User:Tomruen/Truncated 5-orthoplexes


5-orthoplex

Truncated 5-orthoplex

Bitruncated 5-orthoplex

5-cube

Truncated 5-cube

Bitruncated 5-cube
Orthogonal projections in BC5 Coxeter plane

In six-dimensional geometry, a truncated 5-orthoplex is a convex uniform 5-polytope, being a truncation of the regular 5-orthoplex.

There are 4 unique truncations of the 5-orthoplex. Vertices of the truncation 5-orthoplex are located as pairs on the edge of the 5-orthoplex. Vertices of the bitruncated 5-orthoplex are located on the triangular faces of the 5-orthoplex. The third and fourth truncations are more easily constructed as second and first truncations of the 5-cube.

Truncated 5-orthoplex

edit
Truncated 5-orthoplex
Type uniform 5-polytope
Schläfli symbol t{3,3,3,4}
t{3,31,1}
Coxeter-Dynkin diagrams          
       
4-faces 42
Cells 240
Faces 400
Edges 280
Vertices 80
Vertex figure  
Elongated octahedral pyramid
Coxeter groups BC5, [3,3,3,4]
D5, [32,1,1]
Properties convex

Alternate names

edit
  • Truncated pentacross
  • Truncated triacontiditeron (Acronym: tot) (Jonathan Bowers)[1]

Coordinates

edit

Cartesian coordinates for the vertices of a truncated 5-orthoplex, centered at the origin, are all 80 vertices are sign (4) and coordinate (20) permutations of

(±2,±1,0,0,0)

Images

edit

The trunacted 5-orthoplex is constructed by a truncation operation applied to the 5-orthoplex. All edges are shortened, and two new vertices are added on each original edge.

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]

Bitruncated 5-orthoplex

edit
Bitruncated 5-orthoplex
Type uniform 5-polytope
Schläfli symbol 2t{3,3,3,4}
2t{3,31,1}
Coxeter-Dynkin diagrams          
       
4-faces 42
Cells 280
Faces 720
Edges 720
Vertices 240
Vertex figure  
square-pyramidal pyramid
Coxeter groups BC5, [3,3,3,4]
D5, [32,1,1]
Properties convex

The bitruncated 5-orthoplex can tessellate space in the tritruncated 5-cubic honeycomb.

Alternate names

edit
  • Bitruncated pentacross
  • Bitruncated triacontiditeron (acronym: gart) (Jonathan Bowers)[2]

Coordinates

edit

Cartesian coordinates for the vertices of a truncated 5-orthoplex, centered at the origin, are all 80 vertices are sign and coordinate permutations of

(±2,±2,±1,0,0)

Images

edit

The bitrunacted 5-orthoplex is constructed by a bitruncation operation applied to the 5-orthoplex. All edges are shortened, and two new vertices are added on each original edge.

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]
edit

This polytope is one of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.

B5 polytopes
 
β5
 
t1β5
 
t2γ5
 
t1γ5
 
γ5
 
t0,1β5
 
t0,2β5
 
t1,2β5
 
t0,3β5
 
t1,3γ5
 
t1,2γ5
 
t0,4γ5
 
t0,3γ5
 
t0,2γ5
 
t0,1γ5
 
t0,1,2β5
 
t0,1,3β5
 
t0,2,3β5
 
t1,2,3γ5
 
t0,1,4β5
 
t0,2,4γ5
 
t0,2,3γ5
 
t0,1,4γ5
 
t0,1,3γ5
 
t0,1,2γ5
 
t0,1,2,3β5
 
t0,1,2,4β5
 
t0,1,3,4γ5
 
t0,1,2,4γ5
 
t0,1,2,3γ5
 
t0,1,2,3,4γ5

Notes

edit
  1. ^ Klitzing, (x3x3o3o4o - tot)
  2. ^ Klitzing, (x3x3x3o4o - gart)

References

edit
  • 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.
  • Klitzing, Richard. "5D uniform polytopes (polytera)". x3x3o3o4o - tot, x3x3x3o4o - gart
edit
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds

Category:5-polytopes