# Faceted hexacosichoron

Faceted hexacosichoron Rank4
TypeRegular
SpaceSpherical
Notation
Bowers style acronymFix
Coxeter diagramo5/2o5o3x (         )
Schläfli symbol{3,5,5/2}
Elements
Cells120 icosahedra
Faces1200 triangles
Edges720
Vertices120
Vertex figureGreat dodecahedron, edge length 1
Edge figureike 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 angle120°
Central density4
Number of pieces2400
Level of complexity4
Related polytopes
ArmyEx
RegimentEx
CompanyEx
DualSmall stellated hecatonicosachoron
ConjugateGreat faceted hexacosichoron
Convex coreHecatonicosachoron
Abstract properties
Euler characteristic480
Topological properties
OrientableYes
Properties
SymmetryH4, order 14400
ConvexNo
NatureTame

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.

## Vertex coordinates

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