Hilbert's eighteenth problem

Hilbert's eighteenth problem is one of the 23

Hilbert problems set out in a celebrated list compiled in 1900 by mathematician David Hilbert. It asks three separate questions about lattices and sphere packing in Euclidean space.^[1]

Symmetry groups in $n$ dimensions

The first part of the problem asks whether there are only finitely many essentially different space groups in $n$ -dimensional Euclidean space. This was answered affirmatively by Bieberbach.

Anisohedral tiling in 3 dimensions

The second part of the problem asks whether there exists a

transitive) tiling. Such tiles are now known as anisohedral

. In asking the problem in three dimensions, Hilbert was probably assuming that no such tile exists in two dimensions; this assumption later turned out to be incorrect.

The first such tile in three dimensions was found by

infinite cyclic group

of symmetries.

Sphere packing

The third part of the problem asks for the densest sphere packing or packing of other specified shapes. Although it expressly includes shapes other than spheres, it is generally taken as equivalent to the Kepler conjecture.

In 1998, American mathematician

computer-aided proof of the Kepler conjecture. It shows that the most space-efficient way to pack spheres is in a pyramid shape.^[3]