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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}}
| | The name of the writer is Numbers but it's not the most masucline name out there. To gather coins is what his family members and him enjoy. My day job is a librarian. California is our birth place.<br><br>Here is my web page ... std testing at home ([http://www.pinaydiaries.com/user/LConsiden Learn Additional Here]) |
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| ==Symmetry groups in <math>n</math> dimensions==
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| 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]].
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| ==Anisohedral tiling in 3 dimensions==
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| 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. | |
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| 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}}
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| ==Sphere packing==
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| 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]].
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| 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}}
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| ==References==
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| {{reflist}}
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| *{{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}}
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| *{{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}}
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| *{{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}}
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| {{Hilbert's problems}}
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| [[Category:Hilbert's problems|#18]]
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| [[Category:Tessellation]]
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The name of the writer is Numbers but it's not the most masucline name out there. To gather coins is what his family members and him enjoy. My day job is a librarian. California is our birth place.
Here is my web page ... std testing at home (Learn Additional Here)