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	${\displaystyle \sqrt{\frac{3}{7}}}$ 0.654654[1]
Properties	convex,isogonalself-dual

Ingeometry, a 6-simplex is aself-dualregular6-polytope. It has 7vertices, 21edges, 35 trianglefaces, 35tetrahedralcells, 215-cell 4-faces, and 75-simplex 5-faces. Itsdihedral angle is cos⁻¹(1/6), or approximately 80.41°.

It can also be called aheptapeton, orhepta-6-tope, as a 7-facetted polytope in 6-dimensions. Thenameheptapeton is derived fromhepta for sevenfacets inGreek and-peta for having five-dimensional facets, and-on. Jonathan Bowers gives a heptapeton the acronymhop.^[2]

Thisconfiguration 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]

TheCartesian 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 the6-simplex can be more simply positioned in 7-space as permutations of:

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

This construction is based onfacets of the7-orthoplex.

orthographic projections

AkCoxeter 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 35uniform 6-polytopes based on the [3,3,3,3,3]Coxeter group, all shown here in A₆Coxeter planeorthographic 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.
  • 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: 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 fromthe original on 4 February 2007.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary

• 6-polytopes

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