 (6) ◽  
Author(s):  
Heidrun Sowa
Keyword(s):  
Sphere Packing ◽  
Sphere Packings ◽  
Transition Path ◽  
Homogeneous Sphere ◽  
Zinc Blende ◽  
Wyckoff Position ◽  
Silicon And Germanium ◽  
Binary Compounds ◽  
Diamond Configuration ◽  
Diamond Type

In order to find a transition path from the zinc-blende to the NaCl type both structures are described with the aid of heterogeneous sphere packings. If all atoms in such crystal structures are replaced by like ones, atomic arrangements result that correspond to homogeneous sphere packings belonging to the diamond type or forming a cubic primitive lattice, respectively.It is shown, that a diamond configuration may be deformed into a cubic primitive lattice within the Wyckoff position Imma 4(e) mm2 0,¼,z. The corresponding phase transition in binary compounds from the zinc-blende to the NaCl type can be described as a deformation of a heterogeneous sphere packing in the subgroup Imm2 of Imma. Since no bonds have to be broken this type of transition is displacive.In addition, structural relations between high-pressure phases of semiconductors like silicon and germanium and related AB compounds are shown.

Monoclinic sphere packings. III. Trivariant lattice complexes of P2/c and P21/c

2019 ◽  
Vol 75 (2) ◽  
pp. 325-335
Author(s):  
Heidrun Sowa
Keyword(s):  
Sphere Packing ◽  
The Other ◽  
Monoclinic Space Group ◽  
Sphere Packings ◽  
Homogeneous Sphere ◽  
Space Group

All homogeneous sphere packings were derived that refer to the trivariant lattice complexes of monoclinic space-group types P2/c and P21/c. In total, sphere packings of 55 types have been found. The maximal inherent symmetry is monoclinic for 17 types while the other types comprise at least one sphere packing with cubic (four cases), hexagonal (six cases), tetragonal (eight cases) or orthorhombic (20 cases) symmetry.

scholarly journals Uniqueness of Certain Spherical Codes

1981 ◽  
Vol 33 (2) ◽  
pp. 437-449 ◽  
Author(s):  
Eiichi Bannai ◽  
N. J. A. Sloane
Keyword(s):  
Unit Sphere ◽  
Spherical Codes ◽  
Nonoverlapping Unit

In this paper we show that there is essentially only one way of arranging 240 (resp. 196560) nonoverlapping unit spheres in R8 (resp. R24) so that they all touch another unit sphere, and only one way of arranging 56 (resp. 4600) spheres in R8 (resp. R24) so that they all touch two further, touching spheres. The following tight spherical t-designs are unique: the 5-design in Ω7, the 7-designs in Ω8 and Ω23, and the 11-design in Ω24. It was shown in [20] that the maximum number of nonoverlapping unit spheres in R8 (resp. R24) that can touch another unit sphere is 240 (resp. 196560). Arrangements of spheres meeting these bounds can be obtained from the E8 and Leech lattices, respectively. The present paper shows that these are the only arrangements meeting these bounds.

Tetragonal sphere packings

1993 ◽  
Vol 205 (1) ◽  
Author(s):  
Werner Fischer
Keyword(s):  
Degrees Of Freedom ◽  
Sphere Packing ◽  
Sphere Packings ◽  
Tetragonal Lattice ◽  
Three Degrees Of Freedom

AbstractFor tetragonal lattice complexes with three degrees of freedom the sphere-packing conditions, the generation classes and the (topological) types of sphere packings are tabulated. The use of the table is illustrated by means of two structural examples.

Six and Seven Dimensional Non-Lattice Sphere Packings

1969 ◽  
Vol 12 (2) ◽  
pp. 151-155 ◽  
Author(s):  
John Leech
Keyword(s):  
Sphere Packing ◽  
Lattice Packing ◽  
Sphere Packings ◽  
Euclidean Spaces ◽  
Lattice Packings

The densest lattice packings of equal spheres in Euclidean spaces En of n dimensions are known for n ⩽ 8. However, it is not known for any n ⩾ 3 whether there can be any non-lattice sphere packing with density exceeding that of the densest lattice packing. W. Barlow described [1] a non-lattice packing in E3 with the same density as the densest lattice packing, and I described [6] three non-lattice packings in E5 which also have this property.

Orthorhombic sphere packings. V. Trivariant lattice complexes of space groups belonging to crystal class 222

2014 ◽  
Vol 70 (6) ◽  
pp. 591-604 ◽  
Author(s):  
Heidrun Sowa
Keyword(s):  
Sphere Packing ◽  
Tetragonal Symmetry ◽  
Orthorhombic Symmetry ◽  
Sphere Packings ◽  
Homogeneous Sphere ◽  
Space Group ◽  
Space Groups ◽  
Characteristic Space ◽  
Different Types ◽  
Crystal Class

This paper completes the derivation of all types of homogeneous sphere packing with orthorhombic symmetry. The nine orthorhombic trivariant lattice complexes belonging to the space groups of crystal class 222 were examined in regard to the existence of homogeneous sphere packings and of interpenetrating sets of layers of spheres. Altogether, sphere packings of 84 different types have been found; the maximal inherent symmetry is orthorhombic for only 36 of these types. In addition, interpenetrating sets of 63nets occur once. All lattice complexes with orthorhombic characteristic space group give rise to 260 different types of sphere packing in total. The maximal inherent symmetry is orthorhombic for 160 of these types. Sphere packings of 13 types can also be generated with cubic, those of seven types with hexagonal and those of 80 types with tetragonal symmetry. In addition, ten types of interpenetrating sphere packing and two types of sets of interpenetrating sphere layers are obtained. Most of the sphere packings can be subdivided into layer-like subunits perpendicular to one of the orthorhombic main axes.

Sign in / Sign up

Export Citation Format

Share Document