Great grand stellated 120-cell

Short description: Regular Schläfli-Hess 4-polytope with 600 vertices

Great grand stellated 120-cell
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
Symmetry group	H4, [3,3,5]
Dual	Grand 600-cell
Properties	Regular

A Zome model

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.

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.

See also

• List of regular polytopes
• Convex regular 4-polytope – Set of convex regular polychora
• Kepler-Poinsot solids – regular star polyhedron
• Star polygon – regular star polygons

References

• 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-Strauss, The Symmetries of Things 2008, ISBN:978-1-56881-220-5 (Chapter 26, Regular Star-polytopes, pp. 404–408)
• Klitzing, Richard. "4D uniform polytopes (polychora) o3o3o5/2x - gogishi". https://bendwavy.org/klitzing/dimensions/polychora.htm. 

External links

• Regular polychora
• Discussion on names
• Reguläre Polytope
• The Regular Star Polychora
• Zome Model of the Final Stellation of the 120-cell