Cantellated 5-cubes

Error creating thumbnail: 5-cube Template:CDD	Error creating thumbnail: Cantellated 5-cube Template:CDD	File:5-cube t13.svg Bicantellated 5-cube Template:CDD	File:5-cube t24.svg Cantellated 5-orthoplex Template:CDD
File:5-cube t4.svg 5-orthoplex Template:CDD	File:5-cube t012.svg Cantitruncated 5-cube Template:CDD	Error creating thumbnail: Bicantitruncated 5-cube Template:CDD	File:5-cube t234.svg Cantitruncated 5-orthoplex Template:CDD
Orthogonal projections in B5 Coxeter plane

In six-dimensional geometry, a cantellated 5-cube is a convex uniform 5-polytope, being a cantellation of the regular 5-cube.

There are 6 unique cantellation for the 5-cube, including truncations. Half of them are more easily constructed from the dual 5-orthoplex

Cantellated 5-cube

Cantellated 5-cube
Type	Uniform 5-polytope
Schläfli symbol	rr{4,3,3,3} = $r\left\{\begin{array}{c}4\\ 3,3,3\end{array}\right\}$
Coxeter-Dynkin diagram	Template:CDD = Template:CDD
4-faces	122	10 Template:CDD File:Schlegel half-solid cantellated 8-cell.png 80 Template:CDD 32 Template:CDD File:Schlegel half-solid rectified 5-cell.png
Cells	680	40 Template:CDD File:Uniform polyhedron-43-t02.png 320 Template:CDD File:Triangular prism wedge.png 160 Template:CDD File:Uniform polyhedron-33-t1.svg 160 Template:CDD File:Uniform polyhedron-33-t0.png
Faces	1520	80 Template:CDD File:2-cube.svg 480 Template:CDD File:2-cube.svg 320 Template:CDD File:2-simplex t0.svg 640 Template:CDD File:2-simplex t0.svg
Edges	1280	320+960
Vertices	320
Vertex figure
Coxeter group	B5 [4,3,3,3]
Properties	convex, uniform

Alternate names

• Small rhombated penteract (Acronym: sirn) (Jonathan Bowers)

The Cartesian coordinates of the vertices of a cantellated 5-cube having edge length 2 are all permutations of:

$(\pm 1,\pm 1,\pm (1+\sqrt{2}),\pm (1+\sqrt{2}),\pm (1+\sqrt{2}))$

Bicantellated 5-cube

Bicantellated 5-cube
Type	Uniform 5-polytope
Schläfli symbols	2rr{4,3,3,3} = $r\left\{\begin{array}{c}3,4\\ 3,3\end{array}\right\}$ r{32,1,1} = $r\left\{\begin{array}{c}3,3\\ 3\\ 3\end{array}\right\}$
Coxeter-Dynkin diagrams	Template:CDD = Template:CDD Template:CDD
4-faces	122	10 Template:CDD File:Schlegel half-solid cantellated 16-cell.png 80 Template:CDD File:4-3 duoprism.png 32 Template:CDD Error creating thumbnail:
Cells	840	40 Template:CDD File:Uniform polyhedron-43-t1.svg 240 Template:CDD File:Tetragonal prism.png 160 Template:CDD Error creating thumbnail: 320 Template:CDD File:Triangular prism wedge.png 80 Template:CDD File:Uniform polyhedron-33-t1.svg
Faces	2160	240 Template:CDD File:2-cube.svg 320 Template:CDD File:2-simplex t0.svg 960 Template:CDD File:2-cube.svg 320 Template:CDD File:2-simplex t0.svg 320 Template:CDD File:2-simplex t0.svg
Edges	1920	960+960
Vertices	480
Vertex figure	File:Bicantellated penteract verf.png
Coxeter groups	B5, [3,3,3,4] D5, [32,1,1]
Properties	convex, uniform

In five-dimensional geometry, a bicantellated 5-cube is a uniform 5-polytope.

