In mathematics, the 600-cell is the 4-dimensional convex regular polytope with 600 facets. Its Schläfli symbol is {3,3,5}. It may be considered the 4-dimensional analogue of the icosahedron.

The 600 facets are tetrahedral, and 20 meet at each vertex. The number of vertices is 120 as the 600-cell is dual to the 120-cell.

Vertices of a 600-cell centred at the origin of 4-space, with edges of length 1/φ, can be given as follows: 16 vertices of the form

(±½,±½,±½,±½),

and 8 vertices obtained from

(0,0,0,±1)

by permuting coordinates. The final 96 vertices are obtained by taking even permutations of

½(±1,±φ,±1/φ,0).

where φ = (1+√5)/2 is the golden ratio. Note that the first 16 vertices are the vertices of a tesseract, and that these, together with the next 8, form the vertices of the 24-cell.

When interpreted as quaternions, the 120 vertices of the 600-cell form a group under quaternionic multiplication. This group is often called the binary icosahedral group as it is the double cover of the ordinary icosahedral group I. The binary icosahedral group is isomorphic to SL(2,5).

The symmetry group of the 600-cell is the Weyl group of H₄ . This is a group of order 14400.

See also

• List of regular polytopes