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	${\sqrt {\tfrac {3}{7}}}$ 0.654654
Properties	convex,isogonalself-dual

Ingeometry,a 6-simplexis aself-dualregular6-polytope.It has 7vertices,21edges,35 trianglefaces,35tetrahedralcells,215-cell4-faces, and 75-simplex5-faces. Itsdihedral angleis cos⁻¹(1/6), or approximately 80.41°.

Alternate names

It can also be called aheptapeton,orhepta-6-tope,as a 7-facettedpolytope in 6-dimensions. Thenameheptapetonis derived fromheptafor sevenfacetsinGreekand-petafor having five-dimensional facets, and-on.Jonathan Bowers gives a heptapeton the acronymhop.

As a configuration

Thisconfiguration matrixrepresents 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.

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

TheCartesian coordinatesfor an origin-centered regular heptapeton having edge length 2 are:

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

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

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

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

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

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

The vertices of the6-simplexcan be more simply positioned in 7-space as permutations of:

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

This construction is based onfacetsof the7-orthoplex.

Images

orthographic projections

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

Related uniform 6-polytopes

The regular 6-simplex is one of 35uniform 6-polytopesbased on the [3,3,3,3,3]Coxeter group,all shown here in A₆Coxeter planeorthographic projections.

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.ISBN0-486-61480-8.
  • Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; Weiss, Asia Ivic, eds. (1995).Kaleidoscopes: Selected Writings of H.S.M. Coxeter.Wiley.ISBN978-0-471-01003-6.
    • (Paper 22)— (1940)."Regular and Semi Regular Polytopes I".Math. Zeit.46:380–407.doi:10.1007/BF01181449.S2CID186237114.
    • (Paper 23)— (1985)."Regular and Semi-Regular Polytopes II".Math. Zeit.188(4): 559–591.doi:10.1007/BF01161657.S2CID120429557.
    • (Paper 24)— (1988)."Regular and Semi-Regular Polytopes III".Math. Zeit.200:3–45.doi:10.1007/BF01161745.S2CID186237142.
• Conway, John H.;Burgiel, Heidi; Goodman-Strauss, Chaim (2008). "26. Hemicubes: 1_n1".The Symmetries of Things.p. 409.ISBN978-1-56881-220-5.
• Johnson, Norman(1991). "Uniform Polytopes" (Manuscript). Norman Johnson (mathematician).
  • Johnson, N.W. (1966).The Theory of Uniform Polytopes and Honeycombs(PhD). University of Toronto.OCLC258527038.

External links

• Olshevsky, George."Simplex".Glossary for Hyperspace.Archived fromthe originalon 4 February 2007.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary