In fivedimensional geometry, a
5simplex
Contents 1 Alternate names 2 As a configuration 3 Regular hexateron cartesian coordinates 4 Projected images 5 Related uniform 5polytopes 6 Other forms 7 Notes 8 References 9 External links Alternate names[edit] It can also be called a hexateron, or hexa5tope, as a 6facetted polytope in 5dimensions. The name hexateron is derived from hexa for having six facets and teron (with ter being a corruption of tetra) for having fourdimensional facets. By Jonathan Bowers, a hexateron is given the acronym hix.[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] [ 6 5 10 10 5 2 15 4 6 4 3 3 20 3 3 4 6 4 15 2 5 10 10 5 6 ] displaystyle begin bmatrix begin matrix 6&5&10&10&5\2&15&4&6&4\3&3&20&3&3\4&6&4&15&2\5&10&10&5&6end matrix end bmatrix Regular hexateron cartesian coordinates[edit]
The hexateron can be constructed from a
5cell
( 1 15 , 1 10 , 1 6 , 1 3 , ± 1 ) ( 1 15 , 1 10 , 1 6 , − 2 3 , 0 ) ( 1 15 , 1 10 , − 3 2 , 0 , 0 ) ( 1 15 , − 2 2 5 , 0 , 0 , 0 ) ( − 5 3 , 0 , 0 , 0 , 0 ) displaystyle begin aligned &left( tfrac 1 sqrt 15 , tfrac 1 sqrt 10 , tfrac 1 sqrt 6 , tfrac 1 sqrt 3 , pm 1right)\[5pt]&left( tfrac 1 sqrt 15 , tfrac 1 sqrt 10 , tfrac 1 sqrt 6 ,  tfrac 2 sqrt 3 , 0right)\[5pt]&left( tfrac 1 sqrt 15 , tfrac 1 sqrt 10 ,  tfrac sqrt 3 sqrt 2 , 0, 0right)\[5pt]&left( tfrac 1 sqrt 15 ,  tfrac 2 sqrt 2 sqrt 5 , 0, 0, 0right)\[5pt]&left( tfrac sqrt 5 sqrt 3 , 0, 0, 0, 0right)end aligned The vertices of the
5simplex
orthographic projections Ak
Coxeter
Graph Dihedral symmetry [6] [5] Ak
Coxeter
Graph Dihedral symmetry [4] [3]
Stereographic projection
Related uniform 5polytopes[edit]
It is first in a dimensional series of uniform polytopes and
honeycombs, expressed by
Coxeter
13k dimensional figures Space Finite Euclidean Hyperbolic n 4 5 6 7 8 9 Coxeter group A3A1 A5 D6 E7 E ~ 7 displaystyle tilde E _ 7 =E7+ T ¯ 8 displaystyle bar T _ 8 =E7++ Coxeter diagram Symmetry [3−1,3,1] [30,3,1] [31,3,1] [32,3,1] [[33,3,1]] [34,3,1] Order 48 720 23,040 2,903,040 ∞ Graph   Name 13,1 130 131 132 133 134 It is first in a dimensional series of uniform polytopes and
honeycombs, expressed by
Coxeter
3k1 dimensional figures Space Finite Euclidean Hyperbolic n 4 5 6 7 8 9 Coxeter group A3A1 A5 D6 E7 E ~ 7 displaystyle tilde E _ 7 =E7+ T ¯ 8 displaystyle bar T _ 8 =E7++ Coxeter diagram Symmetry [3−1,3,1] [30,3,1] [[31,3,1]] = [4,3,3,3,3] [32,3,1] [33,3,1] [34,3,1] Order 48 720 46,080 2,903,040 ∞ Graph   Name 31,1 310 311 321 331 341 The 5simplex, as 220 polytope is first in dimensional series 22k. 22k figures of n dimensions Space Finite Euclidean Hyperbolic n 5 6 7 8 Coxeter group A5 E6 E ~ 6 displaystyle tilde E _ 6 =E6+ E6++ Coxeter diagram Graph ∞ ∞ Name 220 221 222 223 The regular
5simplex
A5 polytopes t0 t1 t2 t0,1 t0,2 t1,2 t0,3 t1,3 t0,4 t0,1,2 t0,1,3 t0,2,3 t1,2,3 t0,1,4 t0,2,4 t0,1,2,3 t0,1,2,4 t0,1,3,4 t0,1,2,3,4 Other forms[edit]
The
5simplex
^ Klitzing, (x3o3o3o3o  hix) ^ Coxeter, Regular Polytopes, sec 1.8 Configurations ^ Coxeter, Complex Regular Polytopes, p.117 References[edit] T. Gosset: On the Regular and SemiRegular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900 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. "5D uniform polytopes (polytera) x3o3o3o3o  hix". External links[edit] Olshevsky, George. "Simplex". Glossary for Hyperspace. Archived from the original on 4 February 2007. Polytopes of Various Dimensions, Jonathan Bowers 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
