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| | The person who wrote the article is called Jayson Hirano and he totally digs that title. Office supervising is where her main earnings comes from. To perform lacross is the thing I adore most of all. Her family lives in Ohio but her spouse wants them to transfer.<br><br>My web-site [http://findyourflirt.net/index.php?m=member_profile&p=profile&id=117823 psychic phone] |
| !bgcolor=#e7dcc3 colspan=2|Regular octaexon<BR>(7-simplex)
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| |bgcolor=#ffffff align=center colspan=2|[[Image:Uniform polytope 3,3,3,3,3,3 t0.jpg|280px]]<BR>Model created using straws (edges) and plasticine balls (vertices) in [[Triakis tetrahedron|triakis tetrahedral]] envelope
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| |bgcolor=#e7dcc3|Type||Regular [[7-polytope]]
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| |bgcolor=#e7dcc3|Family||[[simplex]]
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| |bgcolor=#e7dcc3|[[Schläfli symbol]]|| {3,3,3,3,3,3}
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| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD||node_1|3|node|3|node|3|node|3|node|3|node|3|node}}
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| |bgcolor=#e7dcc3|6-faces||8 [[6-simplex]][[Image:6-simplex_t0.svg|25px]]
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| |bgcolor=#e7dcc3|5-faces||28 [[5-simplex]][[Image:5-simplex_t0.svg|25px]]
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| |bgcolor=#e7dcc3|4-faces||56 [[5-cell]][[Image:4-simplex_t0.svg|25px]]
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| |bgcolor=#e7dcc3|Cells||70 [[tetrahedron]][[Image:3-simplex_t0.svg|25px]]
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| |bgcolor=#e7dcc3|Faces||56 [[triangle]][[Image:2-simplex_t0.svg|25px]]
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| |bgcolor=#e7dcc3|Edges||28
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| |bgcolor=#e7dcc3|Vertices||8
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| |bgcolor=#e7dcc3|[[Vertex figure]]||[[6-simplex]]
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| |bgcolor=#e7dcc3|[[Petrie polygon]]||[[octagon]]
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| |bgcolor=#e7dcc3|[[Coxeter group]]|| A<sub>7</sub> [3,3,3,3,3,3]
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| |bgcolor=#e7dcc3|Dual||[[Self-dual polytope|Self-dual]]
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| |bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
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| |}
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| In [[seven-dimensional space|7-dimensional]] [[geometry]], a 7-[[simplex]] is a self-dual [[Regular polytope|regular]] [[7-polytope]]. It has 8 [[vertex (geometry)|vertices]], 28 [[Edge (geometry)|edge]]s, 56 triangle [[Face (geometry)|faces]], 70 tetrahedral [[Cell (mathematics)|cells]], 56 [[5-cell]] 5-faces, 28 [[5-simplex]] 6-faces, and 8 [[6-simplex]] 7-faces. Its [[dihedral angle]] is cos<sup>−1</sup>(1/7), or approximately 81.79°.
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| == Alternate names ==
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| It can also be called an '''octaexon''', or '''octa-7-tope''', as an 8-[[facet (geometry)|facetted]] polytope in 7-dimensions. The [[5-polytope#A note on generality of terms for n-polytopes and elements|name]] ''octaexon'' is derived from ''octa'' for eight [[Facet (mathematics)|facets]] in [[Greek language|Greek]] and [[exa|''-ex'']] for having six-dimensional facets, and ''-on''. Jonathan Bowers gives an octaexon the acronym '''oca'''.<ref>{{KlitzingPolytopes|polyexa.htm|7D uniform polytopes (polyexa)|x3o3o3o3o3o - oca}}</ref>
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| == Coordinates ==
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| The [[Cartesian coordinate]]s of the vertices of an origin-centered regular octaexon having edge length 2 are:
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| :<math>\left(\sqrt{1/28},\ \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{1/28},\ \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{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ -\sqrt{3/2},\ 0,\ 0\right)</math>
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| :<math>\left(\sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ -2\sqrt{2/5},\ 0,\ 0,\ 0\right)</math>
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| :<math>\left(\sqrt{1/28},\ \sqrt{1/21},\ -\sqrt{5/3},\ 0,\ 0,\ 0,\ 0\right)</math>
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| :<math>\left(\sqrt{1/28},\ -\sqrt{12/7},\ 0,\ 0,\ 0,\ 0,\ 0\right)</math>
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| :<math>\left(-\sqrt{7/4},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)</math>
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| More simply, the vertices of the ''7-simplex'' can be positioned in 8-space as permutations of (0,0,0,0,0,0,0,1). This construction is based on [[Facet (geometry)|facets]] of the [[8-orthoplex]].
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| == Images ==
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| {{7-simplex Coxeter plane graphs|t0|150}}
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| == Related polytopes ==
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| This polytope is a facet in the uniform tessellation [[3 31 honeycomb|3<sub>31</sub>]] with [[Coxeter-Dynkin diagram]]:
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| :{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}}
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| This polytope is one of 71 [[uniform 7-polytope]]s with A<sub>7</sub> symmetry.
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| {{Octaexon family}}
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| == Notes ==
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| {{reflist}}
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| == External links ==
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| * {{PolyCell | urlname = glossary.html| title = Glossary for hyperspace}}
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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:7-polytopes]]
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| {{Geometry-stub}}
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