Sphere packings, I

@article{Hales1998SpherePI,
  title={Sphere packings, I},
  author={Thomas C. Hales},
  journal={Discrete \& Computational Geometry},
  year={1998},
  volume={17},
  pages={1-51}
}
  • T. Hales
  • Published 11 November 1998
  • Physics, Mathematics
  • Discrete & Computational Geometry
We describe a program to prove the Kepler conjecture on sphere packings. We then carry out the first step of this program. Each packing determines a decomposition of space into Delaunay simplices, which are grouped together into finite configurations called Delaunay stars. A score, which is related to the density of packings, is assigned to each Delaunay star. We conjecture that the score of every Delaunay star is at most the score of the stars in the face-centered cubic and hexagonal close… 

Sphere Packings, II

  • T. Hales
  • Mathematics
    Discret. Comput. Geom.
  • 1997
The second step of a program to prove the Kepler conjecture on sphere packings leads to a decomposition of R3 into polyhedra, which has density at most that of a regular tetrahedron.

Sphere Packings, VI. Tame Graphs and Linear Programs

  • T. Hales
  • Mathematics
    Discret. Comput. Geom.
  • 2006
It is proved that each such graph must be isomorphic to a tame graph, of which there are only finitely many up to isomorphism.

Sphere Packing, IV. Detailed Bounds

  • T. Hales
  • Mathematics
    Discret. Comput. Geom.
  • 2006
The results form the technical heart of the proof of the Kepler conjecture, by giving detailed bounds on the function f, which can be expressed as a sum of terms, indexed by regions on a unit sphere.

Sphere Packings, V. Pentahedral Prisms

This paper proves that decomposition stars associated with the plane graph of arrangements the authors term pentahedral prisms do not contravene, using interval arithmetic methods to prove particular linear relations on components of any such contravening decomposition star.

Dense Sphere Packings: A Blueprint for Formal Proofs

The 400-year-old Kepler conjecture asserts that no packing of congruent balls in three dimensions can have a density exceeding the familiar pyramid-shaped cannonball arrangement. In this book, a new

Sphere Packing, III. Extremal Cases

  • T. Hales
  • Mathematics
    Discret. Comput. Geom.
  • 2006
This paper shows that certain points in the domain were conjectured to give the global maxima and are indeed local maxima, and various approximations to f are developed, that will be used in subsequent papers to bound the value of the function f.

Historical Overview of the Kepler Conjecture

  • T. Hales
  • Physics
    Discret. Comput. Geom.
  • 2006
AbstractThis paper is the first in a series of six papers devoted to the proof of the Kepler conjecture, which asserts that no packing of congruent balls in three dimensions has density greater than

On Self-Similar Finitely Generated Sphere Packings

This paper is first a brief survey on links between Geometry of Numbers and aperiodic crystals in Physics, viewed from the mathematical side. In a second part, we prove the existence of a canonical

A Formulation of the Kepler Conjecture

AbstractThis paper is the second in a series of six papers devoted to the proof of the Kepler conjecture, which asserts that no packing of congruent balls in three dimensions has density greater than

Periodic Planar Disk Packings

Several conditions are given when a packing of equal disks in a torus is locally maximally dense, where the torus is defined as the quotient of the plane by a two-dimensional lattice. Conjectures are
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Sphere Packings, II

  • T. Hales
  • Mathematics
    Discret. Comput. Geom.
  • 1997
The second step of a program to prove the Kepler conjecture on sphere packings leads to a decomposition of R3 into polyhedra, which has density at most that of a regular tetrahedron.

Sphere packings III

This is the fifth in a series of papers giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than

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Using the Delaunay decomposition, a local notion of density for sphere packings in ℝ3 is defined and the face-centered-cubic and hexagonal-close-packings provide local maxima (in a strong sense defined below) to the function which associates to every saturated sphere packing in ™3 its density.

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