Faceted hexacosichoron

Faceted hexacosichoron
(OFF file)
Rank	4
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Fix
Coxeter diagram	o5/2o5o3x ()
Schläfli symbol	{3,5,5/2}
Elements
Cells	120 icosahedra
Faces	1200 triangles
Edges	720
Vertices	120
Vertex figure	Great dodecahedron, edge length 1
Edge figure	ike 3 ike 3 ike 3 ike 3 ike 3
Measures (edge length 1)
Circumradius	$\frac{1+\sqrt5}{2} ≈ 1.61803$
Edge radius	$\frac{\sqrt{5+2\sqrt5}}{2} ≈ 1.53884$
Face radius	$\frac{3\sqrt3+\sqrt{15}}{6} ≈ 1.51152$
Inradius	$\frac{3+\sqrt5}{4} ≈ 1.30902$
Hypervolume	$25\frac{7+3\sqrt5}{4} ≈ 85.67627$
Dichoral angle	120°
Central density	4
Number of pieces	2400
Level of complexity	4
Related polytopes
Army	Ex
Regiment	Ex
Company	Ex
Dual	Small stellated hecatonicosachoron
Conjugate	Great faceted hexacosichoron
Convex core	Hecatonicosachoron
Abstract properties
Euler characteristic	480
Topological properties
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

The faceted hexacosichoron, or fix, also commonly called the icosahedral 120-cell, is one of the 10 Schläfli–Hess polychora. It has 120 icosahedra as cells, joining 5 to an edge and 12 to a vertex in the form of a great dodecahedron.

As the name suggests, it is a faceting of the hexacosichoron, sharing its vertices, edges, and faces. The icosahedral cells are the same as the vertex figures of the hexacosichoron.

Cross-sections

Vertex coordinates

Its vertices are the same as those of its regiment colonel, the hexacosichoron.

External links

• Bowers, Jonathan. "Category 1: Regular Polychora" (#7).

• Klitzing, Richard. "fix".

• Nan Ma. "Icosahedral 120-cell {3, 5, 5/2}".

• Wikipedia Contributors. "Icosahedral 120-cell".