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5-orthoplex

January 01, 1970

Regular 5-orthoplex
(pentacross)

Orthogonal projection
inside Petrie polygon
Type Regular 5-polytope
Family orthoplex
Schläfli symbol {3,3,3,4}
{3,3,31,1}
Coxeter-Dynkin diagrams
4-faces 32 {33}
Cells 80 {3,3}
Faces 80 {3}
Edges 40
Vertices 10
Vertex figure
16-cell
Petrie polygon decagon
Coxeter groups BC5, [3,3,3,4]
D5, [32,1,1]
Dual 5-cube
Properties convex, Hanner polytope

In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.

It has two constructed forms, the first being regular with Schläfli symbol {33,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,31,1} or Coxeter symbol 211.

It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 5-hypercube or 5-cube.

Alternate names edit

  • pentacross, derived from combining the family name cross polytope with pente for five (dimensions) in Greek.
  • Triacontaditeron (or triacontakaiditeron) - as a 32-facetted 5-polytope (polyteron).

As a configuration edit

This configuration matrix represents the 5-orthoplex. The rows and columns correspond to vertices, edges, faces, cells and 4-faces. The diagonal numbers say how many of each element occur in the whole 5-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.[1][2]

 

Cartesian coordinates edit

Cartesian coordinates for the vertices of a 5-orthoplex, centered at the origin are

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

Construction edit

There are three Coxeter groups associated with the 5-orthoplex, one regular, dual of the penteract with the C5 or [4,3,3,3] Coxeter group, and a lower symmetry with two copies of 5-cell facets, alternating, with the D5 or [32,1,1] Coxeter group, and the final one as a dual 5-orthotope, called a 5-fusil which can have a variety of subsymmetries.

Name Coxeter diagram Schläfli symbol Symmetry Order Vertex figure(s)
regular 5-orthoplex           {3,3,3,4} [3,3,3,4] 3840        
Quasiregular 5-orthoplex         {3,3,31,1} [3,3,31,1] 1920      
5-fusil
          {3,3,3,4} [4,3,3,3] 3840        
          {3,3,4}+{} [4,3,3,2] 768        
          {3,4}+{4} [4,3,2,4] 384        
       
          {3,4}+2{} [4,3,2,2] 192        
       
          2{4}+{} [4,2,4,2] 128        
          {4}+3{} [4,2,2,2] 64        
       
          5{} [2,2,2,2] 32        

Other images edit

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph      
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph    
Dihedral symmetry [4] [4]
 
The perspective projection (3D to 2D) of a stereographic projection (4D to 3D) of the Schlegel diagram (5D to 4D) of the 5-orthoplex. 10 sets of 4 edges form 10 circles in the 4D Schlegel diagram: two of these circles are straight lines in the stereographic projection because they contain the center of projection.

Related polytopes and honeycombs edit

2k1 figures in n dimensions
Space Finite Euclidean Hyperbolic
n 3 4 5 6 7 8 9 10
Coxeter
group 		E3=A2A1 E4=A4 E5=D5 E6 E7 E8 E9 =   = E8+ E10 =   = E8++
Coxeter
diagram 		                                                                                         
Symmetry [3−1,2,1] [30,2,1] [[31,2,1]] [32,2,1] [33,2,1] [34,2,1] [35,2,1] [36,2,1]
Order 12 120 384 51,840 2,903,040 696,729,600 ∞
Graph             - -
Name 2−1,1 201 211 221 231 241 251 261

This polytope is one of 31 uniform 5-polytopes generated from the B5 Coxeter plane, including the regular 5-cube and 5-orthoplex.

References edit

  1. ^ Coxeter, Regular Polytopes, sec 1.8 Configurations
  2. ^ Coxeter, Complex Regular Polytopes, p.117
  • H.S.M. Coxeter:
    • 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, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
      • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
      • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
      • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • 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) x3o3o3o4o - tac".

