Logic redundancy

2013-04-15T20:44:40Z

86.142.228.40: linkify consensus theorem

{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Regular decayotton (9-simplex)
|-
|bgcolor=#ffffff align=center colspan=2|[[Image:9-simplex_t0.svg|280px]] [[Orthogonal projection]] inside [[Petrie polygon]]
|-
|bgcolor=#e7dcc3|Type||Regular [[9-polytope]]
|-
|bgcolor=#e7dcc3|Family||[[simplex]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {3,3,3,3,3,3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
|-
|bgcolor=#e7dcc3|8-faces||10 [[8-simplex]][[Image:8-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|7-faces||45 [[7-simplex]][[Image:7-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|6-faces||120 [[6-simplex]][[Image:6-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|5-faces||210 [[5-simplex]][[Image:5-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|4-faces||252 [[5-cell]][[Image:4-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Cells||210 [[tetrahedron]][[Image:3-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Faces||120 [[triangle]][[Image:2-simplex_t0.svg|25px]]
|-
|bgcolor=#e7dcc3|Edges||45
|-
|bgcolor=#e7dcc3|Vertices||10
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[8-simplex]]
|-
|bgcolor=#e7dcc3|[[Petrie polygon]]||[[decagon]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]|| A9 [3,3,3,3,3,3,3,3]
|-
|bgcolor=#e7dcc3|Dual||[[Self-dual polytope|Self-dual]]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
In [[geometry]], a 9-[[simplex]] is a self-dual [[Regular polytope|regular]] [[9-polytope]]. It has 10 [[vertex (geometry)|vertices]], 45 [[Edge (geometry)|edge]]s, 120 triangle [[Face (geometry)|faces]], 210 tetrahedral [[Cell (mathematics)|cells]], 252 [[5-cell]] 4-faces, 210 [[5-simplex]] 5-faces, 120 [[6-simplex]] 6-faces, 45 [[7-simplex]] 7-faces, and 10 [[8-simplex]] 8-faces. Its [[dihedral angle]] is cos−1(1/9), or approximately 83.62°.

It can also be called a '''decayotton''', or '''deca-9-tope''', as a 10-[[facet (geometry)|facetted]] polytope in 9-dimensions.. The [[5-polytope#A note on generality of terms for n-polytopes and elements|name]] ''decayotton'' is derived from ''deca'' for ten [[Facet (mathematics)|facets]] in [[Greek language|Greek]] and [[Yotta|-yott]] (variation of oct for eight), having 8-dimensional facets, and ''-on''.

== Coordinates ==

The [[Cartesian coordinate]]s of the vertices of an origin-centered regular decayotton having edge length 2 are:

:<math>\left(\sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ \sqrt{1/3},\ \pm1\right)</math>
:<math>\left(\sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ -2\sqrt{1/3},\ 0\right)</math>
:<math>\left(\sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ -\sqrt{3/2},\ 0,\ 0\right)</math>
:<math>\left(\sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ -2\sqrt{2/5},\ 0,\ 0,\ 0\right)</math>
:<math>\left(\sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ -\sqrt{5/3},\ 0,\ 0,\ 0,\ 0\right)</math>
:<math>\left(\sqrt{1/45},\ 1/6,\ \sqrt{1/28},\ -\sqrt{12/7},\ 0,\ 0,\ 0,\ 0,\ 0\right)</math>
:<math>\left(\sqrt{1/45},\ 1/6,\ -\sqrt{7/4},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)</math>
:<math>\left(\sqrt{1/45},\ -4/3,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)</math>
:<math>\left(-3\sqrt{1/5},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)</math>

More simply, the vertices of the ''9-simplex'' can be positioned in 10-space as permutations of (0,0,0,0,0,0,0,0,0,1). This construction is based on [[Facet (geometry)|facets]] of the [[10-orthoplex]].

== Images ==

{{A9 Coxeter plane graphs|t0|100}}

== References==
* [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]:
** 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)
** 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)
** '''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]
*** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10]
*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591]
*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
* [[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: 1n1)
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966)
* {{KlitzingPolytopes|polyyotta.htm|9D uniform polytopes (polyyotta)|x3o3o3o3o3o3o3o3o - day}}

== External links ==
* {{PolyCell | urlname = glossary.html| title = Glossary for hyperspace}}
* [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions]
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

[[Category:9-polytopes]]
{{Polytopes}}

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Logic redundancy