8-simplex

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"\|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 25px
style="background:#e7dcc3;"\|6-faces	36 6-simplex 25px
style="background:#e7dcc3;"\|5-faces	84 5-simplex 25px
style="background:#e7dcc3;"\|4-faces	126 5-cell 25px
style="background:#e7dcc3;"\|Cells	126 tetrahedron 25px
style="background:#e7dcc3;"\|Faces	84 triangle 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
style="background:#e7dcc3;"\|Properties	convex

In geometry, an 8-simplex is a self-dual regular 8-polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral 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-facetted polytope in eight-dimensions. The name enneazetton is derived from ennea for nine facets in Greek and -zetta for having seven-dimensional facets, with suffix -on.

Jonathan Bowers gives it the acronym ene.{{harvnb|Klitzing|at=[https://bendwavy.org/klitzing/incmats/ene.htm (x3o3o3o3o3o3o3o – ene)]}}

As a configuration

This configuration matrix represents the 8-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces, 6-faces and 7-faces. The diagonal numbers say how many of each element occur in the whole 8-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}}

$\begin{bmatrix}\begin{matrix}
9 & 8 & 28 & 56 & 70 & 56 & 28 & 8
\\ 2 & 36 & 7 & 21 & 35 & 35 & 21 & 7
\\ 3 & 3 & 84 & 6 & 15 & 20 & 15 & 6
\\ 4 & 6 & 4 & 126 & 5 & 10 & 10 & 5
\\ 5 & 10 & 10 & 5 & 126 & 4 & 6 & 4
\\ 6 & 15 & 20 & 15 & 6 & 84 & 3 & 3
\\ 7 & 21 & 35 & 35 & 21 & 7 & 36 & 2
\\ 8 & 28 & 56 & 70 & 56 & 28 & 8 & 9
\end{matrix}\end{bmatrix}$

Coordinates

The Cartesian coordinates of the vertices of an origin-centered regular enneazetton having edge length 2 are:

: $\left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ \sqrt{1/3},\ \pm1\right)$

: $\left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ -2\sqrt{1/3},\ 0\right)$

: $\left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ -\sqrt{3/2},\ 0,\ 0\right)$

: $\left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ -2\sqrt{2/5},\ 0,\ 0,\ 0\right)$

: $\left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ -\sqrt{5/3},\ 0,\ 0,\ 0,\ 0\right)$

: $\left(1/6,\ \sqrt{1/28},\ -\sqrt{12/7},\ 0,\ 0,\ 0,\ 0,\ 0\right)$

: $\left(1/6,\ -\sqrt{7/4},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)$

: $\left(-4/3,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)$

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 facets of the 9-orthoplex.

Another origin-centered construction uses (1,1,1,1,1,1,1,1)/3 and permutations of (1,1,1,1,1,1,1,-11)/12 for edge length √2.

Images

File:K9-gyroelongated square pyramid.gif, edges colored by length.]]

Related polytopes and honeycombs

This polytope is a facet in the uniform tessellations: 2₅₁, and 5₂₁ with respective Coxeter-Dynkin diagrams:

:{{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 uniform 8-polytopes with A₈ symmetry.

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=[https://archive.org/details/regularpolytopes00coxe_869/page/n319 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}}
{{KlitzingPolytopes|polyzetta.htm|8D uniform polytopes (polyzetta) with acronyms}} (x3o3o3o3o3o3o3o – ene) {{sfn whitelist|CITEREFKlitzing}}

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:8-polytopes

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"\|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 25px
style="background:#e7dcc3;"\|6-faces	36 6-simplex 25px
style="background:#e7dcc3;"\|5-faces	84 5-simplex 25px
style="background:#e7dcc3;"\|4-faces	126 5-cell 25px
style="background:#e7dcc3;"\|Cells	126 tetrahedron 25px
style="background:#e7dcc3;"\|Faces	84 triangle 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
style="background:#e7dcc3;"\|Properties	convex