Hilbert's eighteenth problem
Encyclopedia
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
The first part of the problem asks whether there are only finitely many essentially different space group
s in
-dimensional Euclidean space
. This was answered affirmatively by Bieberbach
.
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.
or packing of other specified shapes. Although it expressly includes shapes other than spheres, it is generally taken as equivalent to the Kepler conjecture
.
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
. 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 groupSpace group
In mathematics and geometry, a space group is a symmetry group, usually for three dimensions, that divides space into discrete repeatable domains.In three dimensions, there are 219 unique types, or counted as 230 if chiral copies are considered distinct...
s in
-dimensional Euclidean spaceEuclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
. This was answered affirmatively by Bieberbach
Ludwig Bieberbach
Ludwig Georg Elias Moses Bieberbach was a German mathematician.-Biography:Born in Goddelau, near Darmstadt, he studied at Heidelberg and under Felix Klein at Göttingen, receiving his doctorate in 1910. His dissertation was titled On the theory of automorphic functions...
.
Anisohedral tiling in 3 dimensions
The second part of the problem asks whether there exists a polyhedronPolyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
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
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
) tiling. Such tiles are now known as anisohedral
Anisohedral tiling
In geometry, a shape is said to be anisohedral if it admits a tiling, but no such tiling is isohedral ; that is, in any tiling by that shape there are two tiles that are not equivalent under any symmetry of the tiling...
. 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
Heinrich Heesch
Heinrich Heesch was a German mathematician. He was born in Kiel and died in Hanover.In Göttingen he worked on Group theory. In 1933 Heesch witnessed the National Socialist purges among the university staff...
in 1935.
Sphere packing
The third part of the problem asks for the densest sphere packingSphere 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...
or packing of other specified shapes. Although it expressly includes shapes other than spheres, it is generally taken as equivalent to the Kepler conjecture
Kepler conjecture
The Kepler conjecture, named after the 17th-century German astronomer Johannes Kepler, is a mathematical conjecture about sphere packing in three-dimensional Euclidean space. It says that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic...
.