Stericated 5cubes
5cube 
Stericated 5cube 
Steritruncated 5cube 
Stericantellated 5cube 
Steritruncated 5orthoplex 
Stericantitruncated 5cube 
Steriruncitruncated 5cube 
Stericantitruncated 5orthoplex 
Omnitruncated 5cube 
Orthogonal projections in B_{5} Coxeter plane 

In fivedimensional geometry, a stericated 5cube is a convex uniform 5polytope with fourthorder truncations (sterication) of the regular 5cube.
There are eight degrees of sterication for the 5cube, including permutations of runcination, cantellation, and truncation. The simple stericated 5cube is also called an expanded 5cube, with the first and last nodes ringed, for being constructible by an expansion operation applied to the regular 5cube. The highest form, the steriruncicantitruncated 5cube, is more simply called an omnitruncated 5cube with all of the nodes ringed.
Contents
Stericated 5cube
Stericated 5cube  
Type  Uniform 5polytope  
Schläfli symbol  2r2r{4,3,3,3}  
CoxeterDynkin diagram 


4faces  242  
Cells  800  
Faces  1040  
Edges  640  
Vertices  160  
Vertex figure  
Coxeter group  B_{5} [4,3,3,3]  
Properties  convex 
Alternate names
 Stericated penteract / Stericated 5orthoplex / Stericated pentacross
 Expanded penteract / Expanded 5orthoplex / Expanded pentacross
 Small cellated penteract (Acronym: scan) (Jonathan Bowers)^{[1]}
Coordinates
The Cartesian coordinates of the vertices of a stericated 5cube having edge length 2 are all permutations of:
Images
The stericated 5cube is constructed by a sterication operation applied to the 5cube.
Coxeter plane  B_{5}  B_{4} / D_{5}  B_{3} / D_{4} / A_{2} 

Graph  
Dihedral symmetry  [10]  [8]  [6] 
Coxeter plane  B_{2}  A_{3}  
Graph  
Dihedral symmetry  [4]  [4] 
Steritruncated 5cube
Steritruncated 5cube  

Type  uniform 5polytope 
Schläfli symbol  t_{0,1,4}{4,3,3,3} 
CoxeterDynkin diagrams  
4faces  242 
Cells  1600 
Faces  2960 
Edges  2240 
Vertices  640 
Vertex figure  
Coxeter groups  B_{5}, [3,3,3,4] 
Properties  convex 
Alternate names
 Steritruncated penteract
 Prismatotruncated penteract (Acronym: capt) (Jonathan Bowers)^{[2]}
Construction and coordinates
The Cartesian coordinates of the vertices of a steritruncated 5cube having edge length 2 are all permutations of:
Images
Coxeter plane  B_{5}  B_{4} / D_{5}  B_{3} / D_{4} / A_{2} 

Graph  
Dihedral symmetry  [10]  [8]  [6] 
Coxeter plane  B_{2}  A_{3}  
Graph  
Dihedral symmetry  [4]  [4] 
Stericantellated 5cube
Stericantellated 5cube  
Type  Uniform 5polytope  
Schläfli symbol  t_{0,2,4}{4,3,3,3}  
CoxeterDynkin diagram 


4faces  242  
Cells  2080  
Faces  4720  
Edges  3840  
Vertices  960  
Vertex figure  
Coxeter group  B_{5} [4,3,3,3]  
Properties  convex 
Alternate names
 Stericantellated penteract
 Stericantellated 5orthoplex, stericantellated pentacross
 Cellirhombated penteractitriacontiditeron (Acronym: carnit) (Jonathan Bowers)^{[3]}
Coordinates
The Cartesian coordinates of the vertices of a stericantellated 5cube having edge length 2 are all permutations of:
Images
Coxeter plane  B_{5}  B_{4} / D_{5}  B_{3} / D_{4} / A_{2} 

Graph  
Dihedral symmetry  [10]  [8]  [6] 
Coxeter plane  B_{2}  A_{3}  
Graph  
Dihedral symmetry  [4]  [4] 
Stericantitruncated 5cube
Stericantitruncated 5cube  
Type  Uniform 5polytope  
Schläfli symbol  t_{0,1,2,4}{4,3,3,3}  
CoxeterDynkin diagram 

4faces  242  
Cells  2400  
Faces  6000  
Edges  5760  
Vertices  1920  
Vertex figure  
Coxeter group  B_{5} [4,3,3,3]  
Properties  convex, isogonal 
Alternate names
 Stericantitruncated penteract
 Steriruncicantellated 16cell / Biruncicantitruncated pentacross
 Celligreatorhombated penteract (cogrin) (Jonathan Bowers)^{[4]}
Coordinates
The Cartesian coordinates of the vertices of an stericantitruncated 5cube having an edge length of 2 are given by all permutations of coordinates and sign of:
Images
Coxeter plane  B_{5}  B_{4} / D_{5}  B_{3} / D_{4} / A_{2} 

