Relation To 4-dimensional Symmetry Groups
The 4-dimensional analog of the icosahedral symmetry group Ih is the symmetry group of the 600-cell (also that of its dual, the 120-cell). Just as the former is the Coxeter group of type H3, the latter is the Coxeter group of type H4, also denoted . Its rotational subgroup, denoted + is a group of order 7200 living in SO(4). SO(4) has a double cover called Spin(4) in much the same way that Spin(3) is the double cover of SO(3). Similar to the isomorphism Spin(3) = Sp(1), the group Spin(4) is isomorphic to Sp(1) × Sp(1).
The preimage of + in Spin(4) (a four-dimensional analogue of 2I) is precisely the product group 2I × 2I of order 14400. The rotational symmetry group of the 600-cell is then
- + = ( 2I × 2I ) / { ±1 }.
Various other 4-dimensional symmetry groups can be constructed from 2I. For details, see (Conway and Smith, 2003).
Read more about this topic: Binary Icosahedral Group
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