Alternate names

• Bicantellated penteract, bicantellated 5-orthoplex, or bicantellated pentacross
• Small birhombated penteractitriacontiditeron (Acronym: sibrant) (Jonathan Bowers)

Coordinates

The Cartesian coordinates of the vertices of a bicantellated 5-cube having edge length 2 are all permutations of:

(0,1,1,2,2)

Images

Template:5-cube Coxeter plane graphs

Cantitruncated 5-cube

Cantitruncated 5-cube
Type	Uniform 5-polytope
Schläfli symbol	tr{4,3,3,3} = $t\left\{\begin{array}{c}4\\ 3,3,3\end{array}\right\}$
Coxeter-Dynkin diagram	Template:CDD = Template:CDD
4-faces	122	10 Template:CDD Error creating thumbnail: 80 Template:CDD Error creating thumbnail: 32 Template:CDD File:Schlegel half-solid truncated pentachoron.png
Cells	680	40 Template:CDD File:Uniform polyhedron-43-t012.png 320 Template:CDD File:Triangular prism wedge.png 160 Template:CDD File:Uniform polyhedron-33-t01.png 160 Template:CDD File:Uniform polyhedron-33-t0.png
Faces	1520	80 Template:CDD File:Regular octagon.svg 480 Template:CDD File:2-cube.svg 320 Template:CDD File:2-simplex t01.svg 640 Template:CDD File:2-simplex t0.svg
Edges	1600	320+320+960
Vertices	640
Vertex figure	File:Canitruncated 5-cube verf.png
Coxeter group	B5 [4,3,3,3]
Properties	convex, uniform

Alternate names

• Tricantitruncated 5-orthoplex / tricantitruncated pentacross
• Great rhombated penteract (girn) (Jonathan Bowers)

Coordinates

The Cartesian coordinates of the vertices of a cantitruncated 5-cube having an edge length of 2 are given by all permutations of coordinates and sign of:

$(1,1+\sqrt{2},1+2\sqrt{2},1+2\sqrt{2},1+2\sqrt{2})$

Related polytopes

It is third in a series of cantitruncated hypercubes: Template:Cantitruncated hypercube polytopes

Bicantitruncated 5-cube

Bicantitruncated 5-cube
Type	uniform 5-polytope
Schläfli symbol	2tr{3,3,3,4} = $t\left\{\begin{array}{c}3,4\\ 3,3\end{array}\right\}$ t{32,1,1} = $t\left\{\begin{array}{c}3,3\\ 3\\ 3\end{array}\right\}$
Coxeter-Dynkin diagrams	Template:CDD = Template:CDD Template:CDD
4-faces	122	10 Template:CDD Error creating thumbnail: 80 Template:CDD File:4-3 duoprism.png 32 Template:CDD File:Schlegel half-solid cantitruncated 5-cell.png
Cells	840	40 Template:CDD 240 Template:CDD File:Tetragonal prism.png 160 Template:CDD Error creating thumbnail: 320 Template:CDD File:Triangular prism wedge.png 80 Template:CDD File:Uniform polyhedron-33-t01.png
Faces	2160	240 Template:CDD File:2-cube.svg 320 Template:CDD File:2-simplex t01.svg 960 Template:CDD File:2-cube.svg 320 Template:CDD File:2-simplex t01.svg 320 Template:CDD File:2-simplex t0.svg
Edges	2400	960+480+960
Vertices	960
Vertex figure	File:Bicanitruncated 5-cube verf.png
Coxeter groups	B5, [3,3,3,4] D5, [32,1,1]
Properties	convex, uniform

Alternate names

• Bicantitruncated penteract
• Bicantitruncated pentacross
• Great birhombated penteractitriacontiditeron (Acronym: gibrant) (Jonathan Bowers)

Coordinates

Cartesian coordinates for the vertices of a bicantitruncated 5-cube, centered at the origin, are all sign and coordinate permutations of

(±3,±3,±2,±1,0)

Related polytopes

These polytopes are from a set of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.

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, Wiley-Interscience Publication, 1995, Template:ISBN [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.
• Template:KlitzingPolytopes o3o3x3o4x - sirn, o3x3o3x4o - sibrant, o3o3x3x4x - girn, o3x3x3x4o - gibrant

External links

• Template:PolyCell
• Polytopes of Various Dimensions, Jonathan Bowers
  • Runcinated uniform polytera (spid), Jonathan Bowers
• Multi-dimensional Glossary