Truncated 5-cube

Truncated 5-cube
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Tan
Coxeter diagram	x4x3o3o3o ()
Elements
Tera	32 pentachora, 10 truncated tesseracts
Cells	160 tetrahedra, 40 truncated cubes
Faces	320 triangles, 80 octagons
Edges	80+320
Vertices	160
Vertex figure	Tetrahedral pyramid, edge lengths 1 (base) and √2+√2 (sides)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {13+8{\sqrt {2}}}}{2}}\approx 2.46545$
Hypervolume	${\frac {1230+869{\sqrt {2}}}{30}}\approx 81.96505$
Diteral angles	Tat–tet–pen: $\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$
	Tat–tic–tat: 90°
Central density	1
Number of external pieces	42
Level of complexity	5
Related polytopes
Army	Tan
Regiment	Tan
Dual	Pentakis triacontaditeron
Conjugate	Quasitruncated penteract
Abstract & topological properties
Flag count	19200
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	B5, order 3840
Convex	Yes
Nature	Tame

The truncated 5-cube, also called the truncated penteract or tan, is a convex uniform 5-polytope. It consists of 32 regular pentachora and 10 truncated tesseracts. One pentachoron and 4 truncated tesseracts join at each vertex. As the name suggests, it is the truncation of the 5-cube.

Vertex coordinates

The vertices of a truncated 5-cube of edge length 1 are given by all permutations of:

• $\left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right)$.

Representations

A truncated 5-cube has the following Coxeter diagrams:

• x4x3o3o3o () (full symmetry)
• xwwx4xoox3oooo3oooo&#xt (B₄ axial, truncated tesseract-first)

Gallery

• 

  B₄ or D₅ orthographic projection

• 

  B₃, D₄, or A₂

• 

  B₂

• 

  A₃

External links

• Bowers, Jonathan. "Category 2: Truncates" (#14).

• Klitzing, Richard. "tan".
• Wikipedia contributors. "Truncated 5-cube".