Hilbert's eighteenth problem

It asks three separate questions about lattices and sphere packing in Euclidean space.

[1] The first part of the problem asks whether there are only finitely many essentially different space groups in

[2] The related einstein problem asks for a shape that can tile space but not with an infinite cyclic group of symmetries.

Although it expressly includes shapes other than spheres, it is generally taken as equivalent to the Kepler conjecture.

In 1998, American mathematician Thomas Callister Hales gave a computer-aided proof of the Kepler conjecture.