Hilbert's eighteenth problem
From Wikipedia, the free encyclopedia
Hilbert's eighteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It asks three separate questions.
Contents |
[edit] 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.
[edit] Anisohedral tiling in 3 dimensions
The second part of the problem asks whether there exists a polyhedron which tiles 3-dimensional Euclidean space but is not the fundamental region of any space group; that is, which tiles but does not admit an isohedral (tile-transitive) tiling. Such tiles are now known as anisohedral. In asking the problem in three dimensions, Hilbert had presumed that no such tile exists in the plane.
The first such tile in three dimensions was found by Reinhardt in 1928. The first example in two dimensions was found by Heesch in 1935.
[edit] 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.
[edit] References
- Browder, Felix E., ed. (1976). Mathematical developments arising from Hilbert problems (Proceedings of symposia in pure mathematics, volume 28). American Mathematical Society. ISBN 0-8218-1428-1.
| Hilbert's problems |
|---|
| I | II | III | IV | V | VI | VII | VIII | IX | X | XI | XII | XIII | XIV | XV| XVI | XVII | XVIII | XIX | XX | XXI | XXII | XXIII |
 |
This geometry-related article is a stub. You can help Wikipedia by expanding it. |
