The Kepler conjecture

TL;DR

This paper completes a proof of the Kepler conjecture, establishing that no sphere packing in three dimensions exceeds the face-centered cubic packing density of approximately 0.74048, resolving a longstanding problem in discrete geometry.

Contribution

It provides the final step in a comprehensive proof confirming the maximum density of sphere packings in three dimensions.

Findings

Proves the Kepler conjecture in full.

Confirms face-centered cubic packing as optimal.

Completes the proof program outlined in prior work.

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 $\pi/\sqrt{18}\approx 0.74048...$. 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 $8 \pt$.