Max-plus algebra: Difference between revisions

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{{Uniform polypeton db|Uniform polypeton stat table|hop}}
In [[geometry]], a 6-[[simplex]] is a [[Duality (mathematics)|self-dual]] [[Regular polytope|regular]] [[6-polytope]]. It has 7 [[vertex (geometry)|vertices]], 21 [[Edge (geometry)|edge]]s, 35 triangle [[Face (geometry)|faces]], 35 [[Tetrahedron|tetrahedral]] [[Cell (mathematics)|cells]],  21 [[5-cell]] 4-faces, and 7 [[5-simplex]] 5-faces. Its [[dihedral angle]] is cos<sup>−1</sup>(1/6), or approximately 80.41°.
 
== Alternate names ==
 
It can also be called a '''heptapeton''', or '''hepta-6-tope''', as a 7-[[facet (geometry)|facetted]] polytope in 6-dimensions. The [[5-polytope#A note on generality of terms for n-polytopes and elements|name]] ''heptapeton'' is derived from ''hepta'' for seven [[Facet (mathematics)|facets]] in [[Greek language|Greek]] and [[Peta-|''-peta'']] for having five-dimensional facets, and ''-on''. Jonathan Bowers gives a heptapeton the acronym '''hop'''.<ref>Klitzing, (x3o3o3o3o3o - hop)</ref>
 
== Coordinates ==
 
The [[Cartesian coordinate]]s for an origin-centered regular heptapeton having edge length&nbsp;2 are:
 
:<math>\left(\sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ \sqrt{1/3},\ \pm1\right)</math>
:<math>\left(\sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ -2\sqrt{1/3},\ 0\right)</math>
:<math>\left(\sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ -\sqrt{3/2},\ 0,\ 0\right)</math>
:<math>\left(\sqrt{1/21},\ \sqrt{1/15},\ -2\sqrt{2/5},\ 0,\ 0,\ 0\right)</math>
:<math>\left(\sqrt{1/21},\ -\sqrt{5/3},\ 0,\ 0,\ 0,\ 0\right)</math>
:<math>\left(-\sqrt{12/7},\ 0,\ 0,\ 0,\ 0,\ 0\right)</math>
 
The vertices of the ''6-simplex'' can be more simply positioned in 7-space as permutations of:
: (0,0,0,0,0,0,1)
 
This construction is based on [[Facet (geometry)|facets]] of the [[7-orthoplex]].
 
== Images ==
 
{{6-simplex Coxeter plane graphs|t0|150}}
 
== Related uniform 6-polytopes ==
The regular 6-simplex is one of 35 [[Uniform 6-polytope#The A6 .5B3.2C3.2C3.2C3.2C3.5D family .286-simplex.29|uniform 6-polytopes]] based on the [3,3,3,3,3] [[Coxeter group]], all shown here in A<sub>6</sub> [[Coxeter plane]] [[orthographic projection]]s.
 
{{Heptapeton family}}
 
==Notes ==
{{reflist}}
 
== 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.&nbsp;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.&nbsp;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.&nbsp;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|polypeta.htm|6D uniform polytopes (polypeta)|x3o3o3o3o - hix}}
 
== External links ==
*{{GlossaryForHyperspace | anchor=Simplex | title=Simplex }}
* [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions]
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]
{{Polytopes}}
 
[[Category:6-polytopes]]

Revision as of 21:17, 9 April 2013

Template:Uniform polypeton db In geometry, a 6-simplex is a self-dual regular 6-polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5-cell 4-faces, and 7 5-simplex 5-faces. Its dihedral angle is cos−1(1/6), or approximately 80.41°.

Alternate names

It can also be called a heptapeton, or hepta-6-tope, as a 7-facetted polytope in 6-dimensions. The name heptapeton is derived from hepta for seven facets in Greek and -peta for having five-dimensional facets, and -on. Jonathan Bowers gives a heptapeton the acronym hop.[1]

Coordinates

The Cartesian coordinates for an origin-centered regular heptapeton having edge length 2 are:

(1/21,1/15,1/10,1/6,1/3,±1)
(1/21,1/15,1/10,1/6,−21/3,0)
(1/21,1/15,1/10,−3/2,0,0)
(1/21,1/15,−22/5,0,0,0)
(1/21,−5/3,0,0,0,0)
(−12/7,0,0,0,0,0)

The vertices of the 6-simplex can be more simply positioned in 7-space as permutations of:

(0,0,0,0,0,0,1)

This construction is based on facets of the 7-orthoplex.

Images

Template:6-simplex Coxeter plane graphs

Related uniform 6-polytopes

The regular 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.

Template:Heptapeton family

Notes

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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

External links

Template:Polytopes

  1. ↑ Klitzing, (x3o3o3o3o3o - hop)
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