Cantellated 5-cubes

Cantellated 5-cube
Type	Uniform 5-polytope
Schläfli symbol	rr{4,3,3,3} = ${\displaystyle r\left\{{\begin{array}{l}4\\3,3,3\end{array}}\right\}}$
Coxeter-Dynkin diagram	=
4-faces	122	10           80           32
Cells	680	40         320         160         160
Faces	1520	80       480       320       640
Edges	1280	320+960
Vertices	320
Vertex figure
Coxeter group	B5 [4,3,3,3]
Properties	convex, uniform

• 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:

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

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
Type	Uniform 5-polytope
Schläfli symbols	2rr{4,3,3,3} = ${\displaystyle r\left\{{\begin{array}{l}3,4\\3,3\end{array}}\right\}}$  r{32,1,1} = ${\displaystyle r\left\{{\begin{array}{l}3,3\\3\\3\end{array}}\right\}}$
Coxeter-Dynkin diagrams	=
4-faces	122	10           80           32
Cells	840	40         240         160         320         80
Faces	2160	240       320       960       320       320
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.

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

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

(0,1,1,2,2)

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
Type	Uniform 5-polytope
Schläfli symbol	tr{4,3,3,3} = ${\displaystyle t\left\{{\begin{array}{l}4\\3,3,3\end{array}}\right\}}$
Coxeter-Dynkin diagram	=
4-faces	122	10           80           32
Cells	680	40         320         160         160
Faces	1520	80       480       320       640
Edges	1600	320+320+960
Vertices	640
Vertex figure
Coxeter group	B5 [4,3,3,3]
Properties	convex, uniform

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

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:

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

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]

It is third in a series of cantitruncated hypercubes:

Petrie polygon projections

Truncated cuboctahedron	Cantitruncated tesseract	Cantitruncated 5-cube	Cantitruncated 6-cube	Cantitruncated 7-cube	Cantitruncated 8-cube

Bicantitruncated 5-cube
Type	uniform 5-polytope
Schläfli symbol	2tr{3,3,3,4} = ${\displaystyle t\left\{{\begin{array}{l}3,4\\3,3\end{array}}\right\}}$  t{32,1,1} = ${\displaystyle t\left\{{\begin{array}{l}3,3\\3\\3\end{array}}\right\}}$
Coxeter-Dynkin diagrams	=
4-faces	122	10           80           32
Cells	840	40         240         160         320         80
Faces	2160	240       320       960       320       320
Edges	2400	960+480+960
Vertices	960
Vertex figure
Coxeter groups	B5, [3,3,3,4] D5, [32,1,1]
Properties	convex, uniform

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

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)

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]

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

• 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)". o3o3x3o4x - sirn, o3x3o3x4o - sibrant, o3o3x3x4x - girn, o3x3x3x4o - gibrant

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