  To each of the symmetry groups of the Platonic solids we adjoin a carefully
designed involution yielding topological generators of PU(2) which have optimal
covering properties as well as efficient navigation. These are a consequence of
optimal strong approximation for integral quadratic forms associated with
certain special quaternion algebras and their arithmetic groups. The generators
give super efficient 1-qubit quantum gates and are natural building blocks for
the design of universal quantum gates.
