by
Thomas C. Hales

Venue: | Discrete Comput. Geom |

Citations: | 23 - 7 self |

@ARTICLE{Hales96spherepackings,

author = {Thomas C. Hales},

title = {Sphere Packings I},

journal = {Discrete Comput. Geom},

year = {1996},

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 packings. This conjecture implies the Kepler conjecture. To complete the first step of the program, we show that every Delaunay star that satisfies a certain regularity condition satisfies the conjecture. Contents: 1. Introduction, 2. The Program, 3. Quasi-regular Tetrahedra, 4. Quadrilaterals, 5. Restrictions, 6. Combinatorics, 7. The Method of Subdivision, 8. Explicit Formulas for Compression, Volume, and Angle, 9. Floating-Point Calculations. Appendix. D. J. Muder's Proof of Theorem 6.1. Sec...

114 | A Proof of the Kepler Conjecture - Hales |

43 | Lagerungen in der Ebene, auf der Kugel und im Raum - TÂ¶oth - 1972 |

36 |
The packing of equal spheres
- Rogers
- 1958
(Show Context)
Citation Context ..., since the structure of the simplicial decomposition of space will be our primary concern. For a proof that the Delaunay decomposition is a dissection of space into simplices, we refer the reader to =-=[R]-=-. The Delaunay decomposition is dual to the well-known Voronoi decomposition. If the vertices of the Delaunay simplices are in nondegenerate position, two vertices are joined by an edge exactly when t... |

16 |
The sphere packing problem
- Hales
- 1992
(Show Context)
Citation Context ...The Kepler conjecture asserts that no packing of spheres in three dimensions has density exceeding that of the face-centered cubic lattice packing. This density is = p 18s0:74048. In an earlier paper =-=[H2]-=-, we showed how to reduce the Kepler conjecture to a finite calculation. That paper also gave numerical evidence in support of the method and conjecture. This finite calculation is a series of optimiz... |

7 | Sphere packings II
- Hales
- 1997
(Show Context)
Citation Context ...hedron determines a triangle in the standard 6 decomposition (Lemma 3.7). We may identify quasi-regular tetrahedra with clusters over triangular regions. We assign a score to each standard cluster in =-=[H4,3]-=-. In this section we define the score of a quasi-regular tetrahedron and describe the properties that the score should have in general. Let S be a quasi-regular tetrahedron. It is a standard cluster i... |

2 |
Remarks on the Density of Sphere
- Hales
- 1993
(Show Context)
Citation Context ...e result is contained in Theorem 1 below. Background to another approach to this problem is found in [H3]. To add more detail to the proposed program, we recall some constructions from earlier papers =-=[H1]-=-, [H2]. Begin with a packing of nonoverlapping spheres of radius 1 in Euclidean three-space. The density of a packing is defined in [H1]. It is defined as a limit of the ratio of the volume of the uni... |

2 |
Computer Approximations
- al
- 1968
(Show Context)
Citation Context ...0. To reduce the calculations to rational expressions r(x 0 ), rational approximations to the functions p x, arctan(x), and arccos(x) with explicit error bounds are required. These were obtained from =-=[CA]-=-. Reliable approximations to various constants (such as , p 2, and ffi oct ) with explicit error bounds are also required. These were obtained in Mathematica and were double checked against Maple. Let... |

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