Kepler conjecture | EPFL Graph Search

Are you an EPFL student looking for a semester project?

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements. The density of these arrangements is around 74.05%. In 1998, Thomas Hales, following an approach suggested by , announced that he had a proof of the Kepler conjecture. Hales' proof is a proof by exhaustion involving the checking of many individual cases using complex computer calculations. Referees said that they were "99% certain" of the correctness of Hales' proof, and the Kepler conjecture was accepted as a theorem. In 2014, the Flyspeck project team, headed by Hales, announced the completion of a formal proof of the Kepler conjecture using a combination of the Isabelle and HOL Light proof assistants. In 2017, the formal proof was accepted by the journal Forum of Mathematics, Pi. Imagine filling a large container with small equal-sized spheres: Say a porcelain gallon jug with identical marbles. The "density" of the arrangement is equal to the total volume of all the marbles, divided by the volume of the jug. To maximize the number of marbles in the jug means to create an arrangement of marbles stacked between the sides and bottom of the jug, that has the highest possible density, so that the marbles are packed together as closely as possible. Experiment shows that dropping the marbles in randomly, with no effort to arrange them tightly, will achieve a density of around 65%. However, a higher density can be achieved by carefully arranging the marbles as follows: For the first layer of marbles, arrange them in a hexagonal lattice (the honeycomb pattern) Put the next layer of marbles in the lowest lying gaps you can find above and between the marbles in the first layer, regardless of pattern Continue with the same procedure of filling in the lowest gaps in the prior layer, for the third and remaining layers, until the marbles reach the top edge of the jug.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (28)

Related people (3)

Related units (2)

Related concepts (6)

Related lectures (13)

, ,

We prove that the Cohn-Elkies linear programming bound for sphere packing is not sharp in dimension 6. The proof uses duality and optimization over a space of modular forms, generalizing a construction of Cohn- Triantafillou [Math. Comp. 91 (2021), pp. 491 ...

Amer Mathematical Soc2024

, , ,

Why are materials with specific characteristics more abundant than others? This is a fundamental question in materials science and one that is traditionally difficult to tackle, given the vastness of compositional and configurational space. We highlight he ...

Nature Portfolio2024

, ,

arXiv2022

Crystal Packing of Conjugated Molecules MSE-486: Organic electronic materials

Explores the crystal packing of conjugated molecules, discussing shapes, structures, and packing arrangements in the solid state.

Crystal Packing of Conjugated Molecules MSE-486: Organic electronic materials

Explores the shapes and crystal structures of conjugated molecules, discussing packing arrangements, polymorphism, and the impact of substituents on crystal packing.

Supramolecular Organisation: Solid State MSE-486: Organic electronic materials

Explores the supramolecular organization of π-conjugated molecules in the solid state, covering crystal structures and packing preferences.

Sphere packing

In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, spaces of other dimensions (where the problem becomes circle packing in two dimensions, or hypersphere packing in higher dimensions) or to non-Euclidean spaces such as hyperbolic space.

Packing problems

Packing problems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers. The goal is to either pack a single container as densely as possible or pack all objects using as few containers as possible. Many of these problems can be related to real-life packaging, storage and transportation issues. Each packing problem has a dual covering problem, which asks how many of the same objects are required to completely cover every region of the container, where objects are allowed to overlap.

László Fejes Tóth

László Fejes Tóth (Fejes Tóth László, ˈfɛjɛʃ ˈtoːt ˈlaːsloː 12 March 1915 – 17 March 2005) was a Hungarian mathematician who specialized in geometry. He proved that a lattice pattern is the most efficient way to pack centrally symmetric convex sets on the Euclidean plane (a generalization of Thue's theorem, a 2-dimensional analog of the Kepler conjecture). He also investigated the sphere packing problem. He was the first to show, in 1953, that proof of the Kepler conjecture can be reduced to a finite case analysis and, later, that the problem might be solved using a computer.