Truncated hecatonicosachoron

Truncated hecatonicosachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Thi
Coxeter diagram	x5x3o3o ()
Elements
Cells	600 tetrahedra, 120 truncated dodecahedra
Faces	2400 triangles, 720 decagons
Edges	1200+3600
Vertices	2400
Vertex figure	Triangular pyramid, edge lengths 1 (base) and √(5+√5)/2 (sides)
Measures (edge length 1)
Circumradius	$\sqrt{34+15\sqrt5} ≈ 8.21833$
Hypervolume	$25\frac{1577+704\sqrt5}{4} ≈ 19694.9491$
Dichoral angles	Tid–3–tet: $\arccos\left(-\frac{\sqrt{7+3\sqrt5}}{4}\right) ≈ 157.76124^\circ$
	Tid–10–tid: 144°
Central density	1
Number of external pieces	720
Level of complexity	4
Related polytopes
Army	Thi
Regiment	Thi
Dual	Tetrakis hexacosichoron
Conjugate	Quasitruncated great grand stellated hecatonicosachoron
Abstract & topological properties
Flag count	57600
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	Yes
Nature	Tame

The truncated hecatonicosachoron, or thi, also commonly called the truncated 120-cell, is a convex uniform polychoron that consists of 600 regular tetrahedra and 120 truncated dodecahedra. 1 tetrahedron and three truncated dodecahedra join at each vertex. As the name suggests, it can be obtained by truncating the hecatonicosachoron.

Cross-sections

Vertex coordinates

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

• $\left(±\frac12,\,±\frac{5+2\sqrt5}2,\,±\frac{5+2\sqrt5}2,\,±\frac{5+2\sqrt5}2\right),$
• $\left(±\frac{2+\sqrt5}2,\,±\frac{2+\sqrt5}2,\,±\frac{2+\sqrt5}2,\,±\frac{8+3\sqrt5}2\right),$
• $\left(±\frac{3+\sqrt5}2,\,±\frac{3+\sqrt5}2,\,±\frac{3+\sqrt5}2,\,±\frac{7+3\sqrt5}2\right),$

along with all even permutations of:

• $\left(0,\,±\frac12,\,±\frac{13+5\sqrt5}4,\,±\frac{11+5\sqrt5}4\right),$
• $\left(0,\,±\frac12,\,±\frac{15+7\sqrt5}4,\,±\frac{5+3\sqrt5}4\right),$
• $\left(0,\,±\frac{3+\sqrt5}4,\,±3\frac{2+\sqrt5}2,\,±\frac{9+5\sqrt5}4\right),$
• $\left(0,\,±\frac{3+\sqrt5}4,\,±\frac{8+3\sqrt5}2,\,±\frac{7+3\sqrt5}4\right),$
• $\left(0,\,±\frac{1+\sqrt5}2,\,±\frac{7+3\sqrt5}2,\,±(2+\sqrt5)\right),$
• $\left(±\frac12,\,±\frac{3+\sqrt5}4,\,±\frac{15+7\sqrt5}4,\,±\frac{3+\sqrt5}2\right),$
• $\left(±\frac12,\,±\frac{2+\sqrt5}2,\,±3\frac{2+\sqrt5}2,\,±\frac{5+2\sqrt5}2\right),$
• $\left(±\frac{3+\sqrt5}4,\,±\frac{1+\sqrt5}2,\,±\frac{15+7\sqrt5}4,\,±\frac{2+\sqrt5}2\right),$
• $\left(±\frac{3+\sqrt5}4,\,±\frac{3+\sqrt5}2,\,±\frac{13+5\sqrt5}4,\,±\frac{5+2\sqrt5}2\right),$
• $\left(±\frac{3+\sqrt5}4,\,±(2+\sqrt5),\,±\frac{9+5\sqrt5}4,\,±\frac{5+2\sqrt5}2\right),$
• $\left(±\frac{1+\sqrt5}2,\,±\frac{7+3\sqrt5}4,\,±\frac{11+5\sqrt5}4,\,±\frac{5+2\sqrt5}2\right),$
• $\left(±\frac{2+\sqrt5}2,\,±\frac{3+\sqrt5}2,\,±\frac{11+5\sqrt5}4,\,±\frac{9+5\sqrt5}4\right),$
• $\left(±\frac{2+\sqrt5}2,\,±\frac{5+3\sqrt5}4,\,±\frac{13+5\sqrt5}4,\,±(2+\sqrt5)\right),$
• $\left(±\frac{3+\sqrt5}2,\,±\frac{5+3\sqrt5}4,\,±3\frac{2+\sqrt5}2,\,±\frac{7+3\sqrt5}4\right).$

Semi-uniform variant

The truncated hecatonicosachoron has a semi-uniform variant of the form x5y3o3o that maintains its full symmetry. This variant uses 600 tetrahedra of size y and 120 semi-uniform truncated dodecahedra of form x5y3o as cells, with 2 edge lengths. With edges of length a (surrounded by truncated dodecahedra only) and b (of tetrahedra), its circumradius is given by $\sqrt{\frac{14a^2+21b^2+33ab+(6a^2+9b^2+15ab)\sqrt5}{2}}$.

External links

• Bowers, Jonathan. "Category 2: Truncates" (#22).

• Klitzing, Richard. "thi".

• Quickfur. "The Truncated 120-cell".

• Wikipedia Contributors. "Truncated 120-cell".