# Truncated hecatonicosachoron

Truncated hecatonicosachoron Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymThi
Coxeter diagramx5x3o3o (       )
Elements
Cells600 tetrahedra, 120 truncated dodecahedra
Faces2400 triangles, 720 decagons
Edges1200+3600
Vertices2400
Vertex figureTriangular 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 anglesTid–3–tet: $\arccos\left(-\frac{\sqrt{7+3\sqrt5}}{4}\right) ≈ 157.76124^\circ$ Tid–10–tid: 144°
Central density1
Number of external pieces720
Level of complexity4
Related polytopes
ArmyThi
RegimentThi
DualTetrakis hexacosichoron
ConjugateQuasitruncated great grand stellated hecatonicosachoron
Abstract & topological properties
Flag count57600
Euler characteristic0
OrientableYes
Properties
SymmetryH4, order 14400
ConvexYes
NatureTame

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.

## 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 $\sqrt{\frac{14a^2+21b^2+33ab+(6a^2+9b^2+15ab)\sqrt5}{2}}$ .