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| {{Uniform polypeton db|Uniform polypeton stat table|hop}}
| | The title of the writer is Garland. Bookkeeping is how he supports his family members and his salary has been really fulfilling. Kansas is where her home is but she requirements to transfer simply because of her family. To perform badminton is something he truly enjoys doing.<br><br>Also visit my webpage :: auto warranty ([http://Newdayz.de/index.php?mod=users&action=view&id=16038 company website]) |
| In [[geometry]], a 6-[[simplex]] is a [[Duality (mathematics)|self-dual]] [[Regular polytope|regular]] [[6-polytope]]. It has 7 [[vertex (geometry)|vertices]], 21 [[Edge (geometry)|edge]]s, 35 triangle [[Face (geometry)|faces]], 35 [[Tetrahedron|tetrahedral]] [[Cell (mathematics)|cells]], 21 [[5-cell]] 4-faces, and 7 [[5-simplex]] 5-faces. Its [[dihedral angle]] is cos<sup>−1</sup>(1/6), or approximately 80.41°.
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| == Alternate names ==
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| It can also be called a '''heptapeton''', or '''hepta-6-tope''', as a 7-[[facet (geometry)|facetted]] polytope in 6-dimensions. The [[5-polytope#A note on generality of terms for n-polytopes and elements|name]] ''heptapeton'' is derived from ''hepta'' for seven [[Facet (mathematics)|facets]] in [[Greek language|Greek]] and [[Peta-|''-peta'']] for having five-dimensional facets, and ''-on''. Jonathan Bowers gives a heptapeton the acronym '''hop'''.<ref>Klitzing, (x3o3o3o3o3o - hop)</ref>
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| == Coordinates ==
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| The [[Cartesian coordinate]]s for an origin-centered regular heptapeton having edge length 2 are:
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| :<math>\left(\sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ \sqrt{1/3},\ \pm1\right)</math>
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| :<math>\left(\sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ -2\sqrt{1/3},\ 0\right)</math>
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| :<math>\left(\sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ -\sqrt{3/2},\ 0,\ 0\right)</math>
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| :<math>\left(\sqrt{1/21},\ \sqrt{1/15},\ -2\sqrt{2/5},\ 0,\ 0,\ 0\right)</math>
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| :<math>\left(\sqrt{1/21},\ -\sqrt{5/3},\ 0,\ 0,\ 0,\ 0\right)</math>
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| :<math>\left(-\sqrt{12/7},\ 0,\ 0,\ 0,\ 0,\ 0\right)</math>
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| The vertices of the ''6-simplex'' can be more simply positioned in 7-space as permutations of:
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| : (0,0,0,0,0,0,1)
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| This construction is based on [[Facet (geometry)|facets]] of the [[7-orthoplex]].
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| == Images ==
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| {{6-simplex Coxeter plane graphs|t0|150}}
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| == Related uniform 6-polytopes ==
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| The regular 6-simplex is one of 35 [[Uniform 6-polytope#The A6 .5B3.2C3.2C3.2C3.2C3.5D family .286-simplex.29|uniform 6-polytopes]] based on the [3,3,3,3,3] [[Coxeter group]], all shown here in A<sub>6</sub> [[Coxeter plane]] [[orthographic projection]]s.
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| {{Heptapeton family}}
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| ==Notes ==
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| {{reflist}}
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| == References==
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| * [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]:
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| ** Coxeter, ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
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| ** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
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| ** '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
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| *** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10]
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| *** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591]
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| *** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
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| * [[John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1<sub>n1</sub>)
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| * [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
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| ** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966)
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| *{{KlitzingPolytopes|polypeta.htm|6D uniform polytopes (polypeta)|x3o3o3o3o - hix}}
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| == External links == | |
| *{{GlossaryForHyperspace | anchor=Simplex | title=Simplex }}
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| * [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions]
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| * [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]
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| {{Polytopes}}
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| [[Category:6-polytopes]]
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