- Source: 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.
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.
[
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
]
{\displaystyle {\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:
(
1
/
21
,
1
/
15
,
1
/
10
,
1
/
6
,
1
/
3
,
±
1
)
{\displaystyle \left({\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ {\sqrt {1/6}},\ {\sqrt {1/3}},\ \pm 1\right)}
(
1
/
21
,
1
/
15
,
1
/
10
,
1
/
6
,
−
2
1
/
3
,
0
)
{\displaystyle \left({\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ {\sqrt {1/6}},\ -2{\sqrt {1/3}},\ 0\right)}
(
1
/
21
,
1
/
15
,
1
/
10
,
−
3
/
2
,
0
,
0
)
{\displaystyle \left({\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ -{\sqrt {3/2}},\ 0,\ 0\right)}
(
1
/
21
,
1
/
15
,
−
2
2
/
5
,
0
,
0
,
0
)
{\displaystyle \left({\sqrt {1/21}},\ {\sqrt {1/15}},\ -2{\sqrt {2/5}},\ 0,\ 0,\ 0\right)}
(
1
/
21
,
−
5
/
3
,
0
,
0
,
0
,
0
)
{\displaystyle \left({\sqrt {1/21}},\ -{\sqrt {5/3}},\ 0,\ 0,\ 0,\ 0\right)}
(
−
12
/
7
,
0
,
0
,
0
,
0
,
0
)
{\displaystyle \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.:
— (1973). "Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)". Regular Polytopes (3rd ed.). Dover. p. 296. ISBN 0-486-61480-8.
Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; Weiss, Asia Ivic, eds. (1995). Kaleidoscopes: Selected Writings of H.S.M. Coxeter. Wiley. ISBN 978-0-471-01003-6.
(Paper 22) — (1940). "Regular and Semi Regular Polytopes I". Math. Zeit. 46: 380–407. doi:10.1007/BF01181449. S2CID 186237114.
(Paper 23) — (1985). "Regular and Semi-Regular Polytopes II". Math. Zeit. 188 (4): 559–591. doi:10.1007/BF01161657. S2CID 120429557.
(Paper 24) — (1988). "Regular and Semi-Regular Polytopes III". Math. Zeit. 200: 3–45. doi:10.1007/BF01161745. S2CID 186237142.
Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). "26. Hemicubes: 1n1". The Symmetries of Things. p. 409. ISBN 978-1-56881-220-5.
Johnson, Norman (1991). "Uniform Polytopes" (Manuscript). Norman Johnson (mathematician).
Johnson, N.W. (1966). The Theory of Uniform Polytopes and Honeycombs (PhD). University of Toronto. OCLC 258527038.
External links
Olshevsky, George. "Simplex". Glossary for Hyperspace. Archived from the original on 4 February 2007.
Polytopes of Various Dimensions
Multi-dimensional Glossary