In fivedimensional geometry, a fivedimensional polytope or
5polytope
Contents 1 Definition 2 Characteristics 3 Classification 4 Regular 5polytopes 5 Uniform 5polytopes 6 Pyramids 7 See also 8 References 9 External links Definition[edit]
A
5polytope
Each cell must join exactly two 4faces. Adjacent 4faces are not in the same fourdimensional hyperplane. The figure is not a compound of other figures which meet the requirements. Characteristics[edit]
The topology of any given
5polytope
A
5polytope
Main article: Uniform 5polytope A semiregular
5polytope
Main article: List_of_regular_polytopes § Convex_4 A prismatic
5polytope
Regular 5polytopes[edit]
Regular 5polytopes can be represented by the
Schläfli symbol
3,3,3,3  5simplex 4,3,3,3  5cube 3,3,3,4  5orthoplex For the 3 convex regular 5polytopes and three semiregular 5polytope, their elements are: Name
Schläfli
symbol(s)
Coxeter
diagram(s)
Vertices
Edges
Faces
Cells
4faces
Symmetry
5simplex 3,3,3,3 6 15 20 15 6 A5, (120) 5cube 4,3,3,3 32 80 80 40 10 BC5, (3820) 5orthoplex 3,3,3,4 3,3,31,1 10 40 80 80 32 BC5, (3840) 2×D5 Uniform 5polytopes[edit] Main article: Uniform 5polytope For three of the semiregular 5polytope, their elements are: Name
Schläfli
symbol(s)
Coxeter
diagram(s)
Vertices
Edges
Faces
Cells
4faces
Symmetry
Expanded 5simplex t0,4 3,3,3,3 30 120 210 180 162 2×A5, (240) 5demicube 3,32,1 h 4,3,3,3 16 80 160 120 26 D5, (1920) ½BC5 Rectified 5orthoplex t1 3,3,3,4 t1 3,3,31,1 40 240 400 240 42 BC5, (3840) 2×D5 The expanded
5simplex
List of regular polytopes#Fivedimensional regular polytopes and higher References[edit] ^ 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. "5D uniform polytopes (polytera)". External links[edit] Polytopes of Various Dimensions, Jonathan Bowers Uniform Polytera, Jonathan Bowers Multidimensional Glossary, Garrett Jones 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
