Hexacosihecatonicosachoron

Hexacosihecatonicosachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Xhi
Coxeter diagram	o5x3x3o ()
Elements
Cells	600 truncated tetrahedra, 120 truncated icosahedra
Faces	1200 triangles, 720 pentagons, 2400 hexagons
Edges	3600+3600
Vertices	3600
Vertex figure	Digonal disphenoid, edge lengths 1 (base 1), (1+√5)/2 (base 2) and √3 (sides)
Measures (edge length 1)
Circumradius	$\sqrt{\frac{59+25\sqrt5}{2}} ≈ 7.57963$
Hypervolume	$5\frac{5845+2726\sqrt5}{4} ≈ 14925.6516$
Dichoral angles	Tut–3–tut: $\arccos\left(-\frac{1+3\sqrt5}{8}\right) ≈ 164.47751^\circ$
	Ti–6–tut: $\arccos\left(-\frac{\sqrt{7+3\sqrt5}}{4}\right) ≈ 157.76124^\circ$
	Ti–5–ti: 144°
Central density	1
Number of external pieces	720
Level of complexity	6
Related polytopes
Army	Xhi
Regiment	Xhi
Dual	Disphenoidal trischiliahexacosichoron
Conjugate	Great hexacosihecatonicosachoron
Abstract & topological properties
Flag count	86400
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	Yes
Nature	Tame

The hexacosihecatonicosachoron, or xhi, also commonly called the bitruncated 120-cell, is a convex uniform polychoron that consists of 600 truncated tetrahedra and 120 truncated icosahedra. 2 truncated tetrahedra and 2 truncated icosahedra join at each vertex. It is the medial stage of the truncation series between a hecatonicosachoron and its dual hexacosichoron. As such, it could also be called a bitruncated 600-cell.

Cross-sections

Vertex coordinates

Coordinates for the vertices of a hexacosihecatonicosachoron of edge length 1 are given by all permutations of:

• $\left(0,\,0,\,±(1+\sqrt5),\,±\frac{7+3\sqrt5}{2}\right),$
• $\left(±\frac{5+\sqrt5}{4},\,±\frac{5+\sqrt5}{4},\,±\frac{9+5\sqrt5}{4},\,±\frac{9+5\sqrt5}{4}\right),$

together with all even permutations of:

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

Semi-uniform variant

The hexacosihecatonicosachoron has a semi-uniform variant of the form o5x3y3o that maintains its full symmetry. This variant uses 600 semi-uniform truncated tetrahedra of form x3y3o and 120 semi-uniform truncated icosahedra of form o5x3y as cells, with 2 edge lengths.

With edges of length a (of pentagonal faces) and b (of triangular faces), its circumradius is given by $\sqrt{\frac{21a^2+10b^2+28ab+(9a^2+4b^2+12ab)\sqrt5}{2}}$.

External links

• Bowers, Jonathan. "Category 7: Bitruncates" (#301).

• Klitzing, Richard. "xhi".

• Quickfur. "The Bitruncated 120-cell".

• Wikipedia Contributors. "Bitruncated 120-cell".