Point Groups in Three Dimensions - Binary Polyhedral Groups

Binary Polyhedral Groups

The map Spin(3) → SO(3) is the double cover of the rotation group by the spin group in 3 dimensions. (This is the only connected cover of SO(3), since Spin(3) is simply connected.) By the lattice theorem, there is a Galois connection between subgroups of Spin(3) and subgroups of SO(3) (rotational point groups): the image of a subgroup of Spin(3) is a rotational point group, and the preimage of a point group is a subgroup of Spin(3).

The preimage of a finite point group is called a binary polyhedral group, represented as , and is called by the same name as its point group, with the prefix binary, with double the order of the related polyhedral group (l,m,n). For instance, the preimage of the icosahedral group (2,3,5) is the binary icosahedral group, <2,3,5>.

The binary polyhedral groups are:

• : binary cyclic group of an (n + 1)-gon
• : binary dihedral group of an n-gon, <2,2,n>
• : binary tetrahedral group, <2,3,3>
• : binary octahedral group, <2,3,4>
• : binary icosahedral group, <2,3,5>

These are classified by the ADE classification, and the quotient of C2 by the action of a binary polyhedral group is a Du Val singularity.

For point groups that reverse orientation, the situation is more complicated, as there are two pin groups, so there are two possible binary groups corresponding to a given point group.

Note that this is a covering of groups, not a covering of spaces – the sphere is simply connected, and thus has no covering spaces. There is thus no notion of a "binary polyhedron" that covers a 3-dimensional polyhedron. Binary polyhedral groups are discrete subgroups of a Spin group, and under a representation of the spin group act on a vector space, and may stabilize a polyhedron in this representation – under the map Spin(3) → SO(3) they act on the same polyhedron that the underlying (non-binary) group acts on, while under spin representations or other representations they may stabilize other polyhedra.

This is in contrast to projective polyhedra – the sphere does cover projective space (and also lens spaces), and thus a tessellation of projective space or lens space yields a distinct notion of polyhedron.