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'''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.{{sfn|Milnor|1976}}

==Symmetry groups in <math>n</math> dimensions==
The first part of the problem asks whether there are only finitely many essentially different [[space group]]s in <math>n</math>-dimensional [[Euclidean space]].  This was answered affirmatively by [[Ludwig Bieberbach|Bieberbach]].

==Anisohedral tiling in 3 dimensions==
The second part of the problem asks whether there exists a [[polyhedron]] which [[tessellation of space|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-[[group action|transitive]]) tiling.  Such tiles are now known as [[anisohedral tiling|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 (mathematician)|Karl Reinhardt]] in 1928.  The first example in two dimensions was found by [[Heinrich Heesch|Heesch]] in 1935.{{sfn|Edwards|2003}}

==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]].

American mathematician [[Thomas Callister Hales]] has given a [[computer-aided proof]] of the Kepler conjecture. It shows that the most space-efficient way to pack spheres is in a pyramid shape.{{sfn|Hales|2005}}

==References==
{{reflist}}

*{{citation|first=Steve|last=Edwards|title=Heesch's Tiling|year=2003|url=http://web.archive.org/web/20110718054857/http://www.spsu.edu/math/tiling/17.html}}

*{{citation|last=Hales|first=Thomas C.|title=A proof of the Kepler conjecture|journal=Annals of Mathematics|year=2005|volume=162|issue=3|pages=1065–1185|doi=10.4007/annals.2005.162.1065|url=http://annals.math.princeton.edu/wp-content/uploads/annals-v162-n3-p01.pdf}}

*{{citation | last=Milnor|first=J.|chapter=Hilbert's problem 18|editor-last=Browder|editor-first= Felix E.  | title=Mathematical developments arising from Hilbert problems |series=Proceedings of symposia in pure mathematics|volume= 28 | publisher=[[American Mathematical Society]] | year=1976 | isbn=0-8218-1428-1}}

{{Hilbert's problems}}
[[Category:Hilbert's problems|#18]]
[[Category:Tessellation]]