Truncated hexateron

Truncated hexateron
(OFF file)
Rank	5
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Tix
Coxeter diagram	x3x3o3o3o ()
Elements
Tera	6 pentachora, 6 truncated pentachora
Cells	30 tetrahedra, 15 truncated tetrahedra
Faces	60 triangles, 20 hexagons
Edges	15+60
Vertices	30
Vertex figure	Tetrahedral pyramid, edge lengths 1 (base) and √3 (sides)
Measures (edge length 1)
Circumradius	$\frac{\sqrt7}{2} ≈ 1.32288$
Hypervolume	$\frac{79\sqrt3}{160} ≈ 0.85520$
Diteral angles	Tip–tet–pen: $\arccos\left(-\frac15\right) ≈ 101.53696^\circ$
	Tip–tut–tip: $\arccos\left(\frac15\right) ≈ 78.46304^\circ$
Central density	1
Number of external pieces	12
Level of complexity	5
Related polytopes
Army	Tix
Regiment	Tix
Dual	Pentakis hexateron
Conjugate	None
Abstract & topological properties
Flag count	3600
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A5, order 720
Convex	Yes
Nature	Tame

The truncated hexateron, or tix, also called the truncated 5-simplex, is a convex uniform polyteron. It consists of 6 regular pentachora and 6 truncated pentachora. One pentachoron and 4 truncated pentachora join at each vertex. As the name suggests, it is the truncation of the hexateron.

Vertex coordinates

The vertices of a truncated hexateron of edge length 1 can be given in 6 dimensions as all permutations of:

• $\left(\sqrt2,\,\frac{\sqrt2}{2},\,0,\,0,\,0,\,0\right).$

Representations

A truncated hexateron has the following Coxeter diagrams:

• x3x3o3o3o (full symmetry)
• xux3oox3ooo3ooo&#xt (A₄ axial, pentachoron-first)
• xuxo oxux3ooox3oooo&#xt (A₃×A₁ axial, edge-first)
• xuxo3xooo ooox3oxux&#xt (A₂×A₂ axial, hexagon-first)

Gallery

• 

  A₄ orthographic projection

• 

  A₃

• 

  A₂

External links

• Bowers, Jonathan. "Category 2: Truncates" (#13).

• Klitzing, Richard. "tix".

• Wikipedia Contributors. "Truncated 5-simplex".