# Grand hexacosichoron

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Grand hexacosichoron
Rank4
TypeRegular
Notation
Bowers style acronymGax
Coxeter diagramo5/2o3o3x ()
Schläfli symbol{3,3,5/2}
Elements
Cells600 tetrahedra
Faces1200 triangles
Edges720
Vertices120
Vertex figureGreat icosahedron, edge length 1
Edge figuretet 3 tet 3 tet 3 tet 3 tet 3
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${\displaystyle {\frac {{\sqrt {10}}-2{\sqrt {2}}}{4}}\approx 0.083463}$
Hypervolume${\displaystyle 25{\frac {{\sqrt {5}}-2}{4}}\approx 1.47542}$
Dichoral angle${\displaystyle \arccos \left({\frac {3{\sqrt {5}}-1}{8}}\right)\approx 44.47751^{\circ }}$
Central density191
Number of external pieces36000
Level of complexity118
Related polytopes
ArmyEx, edge length ${\displaystyle {\frac {3-{\sqrt {5}}}{2}}}$
RegimentGishi
CompanyGofix
DualGreat grand stellated hecatonicosachoron
ConjugateHexacosichoron
Convex coreHexacosichoron
Abstract & topological properties
Flag count14400
Euler characteristic0
Schläfli type{3,3,5}
OrientableYes
Properties
SymmetryH4, order 14400
ConvexNo
NatureTame

The grand hexacosichoron, or gax, also commonly called the grand 600-cell, is one of the 10 regular Schläfli–Hess polychora. It has 600 regular tetrahedra as cells, joining 5 to an edge and 20 to a vertex in the form of a great icosahedron.

It is a faceting of the great stellated hecatonicosachoron, sharing its vertices and edges, and the faces of the great faceted hexacosichoron.

## Vertex coordinates

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

## Related polychora

Uniform polychoron compounds composed of grand hexacosichora include: