The Kepler conjecture

Abstract

Kepler’s conjecture asserts that the highest possible density an arrangement of congruent balls can have is the one of the face-centered cubic packing. That is, the pyramid arrangement of balls on a square base, or on a triangular base, like oranges are usually arranged at fruit stands. In this project, we study the proof of this problem presented by Thomas Hales in 1998. It will be obvious that in some parts (specially in the end) we do not go into detail when we study the properties of the elements that take place in the proof. The reason is that the notation gets very cumbersome as we go along and the study of these details will not give us a better understanding of the proof. They are necessary steps to prove the conjecture, but our aim is to understand the proof as a whole and to see what strategy Thomas Hales followed. It is also important to note that a big part of the proof relies in computer calculations. All the programs and algorithms can be found online on the documentation of the Flyspeck project. It took years to finish and verify this part of the proof (the project was finally completed on August 2014) and we will not study this part of the proof.