# Great grand stellated 120-cell

In geometry, the great grand stellated 120-cell or great grand stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol {5/2,3,3}, one of 10 regular Schläfli-Hess 4-polytopes. It is unique among the 10 for having 600 vertices, and has the same vertex arrangement as the regular convex 120-cell.

Great grand stellated 120-cell

Orthogonal projection
TypeSchläfli-Hess polychoron
Cells120 {5/2,3}
Faces720 {5/2}
Edges1200
Vertices600
Vertex figure{3,3}
Schläfli symbol{5/2,3,3}
Coxeter-Dynkin diagram
Symmetry groupH4, [3,3,5]
DualGrand 600-cell
PropertiesRegular

It is one of four regular star polychora discovered by Ludwig Schläfli. It is named by John Horton Conway, extending the naming system by Arthur Cayley for the Kepler-Poinsot solids, and the only one containing all three modifiers in the name.

With its dual, it forms the compound of great grand stellated 120-cell and grand 600-cell.

## Images

Coxeter plane images
H4 A2 / B3 A3 / B2
Great grand stellated 120-cell, {5/2,3,3}
[10] [6] [4]
120-cell, {5,3,3}

## As a stellation

The great grand stellated 120-cell is the final stellation of the 120-cell, and is the only Schläfli-Hess polychoron to have the 120-cell for its convex hull. In this sense it is analogous to the three-dimensional great stellated dodecahedron, which is the final stellation of the dodecahedron and the only Kepler-Poinsot polyhedron to have the dodecahedron for its convex hull. Indeed, the great grand stellated 120-cell is dual to the grand 600-cell, which could be taken as a 4D analogue of the great icosahedron, dual of the great stellated dodecahedron.

The edges of the great grand stellated 120-cell are τ6 as long as those of the 120-cell core deep inside the polychoron, and they are τ3 as long as those of the small stellated 120-cell deep within the polychoron.