Coordinates
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 remaining 96 vertices are obtained by taking even permutations of
- ½(±φ,±1,±1/φ,0).
Note that the first 16 vertices are the vertices of a tesseract, the second eight are the vertices of a 16-cell, and that all 24 vertices together are vertices of a 24-cell. The final 96 vertices are the vertices of a snub 24-cell, which can be found by partitioning each of the 96 edges of another 24-cell (dual to the first) in the golden ratio in a consistent manner.
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 and denoted by 2I as it is the double cover of the ordinary icosahedral group I. It occurs twice in the rotational symmetry group RSG of the 600-cell as an invariant subgroup, namely as the subgroup 2IL of quaternion left-multiplications and as the subgroup 2IR of quaternion right-multiplications. Each rotational symmetry of the 600-cell is generated by specific elements of 2IL and 2IR; the pair of opposite elements generate the same element of RSG. The centre of RSG consists of the non-rotation Id and the central inversion -Id. We have the isomorphism RSG ≅ (2IL × 2IR) / {Id, -Id}. The order of RSG equals 120 × 120 / 2 = 7200.
The binary icosahedral group is isomorphic to SL(2,5).
The full symmetry group of the 600-cell is the Weyl group of H4. This is a group of order 14400. It consists of 7200 rotations and 7200 rotation-reflections. The rotations form an invariant subgroup of the full symmetry group. The rotational symmetry group was described by S.L. van Oss (1899); see References.
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