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.

Five Dimensional Non-Lattice Sphere Packings

1967 ◽  
Vol 10 (3) ◽  
pp. 387-393 ◽  
Author(s):  
John Leech
Keyword(s):  
Sphere Packing ◽  
Lattice Packing ◽  
Sphere Packings ◽  
Euclidean Spaces ◽  
Lattice Packings

The densest lattice packings of spheres in Euclidean spaces En of n dimensions are known for n ≤ 8 (for full n — references see [6]). However, it i s not known for any n ≥ 3 whether there can be any non-lattice sphere packing with density exceeding that of the corresponding densest lattice packing.

Some Sphere Packings in Higher Space

1964 ◽  
Vol 16 ◽  
pp. 657-682 ◽  
Author(s):  
John Leech
Keyword(s):  
Two Dimensions ◽  
Lattice Packing ◽  
Sphere Packings ◽  
Euclidean Spaces ◽  
Lattice Packings

This paper is concerned with the packing of equal spheres in Euclidean spaces [n] of n > 8 dimensions. To be precise, a packing is a distribution of spheres any two of which have at most a point of contact in common. If the centres of the spheres form a lattice, the packing is said to be a lattice packing. The densest lattice packings are known for spaces of up to eight dimensions (1, 2), but not for any space of more than eight dimensions. Further, although non-lattice packings are known in [3] and [5] which have the same density as the densest lattice packings, none is known which has greater density than the densest lattice packings in any space of up to eight dimensions, neither, for any space of more than two dimensions, has it been shown that they do not exist.

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.

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.

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.

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.

Sphere Packings and Error-Correcting Codes

1971 ◽  
Vol 23 (4) ◽  
pp. 718-745 ◽  
Author(s):  
John Leech ◽  
N. J. A. Sloane
Keyword(s):  
Error Correcting Codes ◽  
Sphere Packings ◽  
Euclidean Spaces ◽  
Image Position ◽  
Systematic Derivation

Error-correcting codes are used in several constructions for packings of equal spheres in n-dimensional Euclidean spaces En. 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 2m for m ≧ 6. Most of the new packings are nonlattice packings. These new packings increase the previously greatest known numbers of spheres which one sphere may touch, and, except in dimensions 9, 12, 14, 15, they include denser packings than any previously known. The density Δ of the packings in En for n = 2m satisfiesIn this paper we make systematic use of error-correcting codes to obtain sphere packings in En, including several of the densest packings known and several new packings.

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.

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