Elements
Explicitly, the binary icosahedral group is given as the union of the 24 Hurwitz units
- { ±1, ±i, ±j, ±k, ½ ( ±1 ± i ± j ± k ) }
with all 96 quaternions obtained from
- ½ ( 0 ± i ± φ−1j ± φk )
by an even permutation of coordinates (all possible sign combinations). Here φ = ½ (1 + √5) is the golden ratio.
In total there are 120 elements, namely the unit icosians. They all have unit magnitude and therefore lie in the unit quaternion group Sp(1). The convex hull of these 120 elements in 4-dimensional space form a regular polychoron, known as the 600-cell.
Read more about this topic: Binary Icosahedral Group
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