Cantellated 5-Cubes

5-cube	Cantellated 5-cube	Bicantellated 5-cube	Cantellated 5-orthoplex
5-orthoplex	Cantitruncated 5-cube	Bicantitruncated 5-cube	Cantitruncated 5-orthoplex
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} = [math]\displaystyle{ r\left\{\begin{array}{l}4\\3, 3, 3\end{array}\right\} }[/math]
Coxeter-Dynkin diagram	=
4-faces	122	10 80 36px|link=Tetrahedral prism 32
Cells	680	40 320 25px|link=Triangular prism 160 25px|link=Octahedron 160
Faces	1520	80 25px|link=Square 480 25px|link=Square 320 25px|link=Triangle 640 25px|link=Triangle
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)

Coordinates

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

[math]\displaystyle{ \left(\pm1,\ \pm1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2})\right) }[/math]

Images

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]

Bicantellated 5-cube

Bicantellated 5-cube
Type	Uniform 5-polytope
Schläfli symbols	2rr{4,3,3,3} = [math]\displaystyle{ r\left\{\begin{array}{l}3, 4\\3, 3\end{array}\right\} }[/math] r{32,1,1} = [math]\displaystyle{ r\left\{\begin{array}{l}3, 3\\ 3\\3\end{array}\right\} }[/math]
Coxeter-Dynkin diagrams	=
4-faces	122	10 80 36px|link=3-4 duoprism 32
Cells	840	40 240 25px|link=Cube 160 25px|link=Cuboctahedron 320 25px|link=Triangular prism 80
Faces	2160	240 25px|link=Square 320 25px|link=Triangle 960 25px|link=Square 320 25px|link=Triangle 320 25px|link=Triangle
Edges	1920	960+960
Vertices	480
Vertex figure
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

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]

Cantitruncated 5-cube

Cantitruncated 5-cube
Type	Uniform 5-polytope
Schläfli symbol	tr{4,3,3,3} = [math]\displaystyle{ t\left\{\begin{array}{l}4\\3, 3, 3\end{array}\right\} }[/math]
Coxeter-Dynkin diagram	=
4-faces	122	10 80 36px|link=Tetrahedral prism 32
Cells	680	40 320 25px|link=Triangular prism 160 25px|link=Truncated tetrahedron 160
Faces	1520	80 480 25px|link=Square 320 25px|link=Hexagon 640
Edges	1600	320+320+960
Vertices	640
Vertex figure
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:

[math]\displaystyle{ \left(1,\ 1+\sqrt{2},\ 1+2\sqrt{2},\ 1+2\sqrt{2},\ 1+2\sqrt{2}\right) }[/math]

Images

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]

Related polytopes

It is third in a series of cantitruncated hypercubes:

Bicantitruncated 5-cube

Bicantitruncated 5-cube
Type	uniform 5-polytope
Schläfli symbol	2tr{3,3,3,4} = [math]\displaystyle{ t\left\{\begin{array}{l}3, 4\\3, 3\end{array}\right\} }[/math] t{32,1,1} = [math]\displaystyle{ t\left\{\begin{array}{l}3, 3\\ 3\\3\end{array}\right\} }[/math]
Coxeter-Dynkin diagrams	=
4-faces	122	10 80 36px|link=3-4 duoprism 32
Cells	840	40 240 25px|link=Cube 160 25px|link=Truncated octahedron 320 25px|link=Triangular prism 80
Faces	2160	240 25px|link=Square 320 25px|link=Hexagon 960 25px|link=Square 320 25px|link=Hexagon 320
Edges	2400	960+480+960
Vertices	960
Vertex figure
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)

Images

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]

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, 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)". https://bendwavy.org/klitzing/dimensions/polytera.htm.  o3o3x3o4x - sirn, o3x3o3x4o - sibrant, o3o3x3x4x - girn, o3x3x3x4o - gibrant

External links

• Glossary for hyperspace, George Olshevsky.
• Polytopes of Various Dimensions, Jonathan Bowers
  • Runcinated uniform polytera (spid), Jonathan Bowers
• Multi-dimensional Glossary

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