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| {{Uniform polyteron db|Uniform polyteron stat table|hix}}
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| In [[Five-dimensional space|five-dimensional]] [[geometry]], a 5-[[simplex]] is a self-dual [[Regular polytope|regular]] [[5-polytope]]. It has 6 [[vertex (geometry)|vertices]], 15 [[Edge (geometry)|edge]]s, 20 triangle [[Face (geometry)|faces]], 15 tetrahedral [[Cell (mathematics)|cells]], and 6 [[pentachoron]] [[Facet (geometry)|facets]]. It has a [[dihedral angle]] of cos<sup>−1</sup>(1/5), or approximately 78.46°.
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| == Alternate names ==
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| It can also be called a '''hexateron''', or '''hexa-5-tope''', as a 6-[[facet (geometry)|facetted]] polytope in 5-dimensions. The [[5-polytope#A note on generality of terms for n-polytopes and elements|name]] ''hexateron'' is derived from ''hexa-'' for having six [[Facet (mathematics)|facets]] and ''[[polyteron|teron]]'' (with ''ter-'' being a corruption of ''[[tetra-]]'') for having four-dimensional facets.
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| By Jonathan Bowers, a hexateron is given the acronym '''hix'''.<ref>Klitzing, (x3o3o3o3o - hix)</ref>
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| == Regular hexateron cartesian coordinates ==
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| The ''hexateron'' can be constructed from a [[5-cell]] by adding a 6th vertex such that it is equidistant from all the other vertices of the 5-cell.
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| The [[Cartesian coordinates]] for the vertices of an origin-centered regular hexateron having edge length 2 are:
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| :<math>\left(\sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ \sqrt{1/3},\ \pm1\right)</math>
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| :<math>\left(\sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ -2\sqrt{1/3},\ 0\right)</math>
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| :<math>\left(\sqrt{1/15},\ \sqrt{1/10},\ -\sqrt{3/2},\ 0,\ 0\right)</math>
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| :<math>\left(\sqrt{1/15},\ -2\sqrt{2/5},\ 0,\ 0,\ 0\right)</math>
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| :<math>\left(-\sqrt{5/3},\ 0,\ 0,\ 0,\ 0\right)</math>
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| The vertices of the ''5-simplex'' can be more simply positioned on a [[hyperplane]] in 6-space as permutations of (0,0,0,0,0,1) ''or'' (0,1,1,1,1,1). These construction can be seen as facets of the [[hexacross]] or [[rectified 6-cube]] respectively.
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| == Projected images ==
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| {{5-simplex Coxeter plane graphs|t0|100}}
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| {| class=wikitable width=320
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| |[[Image:hexateron.png|320px]]<BR>[[Stereographic projection]] 4D to 3D of [[Schlegel diagram]] 5D to 4D of hexateron.
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| |}
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| == Related uniform 5-polytopes ==
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| It is first in a dimensional series of uniform polytopes and honeycombs, expressed by [[Coxeter]] as 1<sub>3k</sub> series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral [[dihedron]].
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| {{1 3k polytopes}}
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| It is first in a dimensional series of uniform polytopes and honeycombs, expressed by [[Coxeter]] as 3<sub>k1</sub> series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral [[hosohedron]].
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| {{3_k1_polytopes}}
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| The regular 5-simplex is one of 19 [[Uniform_polyteron#The_A5_.5B3.2C3.2C3.2C3.5D_family_.285-simplex.29|uniform polytera]] based on the [3,3,3,3] [[Coxeter group]], all shown here in A<sub>5</sub> [[Coxeter plane]] [[orthographic projection]]s. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)
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| {{Hexateron family}}
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| == Other forms ==
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| The '''hexateron''' can also be considered a [[Polychoral pyramid|pentachoral pyramid]], constructed as a [[pentachoron]] base in a 4-space [[hyperplane]], and an [[Apex (geometry)|apex]] point ''above'' the hyperplane. The five ''sides'' of the pyramid are made of pentachoral cells.
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| == Notes ==
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| {{reflist}}
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| == References ==
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| * [[Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics, Macmillan, 1900
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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|polytera.htm|5D uniform polytopes (polytera)|x3o3o3o3o - hix}}
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| == External links ==
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| *{{GlossaryForHyperspace | anchor=Simplex | title=Simplex }}
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| * [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions], Jonathan Bowers
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| * [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]
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| {{Polytopes}}
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| [[Category:5-polytopes]]
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The title of the author is Nestor. One of the issues I love most is greeting card gathering but I don't have the time recently. Managing people is how I make cash and it's something I really enjoy. He currently life in Idaho and his mothers and fathers reside nearby.
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