External links edit

  • Olshevsky, George. . Glossary for Hyperspace. Archived from the original on 4 February 2007.
  • Polytopes of Various Dimensions
  • Multi-dimensional Glossary
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 p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

January 01, 1970
orthoplex, regular, pentacross, orthogonal, projectioninside, petrie, polygon, type, regular, polytope, family, orthoplex, schläfli, symbol, coxeter, dynkin, diagrams, faces, cells, faces, edges, vertices, vertex, figure, cell, petrie, polygon, decagon, coxete. Regular 5 orthoplex pentacross Orthogonal projectioninside Petrie polygon Type Regular 5 polytope Family orthoplex Schlafli symbol 3 3 3 4 3 3 31 1 Coxeter Dynkin diagrams 4 faces 32 33 Cells 80 3 3 Faces 80 3 Edges 40 Vertices 10 Vertex figure 16 cell Petrie polygon decagon Coxeter groups BC5 3 3 3 4 D5 32 1 1 Dual 5 cube Properties convex Hanner polytope In five dimensional geometry a 5 orthoplex or 5 cross polytope is a five dimensional polytope with 10 vertices 40 edges 80 triangle faces 80 tetrahedron cells 32 5 cell 4 faces It has two constructed forms the first being regular with Schlafli symbol 33 4 and the second with alternately labeled checkerboarded facets with Schlafli symbol 3 3 31 1 or Coxeter symbol 211 It is a part of an infinite family of polytopes called cross polytopes or orthoplexes The dual polytope is the 5 hypercube or 5 cube Contents 1 Alternate names 2 As a configuration 3 Cartesian coordinates 4 Construction 5 Other images 6 Related polytopes and honeycombs 7 References 8 External linksAlternate names editpentacross derived from combining the family name cross polytope with pente for five dimensions in Greek Triacontaditeron or triacontakaiditeron as a 32 facetted 5 polytope polyteron As a configuration editThis configuration matrix represents the 5 orthoplex The rows and columns correspond to vertices edges faces cells and 4 faces The diagonal numbers say how many of each element occur in the whole 5 orthoplex The nondiagonal numbers say how many of the column s element occur in or at the row s element 1 2 10 8 24 32 16 2 40 6 12 8 3 3 80 4 4 4 6 4 80 2 5 10 10 5 32 displaystyle begin bmatrix begin matrix 10 amp 8 amp 24 amp 32 amp 16 2 amp 40 amp 6 amp 12 amp 8 3 amp 3 amp 80 amp 4 amp 4 4 amp 6 amp 4 amp 80 amp 2 5 amp 10 amp 10 amp 5 amp 32 end matrix end bmatrix nbsp Cartesian coordinates editCartesian coordinates for the vertices of a 5 orthoplex centered at the origin are 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 Construction editThere are three Coxeter groups associated with the 5 orthoplex one regular dual of the penteract with the C5 or 4 3 3 3 Coxeter group and a lower symmetry with two copies of 5 cell facets alternating with the D5 or 32 1 1 Coxeter group and the final one as a dual 5 orthotope called a 5 fusil which can have a variety of subsymmetries Name Coxeter diagram Schlafli symbol Symmetry Order Vertex figure s regular 5 orthoplex nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 3 3 3 4 3 3 3 4 3840 nbsp nbsp nbsp nbsp nbsp nbsp nbsp Quasiregular 5 orthoplex nbsp nbsp nbsp nbsp nbsp nbsp nbsp 3 3 31 1 3 3 31 1 1920 nbsp nbsp nbsp nbsp nbsp 5 fusil nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 3 3 3 4 4 3 3 3 3840 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 3 3 4 4 3 3 2 768 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 3 4 4 4 3 2 4 384 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 3 4 2 4 3 2 2 