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} = $r\left\{\begin{array}{l}4\\3, 3, 3\end{array}\right\}$
Coxeter-Dynkin diagram	=
4-faces	122
Cells	680
Faces	1520
Edges	1280
Vertices	320
Vertex figure
Coxeter group	B5 [4,3,3,3]
Properties	convex

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:

$\left(\pm1,\ \pm1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2})\right)$

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} = $r\left\{\begin{array}{l}3, 4\\3, 3\end{array}\right\}$ r{3^2,1,1} = $r\left\{\begin{array}{l}3, 3\\ 3\\3\end{array}\right\}$
Coxeter-Dynkin diagrams	=
4-faces	122
Cells	840
Faces	2160
Edges	1920
Vertices	480
Vertex figure
Coxeter group	B5 [4,3,3,3]
Properties	convex

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} = $t\left\{\begin{array}{l}4\\3, 3, 3\end{array}\right\}$
Coxeter-Dynkin diagram	=
4-faces	122
Cells	680
Faces	1520
Edges	1600
Vertices	640
Vertex figure	Irr. 5-cell
Coxeter group	B5 [4,3,3,3]
Properties	convex, isogonal

Alternate names

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

Coordinates

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

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

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]

Bicantitruncated 5-cube

Bicantitruncated 5-cube
Type	uniform 5-polytope
Schläfli symbol	2tr{3,3,3,4} = $t\left\{\begin{array}{l}3, 4\\3, 3\end{array}\right\}$ t{3^2,1,1} = $t\left\{\begin{array}{l}3, 3\\ 3\\3\end{array}\right\}$
Coxeter-Dynkin diagrams	=
4-faces	122
Cells	840
Faces	2160
Edges	2400
Vertices	960
Vertex figure
Coxeter groups	B5, [3,3,3,4] D5, [3^2,1,1]
Properties	convex

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.

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

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
    • (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.
• Richard Klitzing, 5D, uniform polytopes (polytera) 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