Graph  
Dihedral symmetry  [10]  [8]  [6] 
Coxeter plane  B_{2}  A_{3}  
Graph  
Dihedral symmetry  [4]  [4] 
Steriruncitruncated 5cube
Steriruncitruncated 5cube  
Type  Uniform 5polytope  
Schläfli symbol  2t2r{4,3,3,3}  
CoxeterDynkin diagram 


4faces  242  
Cells  2160  
Faces  5760  
Edges  5760  
Vertices  1920  
Vertex figure  
Coxeter group  B_{5} [4,3,3,3]  
Properties  convex, isogonal 
Alternate names
 Steriruncitruncated penteract / Steriruncitruncated 5orthoplex / Steriruncitruncated pentacross
 Celliprismatotruncated penteractitriacontiditeron (captint) (Jonathan Bowers)^{[5]}
Coordinates
The Cartesian coordinates of the vertices of an steriruncitruncated penteract having an edge length of 2 are given by all permutations of coordinates and sign of:
Images
Coxeter plane  B_{5}  B_{4} / D_{5}  B_{3} / D_{4} / A_{2} 

Graph  
Dihedral symmetry  [10]  [8]  [6] 
Coxeter plane  B_{2}  A_{3}  
Graph  
Dihedral symmetry  [4]  [4] 
Steritruncated 5orthoplex
Steritruncated 5orthoplex  

Type  uniform 5polytope 
Schläfli symbol  t_{0,1,4}{3,3,3,4} 
CoxeterDynkin diagrams  
4faces  242 
Cells  1520 
Faces  2880 
Edges  2240 
Vertices  640 
Vertex figure  
Coxeter group  B_{5}, [3,3,3,4] 
Properties  convex 
Alternate names
 Steritruncated pentacross
 Celliprismated penteract (Acronym: cappin) (Jonathan Bowers)^{[6]}
Coordinates
Cartesian coordinates for the vertices of a steritruncated 5orthoplex, centered at the origin, are all permutations of
Images
Coxeter plane  B_{5}  B_{4} / D_{5}  B_{3} / D_{4} / A_{2} 

Graph  
Dihedral symmetry  [10]  [8]  [6] 
Coxeter plane  B_{2}  A_{3}  
Graph  
Dihedral symmetry  [4]  [4] 
Stericantitruncated 5orthoplex
Stericantitruncated 5orthoplex  
Type  Uniform 5polytope  
Schläfli symbol  t_{0,2,3,4}{4,3,3,3}  
CoxeterDynkin diagram 

4faces  242  
Cells  2320  
Faces  5920  
Edges  5760  
Vertices  1920  
Vertex figure  
Coxeter group  B_{5} [4,3,3,3]  
Properties  convex, isogonal 
Alternate names
 Stericantitruncated pentacross
 Celligreatorhombated pentacross (cogart) (Jonathan Bowers)^{[7]}
Coordinates
The Cartesian coordinates of the vertices of an stericantitruncated 5orthoplex having an edge length of 2 are given by all permutations of coordinates and sign of:
Images
Coxeter plane  B_{5}  B_{4} / D_{5}  B_{3} / D_{4} / A_{2} 

Graph  
Dihedral symmetry  [10]  [8]  [6] 
Coxeter plane  B_{2}  A_{3}  
Graph  
Dihedral symmetry  [4]  [4] 
Omnitruncated 5cube
Omnitruncated 5cube  
Type  Uniform 5polytope  
Schläfli symbol  tr2r{4,3,3,3}  
CoxeterDynkin diagram 


4faces  242  
Cells  2640  
Faces  8160  
Edges  9600  
Vertices  3840  
Vertex figure 
irr. {3,3,3} 

Coxeter group  B_{5} [4,3,3,3]  
Properties  convex, isogonal 
Alternate names
 Steriruncicantitruncated 5cube (Full expansion of omnitruncation for 5polytopes by Johnson)
 Omnitruncated penteract
 Omnitruncated 16cell / omnitruncated pentacross
 Great cellated penteractitriacontiditeron (Jonathan Bowers)^{[8]}
Coordinates
The Cartesian coordinates of the vertices of an omnitruncated tesseract having an edge length of 2 are given by all permutations of coordinates and sign of:
Images
Coxeter plane  B_{5}  B_{4} / D_{5}  B_{3} / D_{4} / A_{2} 

Graph  
Dihedral symmetry  [10]  [8]  [6] 
Coxeter plane  B_{2}  A_{3}  
Graph  
Dihedral symmetry  [4]  [4] 
Related polytopes
This polytope is one of 31 uniform 5polytopes generated from the regular 5cube or 5orthoplex.
Notes
References

H.S.M. Coxeter:
 H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973

Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied 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]

Norman Johnson Uniform Polytopes, Manuscript (1991)
 N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
 Klitzing, Richard. "5D uniform polytopes (polytera)". x3o3o3o4x  scan, x3o3o3x4x  capt, x3o3x3o4x  carnit, x3o3x3x4x  cogrin, x3x3o3x4x  captint, x3x3x3x4x  gacnet, x3x3x3o4x  cogart
External links
 Glossary for hyperspace, George Olshevsky.
 Polytopes of Various Dimensions, Jonathan Bowers
 Multidimensional Glossary