6-simplex

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.{{KlitzingPolytopes|polypeta.htm|6D uniform polytopes (polypeta)|x3o3o3o3o3o — hop}}

As a configuration

This configuration matrix represents the 6-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual simplex's matrix is identical to its 180 degree rotation.{{harvnb|Coxeter|1973|loc=§1.8 Configurations}}{{cite book |first=H.S.M. |last=Coxeter |title=Regular Complex Polytopes |url=https://books.google.com/books?id=9BY9AAAAIAAJ&pg=PA117 |pages=117 |edition=2nd |publisher=Cambridge University Press |year=1991 |isbn=9780521394901}}

$\begin{bmatrix}\begin{matrix}7 & 6 & 15 & 20 & 15 & 6 \\ 2 & 21 & 5 & 10 & 10 & 5 \\ 3 & 3 & 35 & 4 & 6 & 4 \\ 4 & 6 & 4 & 35 & 3 & 3 \\ 5 & 10 & 10 & 5 & 21 & 2 \\ 6 & 15 & 20 & 15 & 6 & 7 \end{matrix}\end{bmatrix}$

Coordinates

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

: $\left(\sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ \sqrt{1/3},\ \pm1\right)$

: $\left(\sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ -2\sqrt{1/3},\ 0\right)$

: $\left(\sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ -\sqrt{3/2},\ 0,\ 0\right)$

: $\left(\sqrt{1/21},\ \sqrt{1/15},\ -2\sqrt{2/5},\ 0,\ 0,\ 0\right)$

: $\left(\sqrt{1/21},\ -\sqrt{5/3},\ 0,\ 0,\ 0,\ 0\right)$

: $\left(-\sqrt{12/7},\ 0,\ 0,\ 0,\ 0,\ 0\right)$

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

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.

Notes

References

Coxeter, H.S.M.:
{{cite book |title-link=Regular Polytopes (book) |author-mask=1 |first=H.S.M. |last=Coxeter |chapter=Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) |title=Regular Polytopes |publisher=Dover |edition=3rd |year=1973 |isbn=0-486-61480-8 |pages=296 }}
{{cite book |editor-first=F. Arthur |editor-last=Sherk |editor2-first=Peter |editor2-last=McMullen |editor3-first=Anthony C. |editor3-last=Thompson |editor4-first=Asia Ivic |editor4-last=Weiss |title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter |publisher=Wiley |year=1995 |isbn=978-0-471-01003-6 |url=https://books.google.com/books?id=fUm5Mwfx8rAC&pg=PP1}}
(Paper 22) {{cite journal |author-mask=1 |first=H.S.M. |last=Coxeter |title=Regular and Semi Regular Polytopes I |journal=Math. Zeit. |volume=46 |pages=380–407 |year=1940 |doi=10.1007/BF01181449 |s2cid=186237114 |url=https://books.google.com/books?id=fUm5Mwfx8rAC&pg=PA251 |url-access=subscription }}
(Paper 23) {{cite journal |author-mask=1 |first=H.S.M. |last=Coxeter |title=Regular and Semi-Regular Polytopes II |journal=Math. Zeit. |volume=188 |pages=559–591 |year=1985 |issue=4 |doi=10.1007/BF01161657 |s2cid=120429557 |url=https://books.google.com/books?id=fUm5Mwfx8rAC&pg=PA279|url-access=subscription }}
(Paper 24) {{cite journal |author-mask=1 |first=H.S.M. |last=Coxeter |title=Regular and Semi-Regular Polytopes III |journal=Math. Zeit. |volume=200 |pages=3–45 |year=1988 |doi=10.1007/BF01161745 |s2cid=186237142 |url=https://books.google.com/books?id=fUm5Mwfx8rAC&pg=PA313|url-access=subscription }}
{{cite book |author-link=John Horton Conway |first1=John H. |last1=Conway |first2=Heidi |last2=Burgiel |first3=Chaim |last3=Goodman-Strauss |chapter=26. Hemicubes: 1_n1 |title=The Symmetries of Things |year=2008 |isbn=978-1-56881-220-5 |pages=409 }}
{{cite document |author-link=Norman Johnson (mathematician) |first=Norman |last=Johnson |title=Uniform Polytopes |date=1991|publisher= Norman Johnson (mathematician) |type=Manuscript }}
{{cite thesis |first=N.W. |last=Johnson |title=The Theory of Uniform Polytopes and Honeycombs |date=1966 |type=PhD |publisher=University of Toronto |url=https://search.library.utoronto.ca/details?402790 |oclc=258527038}}

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]

Category:6-polytopes