2A02:120B:2C4D:A600:F5F1:361A:3F0D:6640: /* Definition of function symbols */ type be -> by

2014-10-29T22:35:21Z

Definition of function symbols: type be -> by

en>Michael Hardy at 16:00, 5 October 2011

2011-10-05T16:00:24Z

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{| class="wikitable" align="right" style="margin-left:10px" width="250"
! style="background:#e7dcc3;" colspan="2"|Regular enneazetton (8-simplex)
|-
| style="background:#fff; text-align:center;" colspan="2"|[[File:8-simplex t0.svg|280px]] [[Orthogonal projection]] inside [[Petrie polygon]]
|-
| style="background:#e7dcc3;"|Type||Regular [[8-polytope]]
|-
| style="background:#e7dcc3;"|Family||[[simplex]]
|-
| style="background:#e7dcc3;"|[[Schläfli symbol]]|| {3,3,3,3,3,3,3}
|-
| style="background:#e7dcc3;"|[[Coxeter-Dynkin diagram]]||{{CDD||node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
|-
| style="background:#e7dcc3;"|7-faces||9 [[7-simplex]][[File:7-simplex t0.svg|25px]]
|-
| style="background:#e7dcc3;"|6-faces||36 [[6-simplex]][[File:6-simplex t0.svg|25px]]
|-
| style="background:#e7dcc3;"|5-faces||84 [[5-simplex]][[File:5-simplex t0.svg|25px]]
|-
| style="background:#e7dcc3;"|4-faces||126 [[5-cell]][[File:4-simplex t0.svg|25px]]
|-
| style="background:#e7dcc3;"|Cells||126 [[tetrahedron]][[File:3-simplex t0.svg|25px]]
|-
| style="background:#e7dcc3;"|Faces||84 [[triangle]][[File:2-simplex t0.svg|25px]]
|-
| style="background:#e7dcc3;"|Edges||36
|-
| style="background:#e7dcc3;"|Vertices||9
|-
| style="background:#e7dcc3;"|[[Vertex figure]]||[[7-simplex]]
|-
| style="background:#e7dcc3;"|[[Petrie polygon]]||[[enneagon]]
|-
| style="background:#e7dcc3;"|[[Coxeter group]]|| A8 [3,3,3,3,3,3,3]
|-
| style="background:#e7dcc3;"|Dual||[[Self-dual polytope|Self-dual]]
|-
| style="background:#e7dcc3;"|Properties||[[Convex polytope|convex]]
|}
In [[geometry]], an 8-[[simplex]] is a self-dual [[Regular polytope|regular]] [[8-polytope]]. It has 9 [[vertex (geometry)|vertices]], 36 [[Edge (geometry)|edges]], 84 triangle [[Face (geometry)|faces]], 126 tetrahedral [[Cell (mathematics)|cells]], 126 [[5-cell]] 4-faces, 84 [[5-simplex]] 5-faces, 36 [[6-simplex]] 6-faces, and 9 [[7-simplex]] 7-faces. Its [[dihedral angle]] is cos−1(1/8), or approximately 82.82°.

It can also be called an '''enneazetton''', or '''ennea-8-tope''', as a 9-[[facet (geometry)|facetted]] polytope in 8-dimensions.. The [[5-polytope#A note on generality of terms for n-polytopes and elements|name]] ''enneazetton'' is derived from ''ennea'' for nine [[Facet (mathematics)|facets]] in [[Greek language|Greek]] and [[Zetta|''-zetta'']] for having seven-dimensional facets, and ''-on''.

== Coordinates ==

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

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

More simply, the vertices of the ''8-simplex'' can be positioned in 9-space as permutations of (0,0,0,0,0,0,0,0,1). This construction is based on [[Facet (geometry)|facets]] of the [[9-orthoplex]].

== Images ==

{{8-simplex Coxeter plane graphs|t0|100}}

== Related polytopes and honeycombs==

This polytope is a facet in the uniform tessellations: [[2 51 honeycomb|251]], and [[5 21 honeycomb|521]] with respective [[Coxeter-Dynkin diagram]]s:
:{{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}, {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea_1}}

This polytope is one of 135 [[8-polytope#The A8 .5B3,3,3,3,3,3,3.5D family (8-simplex)|uniform 8-polytopes]] with A8 symmetry.
{{Enneazetton family}}

== 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|polyzetta.htm|8D uniform polytopes (polyzetta)|x3o3o3o3o3o3o3o - ene}}

== 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]

{{Polytopes}}

[[Category:8-polytopes]]

Extension by definitions - Revision history

2A02:120B:2C4D:A600:F5F1:361A:3F0D:6640: /* Definition of function symbols */ type be -> by

en>Michael Hardy at 16:00, 5 October 2011