Bicantitruncated_5-cube

Cantellated 5-cubes

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

More information Orthogonal projections in B5 Coxeter plane ...

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	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

Coordinates

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

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

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	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, [3^2,1,1]
Properties	convex, uniform

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

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	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

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:

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