192 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 2 4 4 2 4 2 128 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 4 3 4 2 2 2 64 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 5 2 2 2 2 32 nbsp nbsp nbsp nbsp nbsp nbsp nbsp Other images editorthographic projections Coxeter plane B5 B4 D5 B3 D4 A2 Graph nbsp nbsp nbsp Dihedral symmetry 10 8 6 Coxeter plane B2 A3 Graph nbsp nbsp Dihedral symmetry 4 4 nbsp The perspective projection 3D to 2D of a stereographic projection 4D to 3D of the Schlegel diagram 5D to 4D of the 5 orthoplex 10 sets of 4 edges form 10 circles in the 4D Schlegel diagram two of these circles are straight lines in the stereographic projection because they contain the center of projection Related polytopes and honeycombs edit2k1 figures in n dimensions Space Finite Euclidean Hyperbolic n 3 4 5 6 7 8 9 10 Coxetergroup E3 A2A1 E4 A4 E5 D5 E6 E7 E8 E9 E 8 displaystyle tilde E 8 nbsp E8 E10 T 8 displaystyle bar T 8 nbsp E8 Coxeterdiagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Symmetry 3 1 2 1 30 2 1 31 2 1 32 2 1 33 2 1 34 2 1 35 2 1 36 2 1 Order 12 120 384 51 840 2 903 040 696 729 600 Graph nbsp nbsp nbsp nbsp nbsp nbsp Name 2 1 1 201 211 221 231 241 251 261 This polytope is one of 31 uniform 5 polytopes generated from the B5 Coxeter plane including the regular 5 cube and 5 orthoplex B5 polytopes nbsp b5 nbsp t1b5 nbsp t2g5 nbsp t1g5 nbsp g5 nbsp t0 1b5 nbsp t0 2b5 nbsp t1 2b5 nbsp t0 3b5 nbsp t1 3g5 nbsp t1 2g5 nbsp t0 4g5 nbsp t0 3g5 nbsp t0 2g5 nbsp t0 1g5 nbsp t0 1 2b5 nbsp t0 1 3b5 nbsp t0 2 3b5 nbsp t1 2 3g5 nbsp t0 1 4b5 nbsp t0 2 4g5 nbsp t0 2 3g5 nbsp t0 1 4g5 nbsp t0 1 3g5 nbsp t0 1 2g5 nbsp t0 1 2 3b5 nbsp t0 1 2 4b5 nbsp t0 1 3 4g5 nbsp t0 1 2 4g5 nbsp t0 1 2 3g5 nbsp t0 1 2 3 4g5References edit Coxeter Regular Polytopes sec 1 8 Configurations Coxeter Complex Regular Polytopes p 117 H S M Coxeter 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 Wiley Interscience Publication 1995 ISBN 978 0 471 01003 6 1 Paper 22 H S M Coxeter Regular and Semi Regular Polytopes I Math Zeit 46 1940 380 407 MR 2 10 Paper 23 H S M Coxeter Regular and Semi Regular Polytopes II Math Zeit 188 1985 559 591 Paper 24 H S M Coxeter Regular and Semi Regular Polytopes III Math Zeit 200 1988 3 45 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 x3o3o3o4o tac External links editOlshevsky George Cross polytope Glossary for Hyperspace Archived from the original on 4 February 2007 Polytopes of Various Dimensions Multi dimensional Glossary vteFundamental 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 p gon Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron Cube Demicube Dodecahedron Icosahedron Uniform polychoron Pentachoron 16 cell Tesseract Demitesseract 24 cell 120 cell 600 cell Uniform 5 polytope 5 simplex 5 orthoplex 5 cube 5 demicube Uniform 6 polytope 6 simplex 6 orthoplex 6 cube 6 demicube 122 221 Uniform 7 polytope 7 simplex 7 orthoplex 7 cube 7 demicube 132 231 321 Uniform 8 polytope 8 simplex 8 orthoplex 8 cube 8 demicube 142 241 421 Uniform 9 polytope 9 simplex 9 orthoplex 9 cube 9 demicube Uniform 10 polytope 10 simplex 10 orthoplex 10 cube 10 demicube Uniform n polytope n simplex n orthoplex n cube n demicube 1k2 2k1 k21 n pentagonal polytope Topics Polytope families Regular polytope List of regular polytopes and compounds Retrieved from https en wikipedia org w index php title 5 orthoplex amp oldid 1122323249, wikipedia, wiki, book, books, library,

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