# Great faceted hexacosichoron

Great faceted hexacosichoron
Rank4
TypeRegular
Notation
Bowers style acronymGofix
Coxeter diagramo5o5/2o3x ()
Schläfli symbol${\displaystyle \{3,5,5\mid 3\}}$
${\displaystyle \{3,5/2,5\}}$
Elements
Cells120 great icosahedra
Faces1200 triangles
Edges720
Vertices120
Vertex figureSmall stellated dodecahedron, edge length 1
Edge figuregike 3 gike 3 gike 3 gike 3 gike 3
Deep holesTriangles
Measures (edge length 1)
Circumradius${\displaystyle {\frac {{\sqrt {5}}-1}{2}}\approx 0.61803}$
Edge radius${\displaystyle {\frac {\sqrt {5-2{\sqrt {5}}}}{2}}\approx 0.36327}$
Face radius${\displaystyle {\frac {3{\sqrt {3}}-{\sqrt {15}}}{6}}\approx 0.22053}$
Inradius${\displaystyle {\frac {3-{\sqrt {5}}}{4}}\approx 0.19098}$
Hypervolume${\displaystyle 25{\frac {7-3{\sqrt {5}}}{4}}\approx 1.82373}$
Dichoral angle120°
Central density76
Number of external pieces7200
Level of complexity12
Related polytopes
ArmyEx, edge length ${\displaystyle {\frac {3-{\sqrt {5}}}{2}}}$
RegimentGishi
CompanyGofix
DualGreat grand hecatonicosachoron
ConjugateFaceted hexacosichoron
Convex coreHecatonicosachoron
Abstract & topological properties
Flag count14400
Euler characteristic480
Schläfli type{3,5,5}
OrientableYes
Properties
SymmetryH4, order 14400
Flag orbits1
ConvexNo
NatureTame

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

The great faceted hexacosichoron is a faceting of the great stellated hecatonicosachoron, sharing its vertices and edges. The cells are in the same planes as the small stellated dodecahedra of the grand stellated hecatonicosachoron.

## Vertex coordinates

Its vertices are the same as those of its regiment colonel, the great stellated hecatonicosachoron.

## Related polychora

The great faceted hexacosichoron is the captain of a company in the regiment of the great stellated hecatonicosachoron, having the same faces as the grand hexacosichoron.

Uniform polychoron compounds composed of faceted hexacosichora include: