Hilbert's eighteenth problem
- Hilbert's eighteenth problem
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.
Symmetry groups in dimensions
The first part of the problem asks whether there are only finitely many essentially different space groups in -dimensional Euclidean space. This was answered affirmatively by Bieberbach.
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 Karl Reinhardt in 1928. The first example in two dimensions was found by Heesch in 1935.
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.
References
*cite book | author=Browder, Felix E., ed. | title=Mathematical developments arising from Hilbert problems (Proceedings of symposia in pure mathematics, volume 28) | publisher=American Mathematical Society | year=1976 | id=ISBN 0-8218-1428-1
Wikimedia Foundation.
2010.
Look at other dictionaries:
List of mathematics articles (H) — NOTOC H H cobordism H derivative H index H infinity methods in control theory H relation H space H theorem H tree Haag s theorem Haagerup property Haaland equation Haar measure Haar wavelet Haboush s theorem Hackenbush Hadamard code Hadamard… … Wikipedia
Liste de théorèmes — par ordre alphabétique. Pour l établissement de l ordre alphabétique, il a été convenu ce qui suit : Si le nom du théorème comprend des noms de mathématiciens ou de physiciens, on se base sur le premier nom propre cité. Si le nom du théorème … Wikipédia en Français
Anisohedral tiling — In geometry, a shape is said to be anisohedral if it admits a tiling, but no such tiling is isohedral (tile transitive); that is, in any tiling by that shape there are two tiles that are not equivalent under any symmetry of the tiling. A tiling… … Wikipedia
Kepler conjecture — In mathematics, the Kepler conjecture is a 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 close packing … Wikipedia
Aperiodic tiling — are an aperiodic set of tiles, since they admit only non periodic tilings of the plane:] Any of the infinitely many tilings by the Penrose tiles is non periodic. More informally, many refer to the Penrose tilings as being aperiodic tilings , but… … Wikipedia
Number theory — A Lehmer sieve an analog computer once used for finding primes and solving simple diophantine equations. Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers (the… … Wikipedia
Logic and the philosophy of mathematics in the nineteenth century — John Stillwell INTRODUCTION In its history of over two thousand years, mathematics has seldom been disturbed by philosophical disputes. Ever since Plato, who is said to have put the slogan ‘Let no one who is not a geometer enter here’ over the… … History of philosophy
Immanuel Kant — Kant redirects here. For other uses, see Kant (disambiguation). See also: Kant (surname) Immanuel Kant Immanuel Kant Full name Immanuel Kant Born 22 April 1724 … Wikipedia
Glossary of philosophical isms — This is a list of topics relating to philosophy that end in ism . compactTOC NOTOC A * Absolutism – the position that in a particular domain of thought, all statements in that domain are either absolutely true or absolutely false: none is true… … Wikipedia
