Great grand stellated 120-cellOrthogonal projectionTypeSchläfli-Hess polychoronCells120 {5/2,3} Faces720 {5/2} Edges1200Vertices600Vertex figure{3,3} Schläfli symbol {5/2,3,3}Coxeter-Dynkin diagramSymmetry groupH4, [3,3,5]Dual Grand 600-cellProperties Regular
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 + Coxeter plane imagesH3A2 / B3A3 / B2Great grand stellated 120-cell, {5/2,3,3}120-cell, {5,3,3}
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.
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 . 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)
External links Regular polychora Discussion on names Reguläre Polytope The Regular Star Polychora Zome Model of the Final Stellation of the 120-cell