Truncated hexacosichoron

Truncated hexacosichoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Tex
Coxeter diagram	o5o3x3x ()
Elements
Cells	600 truncated tetrahedra, 120 icosahedra
Faces	2400 triangles, 1200 hexagons
Edges	720+3600
Vertices	1440
Vertex figure	Pentagonal pyramid, edge lengths 1 (base) and √3 (sides)
Measures (edge length 1)
Circumradius	$\sqrt{\frac{23+9\sqrt5}{2}} ≈ 4.64352$
Hypervolume	$25\frac{161+80\sqrt5}{4} ≈ 2124.28399$
Dichoral angles	Tut–6–tut: $\arccos\left(-\frac{1+3\sqrt5}{8}\right) ≈ 164.47751^\circ$
	Ike–3–tut: $\arccos\left(-\frac{\sqrt{7+3\sqrt5}}{4}\right) ≈ 157.76124^\circ$
Central density	1
Number of external pieces	729
Level of complexity	4
Related polytopes
Army	Tex
Regiment	Tex
Dual	Dodecakis hecatonicosachoron
Conjugate	Truncated grand hexacosichoron
Abstract & topological properties
Flag count	57600
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	Yes
Nature	Tame

The truncated hexacosichoron, or tex, also commonly called the truncated 600-cell, is a convex uniform polychoron that consists of 120 regular icosahedra and 600 truncated tetrahedra. 1 icosahedron and five truncated tetrahedra join at each vertex. As the name suggests, it can be obtained as the truncation of a hexacosichoron.

It is also isogonal under H4/5 symmetry, with the icosahedra having the symmetry of snub tetrahedra, and 480 of the truncated tetrahedra having trigonal symmetry only.

Cross-sections

Vertex coordinates

The vertices of a truncated hexacosichoron of edge length 1 are given by all even permutations of:

• $\left(0,\,±\frac12,\,±\frac{1+\sqrt5}{4},\,±\frac{7+5\sqrt5}{4}\right),$
• $\left(0,\,±\frac12,\,±3\frac{1+\sqrt5}{4},\,±3\frac{3+\sqrt5}{4}\right),$
• $\left(0,\,±1,\,±\frac{1+\sqrt5}{2},\,±(2+\sqrt5)\right),$
• $\left(0,\,±\frac{5+\sqrt5}{4},\,±\frac{5+3\sqrt5}{4},\,±\frac{4+\sqrt5}{2}\right),$
• $\left(0,\,±\frac{2+\sqrt5}{2},\,±\frac{7+3\sqrt5}{4},\,±\frac{7+\sqrt5}{4}\right),$
• $\left(±\frac12,\,±1,\,±\frac{5+3\sqrt5}{4},\,±\frac{7+3\sqrt5}{4}\right),$
• $\left(±\frac12,\,±\frac{1+\sqrt5}{2},\,±3\frac{3+\sqrt5}{4},\,±\frac{5+\sqrt5}{4}\right),$
• $\left(±\frac{1+\sqrt5}{4},\,±1,\,±\frac{2+\sqrt5}{2},\,±3\frac{3+\sqrt5}{4}\right),$
• $\left(±\frac{1+\sqrt5}{4},\,±\frac{5+\sqrt5}{4},\,±3\frac{1+\sqrt5}{4},\,±\frac{7+3\sqrt5}{4}\right),$
• $\left(±\frac{1+\sqrt5}{2},\,±\frac{2+\sqrt5}{2},\,±3\frac{1+\sqrt5}{4},\,±\frac{5+3\sqrt5}{4}\right).$

Semi-uniform variant

The truncated hexacosichoron has a semi-uniform variant of the form o5o3y3x that maintains its full symmetry. This variant uses 120 icosahedra of size y and 600 semi-uniform truncated tetrahedra of form x3y3o as cells, with 2 edge lengths.

With edges of length a (surrounded by truncated tetrahedra only) and b (of icosahedra), its circumradius is given by $\sqrt{\frac{3a^2+10b^2+10ab+(a^2+4b^2+4ab)\sqrt5}{2}}$.

Related polychora

The truncated hexacosichoron is the colonel of a two-member regiment that also includes the truncated faceted hexacosichoron.

Uniform polychoron compounds composed of truncated hexacosichora include:

• Truncated chirotetrahedral dishexacosichoron (2)
• Truncated snub pentishecatonicosachoron (5)
• Truncated snub decahecatonicosachoron (10)

External links

• Bowers, Jonathan. "Category 2: Truncates" (#23).

• Klitzing, Richard. "tex".

• Quickfur. "The Truncated 600-cell".

• Wikipedia Contributors. "Truncated 600-cell".