# Sphere packings, I

@article{Hales1998SpherePI, title={Sphere packings, I}, author={Thomas C. Hales}, journal={Discrete \& Computational Geometry}, year={1998}, volume={17}, pages={1-51} }

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…

## 1,073 Citations

### Sphere Packings, II

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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.

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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.

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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.

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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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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.

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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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Abstract Since Woodburne (1969) analyzed the three diprotodontid specimens then known from the Mio-Pliocene Beaumaris locality in Victoria, Australia, three more specimens of that group have been…