The Kepler conjecture

Abstract

This is the eighth and final paper in a series 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 $\pi /\sqrt{18}\approx \mathrm{0.74048...}$. This is the oldest problem in discrete geometry and is an important part of Hilbert's 18th problem. An example of a packing achieving this density is the face-centered cubic packing. This paper completes the fourth step of the program outlined in math.MG/9811073: A proof that if some standard region has more than four sides, then the star scores less than $8 \pt$.

62 pages. Eighth and last in a series beginning with math.MG/9811071. Includes an index of terminology and notation for the entire series. The source package of the first version includes the source code of the software used in the proof. The second version has been completely rewritten

References in corpus (1)

• An overview of the Kepler conjecture

Cited by in corpus (24)

• New upper bounds on sphere packings I
• New upper bounds on sphere packings II
• Origin of Scaling Behavior of Protein Packing Density: A Sequential Monte Carlo Study of Compact Long Chain Polymers
• Formal Verification of Nonlinear Inequalities with Taylor Interval Approximations
• Diffusive growth of polydisperse hard-sphere crystals
• A proof of the dodecahedral conjecture
• Jamming III: Characterizing Randomness via the Entropy of Jammed Matter
• A computer verification of the Kepler conjecture
• New upper bounds for the density of translative packings of three-dimensional convex bodies with tetrahedral symmetry
• Jamming I: A volume function for jammed matter
• The geometric stability of Voronoi diagrams with respect to small changes of the sites
• Equality of the Jellium and Uniform Electron Gas next-order asymptotic terms for Coulomb and Riesz potentials
• Developments in Formal Proofs
• Bounds for Multiple Packing and List-Decoding Error Exponents
• Sphere Packings in 3 Dimensions
• A Proof of Fejes Toth's Conjecture on Sphere Packings with Kissing Number Twelve
• Scientists in silico?
• Densest packings of translates of strings and layers of balls
• Upper bound on the disordered density of sphere packing and the Kepler Conjecture
• Multiplicity of Generation, Selection, and Classification Procedures for Jammed Hard-Particle Packings
• Geodesic ball packings generated by regular prism tilings in $\mathbf{Nil}$ geometry
• Visualization of sphere and horosphere packings related to Coxeter tilings generated by simply truncated orthoschemes with parallel faces
• Double Bubbles Minimize
• New Conjectural Lower Bounds on the Optimal Density of Sphere Packings