6-simplex

6-simplex
Type	uniform polypeton
Schläfli symbol	{35}
Coxeter diagrams
Elements	f5 = 7, f4 = 21, C = 35, F = 35, E = 21, V = 7 (χ=0)
Coxeter group	A6, [35], order 5040
Bowers name and (acronym)	Heptapeton (hop)
Vertex figure	5-simplex
Circumradius	${\displaystyle {\sqrt {\tfrac {3}{7}}}}$ 0.654654[1]
Properties	convex, isogonal self-dual

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°.

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.^[2]

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.^[3][4]

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

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

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

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

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

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

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

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

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.

orthographic projections

Ak Coxeter plane	A6	A5	A4
Graph
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

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.

A6 polytopes
t0	t1	t2	t0,1	t0,2	t1,2	t0,3	t1,3	t2,3
t0,4	t1,4	t0,5	t0,1,2	t0,1,3	t0,2,3	t1,2,3	t0,1,4	t0,2,4
t1,2,4	t0,3,4	t0,1,5	t0,2,5	t0,1,2,3	t0,1,2,4	t0,1,3,4	t0,2,3,4	t1,2,3,4
t0,1,2,5	t0,1,3,5	t0,2,3,5	t0,1,4,5	t0,1,2,3,4	t0,1,2,3,5	t0,1,2,4,5	t0,1,2,3,4,5

• 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.
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    • (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: 1_n1". 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.

• Olshevsky, George. "Simplex". Glossary for Hyperspace. Archived from the original on 4 February 2007.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary