A Proof of the Dodecahedral Conjecture

The dodecahedral conjecture states that the volume of the Voronoi polyhedron of a sphere in a packing of equal spheres is at least the volume of a regular dodecahedron with inradius 1. The authors prove the conjecture following the methodology of the proof the Kepler conjecture. (See math.MG/9811071.)

[1]  L. Fejes.  Über die dichteste Kugellagerung , 1942 .

[2]  John Leech,et al.  The Problem of the Thirteen Spheres , 1956, The Mathematical Gazette.

[3]  C. A. Rogers.  The Packing of Equal Spheres , 1958 .

[4]  George E. Collins,et al.  Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975 .

[5]  John F. Hart,et al.  Computer Approximations , 1978 .

[6]  G. Alefeld,et al.  Introduction to Interval Computation , 1983 .

[7]  Douglas J. Muder,et al.  Putting the best face on a Voronoi polyhedron , 1988 .

[8]  David Goldberg,et al.  What every computer scientist should know about floating-point arithmetic , 1991, CSUR.

[9]  T. Hales.  The sphere packing problem , 1992 .

[10]  Douglas J. Muder,et al.  A new bound on the local density of sphere packings , 1993, Discret. Comput. Geom..

[11]  W. Hsiang.  ON THE SPHERE PACKING PROBLEM AND THE PROOF OF KEPLER'S CONJECTURE , 1993 .

[12]  Jian L. Zhou,et al.  User's Guide for CFSQP Version 2.0: A C Code for Solving (Large Scale) Constrained Nonlinear (Minimax) Optimization Problems, Generating Iterates Satisfying All Inequality Constraints , 1994 .

[13]  Lawrence Charles Paulson,et al.  Isabelle: A Generic Theorem Prover , 1994 .

[14]  T. Hales.  The status of the kepler conjecture , 1994 .

[15]  R. B. Kearfott.  Rigorous Global Search: Continuous Problems , 1996 .

[16]  Thomas C. Hales.  Sphere packings, I , 1997, Discret. Comput. Geom..

[17]  K. Bezdek,et al.  Isoperimetric Inequalities and the Dodecahedral Conjecture , 1997 .

[18]  T. Hales.  The Kepler conjecture , 1998, math/9811078.

[19]  Thomas C. Hales.  Some Algorithms Arising in the Proof of the Kepler Conjecture , 2002 .

[20]  William H. Press,et al.  Numerical recipes in C , 2002 .

[21]  Michael Pilato.  Version Control with Subversion , 2004 .

[22]  Georges Gonthier.  A computer-checked proof of the Four Colour Theorem , 2005 .

[23]  Gertrud Josefine Bauer,et al.  Formalizing plane graph theory: towards a formalized proof of the Kepler conjecture , 2006 .

[24]  Roland Zumkeller.  Formal Global Optimisation with Taylor Models , 2006, IJCAR.

[25]  Tobias Nipkow,et al.  Flyspeck I: Tame Graphs , 2006, IJCAR.

[26]  Jorge Nocedal,et al.  Knitro: An Integrated Package for Nonlinear Optimization , 2006 .

[27]  Tobias Nipkow,et al.  Flyspeck II: the basic linear programs , 2009, Annals of Mathematics and Artificial Intelligence.