Truncated 5-cubes

• Truncated penteract (Acronym: tan) (Jonathan Bowers)

The truncated 5-cube may be constructed by truncating the vertices of the 5-cube at ${\displaystyle 1/({\sqrt {2}}+2)}$ of the edge length. A regular 5-cell is formed at each truncated vertex.

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

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

The truncated 5-cube is constructed by a truncation applied to the 5-cube. All edges are shortened, and two new vertices are added on each original edge.

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]

The truncated 5-cube, is fourth in a sequence of truncated hypercubes:

Truncated hypercubes

Image								...
Name	Octagon	Truncated cube	Truncated tesseract	Truncated 5-cube	Truncated 6-cube	Truncated 7-cube	Truncated 8-cube
Coxeter diagram
Vertex figure	( )v( )	( )v{ }	( )v{3}	( )v{3,3}	( )v{3,3,3}	( )v{3,3,3,3}	( )v{3,3,3,3,3}

• Bitruncated penteract (Acronym: bittin) (Jonathan Bowers)

The bitruncated 5-cube may be constructed by bitruncating the vertices of the 5-cube at ${\displaystyle {\sqrt {2}}}$ of the edge length.

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

${\displaystyle \left(0,\ \pm 1,\ \pm 2,\ \pm 2,\ \pm 2\right)}$

The bitruncated 5-cube is third in a sequence of bitruncated hypercubes:

Bitruncated hypercubes

Image							...
Name	Bitruncated cube	Bitruncated tesseract	Bitruncated 5-cube	Bitruncated 6-cube	Bitruncated 7-cube	Bitruncated 8-cube
Coxeter
Vertex figure	( )v{ }	{ }v{ }	{ }v{3}	{ }v{3,3}	{ }v{3,3,3}	{ }v{3,3,3,3}