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== | {| class="wikitable" align="right" style="margin-left:10px" width="250" | ||
!bgcolor=#e7dcc3 colspan=2|Regular decayotton<BR>(9-simplex) | |||
|- | |||
|bgcolor=#ffffff align=center colspan=2|[[Image:9-simplex_t0.svg|280px]]<BR>[[Orthogonal projection]]<BR>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]]|| A<sub>9</sub> [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<sup>−1</sup>(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: 1<sub>n1</sub>) | |||
* [[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}} |
Latest revision as of 22:44, 15 April 2013
Regular decayotton (9-simplex) | |
---|---|
Orthogonal projection inside Petrie polygon | |
Type | Regular 9-polytope |
Family | simplex |
Schläfli symbol | {3,3,3,3,3,3,3,3} |
Coxeter-Dynkin diagram | Template:CDD |
8-faces | 10 8-simplex |
7-faces | 45 7-simplex |
6-faces | 120 6-simplex |
5-faces | 210 5-simplex |
4-faces | 252 5-cell |
Cells | 210 tetrahedron |
Faces | 120 triangle |
Edges | 45 |
Vertices | 10 |
Vertex figure | 8-simplex |
Petrie polygon | decagon |
Coxeter group | A9 [3,3,3,3,3,3,3,3] |
Dual | Self-dual |
Properties | convex |
In geometry, a 9-simplex is a self-dual regular 9-polytope. It has 10 vertices, 45 edges, 120 triangle faces, 210 tetrahedral 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-facetted polytope in 9-dimensions.. The name decayotton is derived from deca for ten facets in Greek and -yott (variation of oct for eight), having 8-dimensional facets, and -on.
Coordinates
The Cartesian coordinates of the vertices of an origin-centered regular decayotton having edge length 2 are:
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 facets of the 10-orthoplex.
Images
Template:A9 Coxeter plane graphs
References
- H.S.M. Coxeter:
- Coxeter, 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 [1]
- (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 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 Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
- Template:KlitzingPolytopes