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.
Symmetry groups in dimensions
The first part of the problem asks whether there are only finitely many essentially different
space group
In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it ...
s in
-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
. This was answered affirmatively by
Bieberbach.
Anisohedral tiling in 3 dimensions
The second part of the problem asks whether there exists a
polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on ...
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 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
Karl Reinhardt in 1928. The first example in two dimensions was found by
Heesch in 1935. The related
einstein problem asks for a shape that can tile space but not with an
infinite cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
of symmetries.
Sphere packing
The third part of the problem asks for the densest
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 pack ...
or packing of other specified shapes. Although it expressly includes shapes other than spheres, it is generally taken as equivalent to the
Kepler conjecture
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 ...
.
In 1998, American mathematician
Thomas Callister Hales
Thomas Callister Hales (born June 4, 1958) is an American mathematician working in the areas of representation theory, discrete geometry, and formal verification. In representation theory he is known for his work on the Langlands program and the p ...
gave a
computer-aided proof
A computer-assisted proof is a mathematical proof that has been at least partially generated by computer.
Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of a mathematical theorem. The idea is to use a ...
of the Kepler conjecture. It shows that the most space-efficient way to pack spheres is in a pyramid shape.
References
*
*
*
{{Hilbert's problems
#18
Tessellation