600-cell
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In geometry, the 600-cell (or hexacosichoron) is the convex regular 4-polytope with Schläfli symbol {3,3,5}. It is sometimes thought of as the 4-dimensional analog of the icosahedron.
The boundary of the 600-cell is composed of 600 tetrahedral cells with 20 meeting at each vertex. Together they form 1200 triangular faces, 720 edges, and 120 vertices. The vertex figure is an icosahedron. The dual polytope of the 600-cell is the 120-cell.
The vertices of a 600-cell centered at the origin of 4-space, with edges of length 1/φ (where φ = (1+√5)/2 is the golden ratio), 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).
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.