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Great grand stellated 120-cell

Regular Schläfli-Hess 4-polytope with 600 vertices

Great grand stellated 120-cell
[[Image:Ortho solid 016-uniform polychoron p33-t0.png	280px]]Orthogonal projection
Type
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Schläfli symbol
Coxeter-Dynkin diagram
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Dual
Properties

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.

Images

H3	A2 / B3	A3 / B2	Great grand stellated 120-cell, {5/2,3,3}	[10]	[6]	[4]	120-cell, {5,3,3}
[[File:Great_grand_stellated_120-cell-ortho-10gon.png	200px]]	[[File:Great_grand_stellated_120-cell-6gon.png	200px]]	[[File:Great_grand_stellated_120-cell-4gon.png	200px]]
[[File:120-cell t0 H3.svg	210px]]	[[File:120-cell t0 A2.svg	210px]]	[[File:120-cell t0 A3.svg	210px]]

As a stellation

The great grand stellated 120-cell is the final regular 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.

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 http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=ABN8623.0001.001.
H. S. M. Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. .
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, (Chapter 26, Regular Star-polytopes, pp. 404–408)

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