# Grand 600-cell

Grand 600-cell

Orthogonal projection
Type Regular star 4-polytope
Cells 600 {3,3}
Faces 1200 {3}
Edges 720
Vertices 120
Vertex figure {3,5/2}
Schläfli symbol {3,3,5/2}
Coxeter-Dynkin diagram
Symmetry group H4, [3,3,5]
Dual Great grand stellated 120-cell
Properties Regular

In geometry, the grand 600-cell or grand polytetrahedron is a regular star 4-polytope with Schläfli symbol {3,3,5/2}. It is one of 10 regular Schläfli-Hess polytopes. It is the only one with 600 cells.

It is one of four regular star 4-polytopes discovered by Ludwig Schläfli. It is 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.

## Related polytopes

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.

Orthographic projections by Coxeter planes
H3 A2 / B3 / D4 A3 / B2

With its dual, it forms the compound of great grand stellated 120-cell and grand 600-cell.