| bgcolor=#e7dcc3 colspan=2 | Grand 600-cell | |
|---|---|---|
| bgcolor=#ffffff align=center colspan=2 | Orthogonal projection | |
| Type | Regular star 4-polytope | |
| Cells | 600 | |
| Faces | 1200 | |
| Edges | 720 | |
| Vertices | 120 | |
| Vertex figure | ||
| Schläfli symbol | ||
| Coxeter-Dynkin diagram | ||
| Symmetry group | H4, [3,3,5] | |
| Dual | Great grand stellated 120-cell | |
| Properties | Regular |
It is one of four regular star 4-polytopes discovered by Ludwig Schläfli. It was named by John Horton Conway, extending the naming system by Arthur Cayley for the Kepler-Poinsot solids.
The grand 600-cell can be seen as the four-dimensional analogue of the great icosahedron (which in turn is analogous to the pentagram); both of these are the only regular n-dimensional star polytopes which are derived by performing stellational operations on the pentagonal polytope which has simplectic faces. It can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of said (n-1)-D simplex faces of the core nD polytope (tetrahedra for the grand 600-cell, equilateral triangles for the great icosahedron, and line segments for the pentagram) until the figure regains regular faces.
The Grand 600-cell is also dual to the great grand stellated 120-cell, mirroring the great icosahedron's duality with the great stellated dodecahedron (which in turn is also analogous to the pentagram); all of these are the final stellations of the n-dimensional "dodecahedral-type" pentagonal polytope.
It has the same edge arrangement as the great stellated 120-cell, and grand stellated 120-cell, and same face arrangement as the great icosahedral 120-cell.