In sixdimensional geometry, a sixdimensional polytope or 6polytope
is a polytope, bounded by
5polytope
Contents 1 Definition 2 Characteristics 3 Classification 4 Regular 6polytopes 5 Uniform 6polytopes 6 References 7 External links Definition[edit]
A
6polytope
Each 4face must join exactly two 5faces (facets). Adjacent facets are not in the same fivedimensional hyperplane. The figure is not a compound of other figures which meet the requirements. Characteristics[edit]
The topology of any given
6polytope
A
6polytope
Main article: List of regular polytopes § Convex_5 A semiregular
6polytope
Main article: Uniform 6polytope A prismatic
6polytope
Regular 6polytopes[edit]
Regular 6polytopes can be generated from Coxeter groups represented
by the
Schläfli symbol
3,3,3,3,3  6simplex 4,3,3,3,3  6cube 3,3,3,3,4  6orthoplex There are no nonconvex regular polytopes of 5 or more dimensions. For the 3 convex regular 6polytopes, their elements are: Name
Schläfli
symbol
Coxeter
diagram
Vertices
Edges
Faces
Cells
4faces
5faces
Symmetry
6simplex 3,3,3,3,3 7 21 35 35 21 7 A6 (720) 6orthoplex 3,3,3,3,4 12 60 160 240 192 64 B6 (46080) 6cube 4,3,3,3,3 64 192 240 160 60 12 B6 (46080) Uniform 6polytopes[edit]
Main article: Uniform 6polytope
Here are six simple uniform convex 6polytopes, including the
6orthoplex
Name
Schläfli
symbol(s)
Coxeter
diagram(s)
Vertices
Edges
Faces
Cells
4faces
5faces
Symmetry
Expanded 6simplex t0,5 3,3,3,3,3 42 210 490 630 434 126 2×A6 (1440) 6orthoplex, 311 (alternate construction) 3,3,3,31,1 12 60 160 240 192 64 D6 (23040) 6demicube 3,33,1 h 4,3,3,3,3 32 240 640 640 252 44 D6 (23040) ½B6 Rectified 6orthoplex t1 3,3,3,3,4 t1 3,3,3,31,1 60 480 1120 1200 576 76 B6 (46080) 2×D6 221 polytope 3,3,32,1 27 216 720 1080 648 99 E6 (51840) 122 polytope 3,32,2 or 72 720 2160 2160 702 54 2×E6 (103680) The expanded
6simplex
^ a b c Richeson, D.; Euler's Gem: The
Polyhedron
T. Gosset: On the Regular and SemiRegular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900 A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910 H.S.M. Coxeter: H.S.M. Coxeter, M.S. LonguetHiggins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954 H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 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] N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966 Klitzing, Richard. "6D uniform polytopes (polypeta)". External links[edit]
Polytope
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
