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Hilbert's eighteenth problem is one of the 23
Hilbert problems Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the pro ...
set out in a celebrated list compiled in 1900 by mathematician
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
. It asks three separate questions about lattices and sphere packing in Euclidean space.


Symmetry groups in n 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 unch ...
s in n-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 s ...
. 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 th ...
which
tiles Tiles are usually thin, square or rectangular coverings manufactured from hard-wearing material such as ceramic, stone, metal, baked clay, or even glass. They are generally fixed in place in an array to cover roofs, floors, walls, edges, or ...
3-dimensional Euclidean space but is not the
fundamental region Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...
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 In plane geometry, the einstein problem asks about the existence of a single prototile that by itself forms an aperiodic set of prototiles, that is, a shape that can tessellate space, but only in a nonperiodic way. Such a shape is called an "eins ...
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 binar ...
of symmetries.


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 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