Great grand stellated 120-cell

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Great grand stellated 120-cell
Ortho solid 016-uniform polychoron p33-t0.png
Orthogonal projection
Type Schläfli-Hess polychoron
Cells 120 {5/2,3}
Faces 720 {5/2}
Edges 1200
Vertices 600
Vertex figure {3,3}
Schläfli symbol {5/2,3,3}
Coxeter-Dynkin diagram CDel node 1.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Symmetry group H4, [3,3,5]
Dual Grand 600-cell
Properties Regular

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

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.

As a stellation[edit]

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.


Great grand stellated 120cell zome 032407.jpg
A Zome model
Schläfli-Hess polychoron-wireframe-1.png
Orthogonal projection as a wireframe

See also[edit]


  • Edmund Hess, (1883) Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder [1].
  • H. S. M. Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8.
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26, Regular Star-polytopes, pp. 404–408)

External links[edit]