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 5simplex Circumradius 0.645497 Properties convex, isogonal selfdual In geometry, a
6simplex
Contents 1 Alternate names 2 As a configuration 3 Coordinates 4 Images 5 Related uniform 6polytopes 6 Notes 7 References 8 External links Alternate names[edit] It can also be called a heptapeton, or hepta6tope, as a 7facetted polytope in 6dimensions. The name heptapeton is derived from hepta for seven facets in Greek and peta for having fivedimensional facets, and on. Jonathan Bowers gives a heptapeton the acronym hop.[1] As a configuration[edit] The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (fvectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.[2][3] [ 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&7end matrix end bmatrix Coordinates[edit] The Cartesian coordinates for an origincentered 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 1right) ( 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 , 0right) ( 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, 0right) ( 1 / 21 , 1 / 15 , − 2 2 / 5 , 0 , 0 , 0 ) displaystyle left( sqrt 1/21 , sqrt 1/15 , 2 sqrt 2/5 , 0, 0, 0right) ( 1 / 21 , − 5 / 3 , 0 , 0 , 0 , 0 ) displaystyle left( sqrt 1/21 ,  sqrt 5/3 , 0, 0, 0, 0right) ( − 12 / 7 , 0 , 0 , 0 , 0 , 0 ) displaystyle left( sqrt 12/7 , 0, 0, 0, 0, 0right) The vertices of the
6simplex
(0,0,0,0,0,0,1) This construction is based on facets of the 7orthoplex. Images[edit] orthographic projections Ak Coxeter plane A6 A5 A4 Graph Dihedral symmetry [7] [6] [5] Ak Coxeter plane A3 A2 Graph Dihedral symmetry [4] [3] Related uniform 6polytopes[edit]
The regular
6simplex
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 Notes[edit] ^ Klitzing, (x3o3o3o3o3o  hop) ^ Coxeter, Regular Polytopes, sec 1.8 Configurations ^ Coxeter, Complex Regular Polytopes, p.117 References[edit] H.S.M. Coxeter: Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0486614808, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in ndimensions (n≥5) H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in ndimensions (n≥5) Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036 [1] (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10] (Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559591] (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345] John H. Conway, Heidi Burgiel, Chaim GoodmanStrass, The Symmetries of Things 2008, ISBN 9781568812205 (Chapter 26. pp. 409: Hemicubes: 1n1) Norman Johnson Uniform Polytopes, Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966) Klitzing, Richard. "6D uniform polytopes (polypeta) x3o3o3o3o  hix". External links[edit] Olshevsky, George. "Simplex". Glossary for Hyperspace. Archived from the original on 4 February 2007. Polytopes of Various Dimensions Multidimensional Glossary v t e Fundamental convex regular and uniform polytopes in dimensions 2–10 Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn Regular polygon Triangle Square pgon Hexagon Pentagon Uniform polyhedron
Tetrahedron
Octahedron
Dodecahedron • Icosahedron Uniform 4polytope
5cell
16cell
Uniform 5polytope
5simplex
5orthoplex
Uniform 6polytope
6simplex
6orthoplex
Uniform 7polytope
7simplex
7orthoplex
Uniform 8polytope
8simplex
8orthoplex
Uniform 9polytope
9simplex
9orthoplex
Uniform 10polytope
10simplex
10orthoplex
Uniform npolytope nsimplex northoplex • ncube ndemicube 1k2 • 2k1 • k21 npentagonal polytope Topics:
Polytope
