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Runcic 5-cubes

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5-cube
Runcic 5-cube
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5-demicube
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Runcicantic 5-cube
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Orthogonal projections in B5 Coxeter plane

In six-dimensional geometry, a runcic 5-cube or (runcic 5-demicube, runcihalf 5-cube) is a convex uniform 5-polytope. There are 2 runcic forms for the 5-cube. Runcic 5-cubes have half the vertices of runcinated 5-cubes.

Runcic 5-cube

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Runcic 5-cube
Type uniform 5-polytope
Schläfli symbol h3{4,3,3,3}
Coxeter-Dynkin diagram        
         
4-faces 42
Cells 360
Faces 880
Edges 720
Vertices 160
Vertex figure
Coxeter groups D5, [32,1,1]
Properties convex

Alternate names

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  • Cantellated 5-demicube/demipenteract
  • Small rhombated hemipenteract (sirhin) (Jonathan Bowers)[1]

Cartesian coordinates

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The Cartesian coordinates for the 960 vertices of a runcic 5-cubes centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±3)

with an odd number of plus signs.

Images

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orthographic projections
Coxeter plane B5
Graph  
Dihedral symmetry [10/2]
Coxeter plane D5 D4
Graph    
Dihedral symmetry [8] [6]
Coxeter plane D3 A3
Graph    
Dihedral symmetry [4] [4]
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It has half the vertices of the runcinated 5-cube, as compared here in the B5 Coxeter plane projections:

 
Runcic 5-cube 		 
Runcinated 5-cube
Runcic n-cubes
n 4 5 6 7 8
[1+,4,3n-2]
= [3,3n-3,1] 		[1+,4,32]
= [3,31,1] 		[1+,4,33]
= [3,32,1] 		[1+,4,34]
= [3,33,1] 		[1+,4,35]
= [3,34,1] 		[1+,4,36]
= [3,35,1]
Runcic
figure 		         
Coxeter        
=       		         
=         		           
=           		             
=             		               
=              
Schläfli h3{4,32} h3{4,33} h3{4,34} h3{4,35} h3{4,36}

Runcicantic 5-cube

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Runcicantic 5-cube
Type uniform 5-polytope
Schläfli symbol t0,1,2{3,32,1}
h3{4,33}
Coxeter-Dynkin diagram                 
4-faces 42
Cells 360
Faces 1040
Edges 1200
Vertices 480
Vertex figure
Coxeter groups D5, [32,1,1]
Properties convex

Alternate names

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  • Cantitruncated 5-demicube/demipenteract
  • Great rhombated hemipenteract (girhin) (Jonathan Bowers)[2]

Cartesian coordinates

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The Cartesian coordinates for the 480 vertices of a runcicantic 5-cube centered at the origin are coordinate permutations:

(±1,±1,±3,±5,±5)

with an odd number of plus signs.

Images

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orthographic projections
Coxeter plane B5
Graph  
Dihedral symmetry [10/2]
Coxeter plane D5 D4
Graph    
Dihedral symmetry [8] [6]
Coxeter plane D3 A3
Graph    
Dihedral symmetry [4] [4]
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It has half the vertices of the runcicantellated 5-cube, as compared here in the B5 Coxeter plane projections:

 
Runcicantic 5-cube 		 
Runcicantellated 5-cube
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This polytope is based on the 5-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

There are 23 uniform 5-polytopes that can be constructed from the D5 symmetry of the 5-demicube, of which are unique to this family, and 15 are shared within the 5-cube family.

D5 polytopes
 
h{4,3,3,3} 		 
h2{4,3,3,3} 		 
h3{4,3,3,3} 		 
h4{4,3,3,3} 		 
h2,3{4,3,3,3} 		 
h2,4{4,3,3,3} 		 
h3,4{4,3,3,3} 		 
h2,3,4{4,3,3,3}

Notes

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  1. ^ Klitzing, (x3o3o *b3x3o - sirhin)
  2. ^ Klitzing, (x3x3o *b3x3o - girhin)

References

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  • 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)". x3o3o *b3x3o - sirhin, x3x3o *b3x3o - girhin
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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
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