Geometry and arithmetic of crystallographic sphere packings

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.

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La conjecture de Manin pour une famille de variétés en dimension supérieure

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500411800004x ◽

2018 ◽

Vol 166 (3) ◽

pp. 433-486 ◽

Cited By ~ 1

Author(s):

KEVIN DESTAGNOL

Keyword(s):

Infinite Family ◽

Strong Form ◽

Higher Dimension ◽

Crepant Resolution ◽

Projective Varieties ◽

Formula Group ◽

Image Position ◽

Manin’S Conjecture

AbstractInspired by a method of La Bretèche relying on some unique factorisation, we generalise work of Blomer, Brüdern and Salberger to prove Manin's conjecture in its strong form conjectured by Peyre for some infinite family of varieties of higher dimension. The varieties under consideration in this paper correspond to the singular projective varieties defined by the following equation$$ x_1 y_2y_3\cdots y_n+x_2y_1y_3 \cdots y_n+ \cdots+x_n y_1 y_2 \cdots y_{n-1}=0 $$in ℙℚ2n−1for alln⩾ 3. This paper comes with an Appendix by Per Salberger constructing a crepant resolution of the above varieties.

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FOX-ARTIN ARC AND CONSTRUCTING KLEINIAN GROUPS ACTING ON S3 WITH LIMIT SET A WILD CANTOR SET

International Journal of Mathematics ◽

10.1142/s0129167x95000031 ◽

1995 ◽

Vol 06 (01) ◽

pp. 19-32 ◽

Cited By ~ 2

Author(s):

NIKOLAY GUSEVSKII ◽

HELEN KLIMENKO

Keyword(s):

Cantor Set ◽

Cantor Sets ◽

Kleinian Groups ◽

Limit Set ◽

Limit Sets ◽

Geometrically Finite ◽

Wild Cantor Set ◽

Wild Cantor Sets

We construct purely loxodromic, geometrically finite, free Kleinian groups acting on S3 whose limit sets are wild Cantor sets. Our construction is closely related to the construction of the wild Fox–Artin arc.

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

Zeitschrift für Kristallographie - Crystalline Materials ◽

10.1524/zkri.2000.215.6.335 ◽

2000 ◽

Vol 215 (6) ◽

Cited By ~ 17

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.

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

Acta Crystallographica Section A Foundations and Advances ◽

10.1107/s2053273318015814 ◽

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.

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Tetragonal sphere packings

Zeitschrift für Kristallographie - Crystalline Materials ◽

10.1524/zkri.1993.205.part-1.9 ◽

1993 ◽

Vol 205 (1) ◽

Cited By ~ 6

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.

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Isoperimetric Inequalities and the Dodecahedral Conjecture

International Journal of Mathematics ◽

10.1142/s0129167x9700038x ◽

1997 ◽

Vol 08 (06) ◽

pp. 759-780 ◽

Cited By ~ 3

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.

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Six and Seven Dimensional Non-Lattice Sphere Packings

Canadian Mathematical Bulletin ◽

10.4153/cmb-1969-014-1 ◽

1969 ◽

Vol 12 (2) ◽

pp. 151-155 ◽

Cited By ~ 11

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.

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

Acta Crystallographica Section A Foundations and Advances ◽

10.1107/s2053273314014193 ◽

2014 ◽

Vol 70 (6) ◽

pp. 591-604 ◽

Cited By ~ 1

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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Five Dimensional Non-Lattice Sphere Packings

Canadian Mathematical Bulletin ◽

10.4153/cmb-1967-037-5 ◽

1967 ◽

Vol 10 (3) ◽

pp. 387-393 ◽

Cited By ~ 7

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.

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ON THE HARD SPHERE MODEL AND SPHERE PACKINGS IN HIGH DIMENSIONS

Forum of Mathematics Sigma ◽

10.1017/fms.2018.25 ◽

2019 ◽

Vol 7 ◽

Cited By ~ 3

Author(s):

MATTHEW JENSSEN ◽

FELIX JOOS ◽

WILL PERKINS

Keyword(s):

Lower Bound ◽

Packing Density ◽

Hard Sphere ◽

Statistical Physics ◽

Sphere Packing ◽

Hard Sphere Model ◽

Sphere Packings ◽

Sphere Model ◽

Single Sphere ◽

Mere Existence

We prove a lower bound on the entropy of sphere packings of $\mathbb{R}^{d}$ of density $\unicode[STIX]{x1D6E9}(d\cdot 2^{-d})$. The entropy measures how plentiful such packings are, and our result is significantly stronger than the trivial lower bound that can be obtained from the mere existence of a dense packing. Our method also provides a new, statistical-physics-based proof of the $\unicode[STIX]{x1D6FA}(d\cdot 2^{-d})$ lower bound on the maximum sphere packing density by showing that the expected packing density of a random configuration from the hard sphere model is at least $(1+o_{d}(1))\log (2/\sqrt{3})d\cdot 2^{-d}$ when the ratio of the fugacity parameter to the volume covered by a single sphere is at least $3^{-d/2}$. Such a bound on the sphere packing density was first achieved by Rogers, with subsequent improvements to the leading constant by Davenport and Rogers, Ball, Vance, and Venkatesh.

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