# Faceted hexacosichoron

Faceted hexacosichoron
Rank4
TypeRegular
Notation
Bowers style acronymFix
Coxeter diagramo5/2o5o3x ()
Schläfli symbol${\displaystyle \{3,5,5\mid 3\}}$
${\displaystyle \{3,5,5/2\}}$
Elements
Cells120 icosahedra
Faces1200 triangles
Edges720
Vertices120
Vertex figureGreat dodecahedron, edge length 1
Edge figureike 3 ike 3 ike 3 ike 3 ike 3
Deep holesTriangles
Measures (edge length 1)
Circumradius${\displaystyle {\frac {1+{\sqrt {5}}}{2}}\approx 1.61803}$
Edge radius${\displaystyle {\frac {\sqrt {5+2{\sqrt {5}}}}{2}}\approx 1.53884}$
Face radius${\displaystyle {\frac {3{\sqrt {3}}+{\sqrt {15}}}{6}}\approx 1.51152}$
Inradius${\displaystyle {\frac {3+{\sqrt {5}}}{4}}\approx 1.30902}$
Hypervolume${\displaystyle 25{\frac {7+3{\sqrt {5}}}{4}}\approx 85.67627}$
Dichoral angle120°
Central density4
Number of external pieces2400
Level of complexity4
Related polytopes
ArmyEx
RegimentEx
CompanyEx
DualSmall stellated hecatonicosachoron
ConjugateGreat faceted hexacosichoron
Convex coreHecatonicosachoron
Abstract & topological properties
Flag count14400
Euler characteristic480
Schläfli type{3,5,5}
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.

## Related polychora

Uniform polychoron compounds composed of faceted hexacosichora include: