  In his famous book, Felix Klein describes a complex variable for the
quotients of the ordinary sphere by the finite groups of rotations and in
particular for the most complex situation of the quotient by the symmetry group
of the icosahedron. The purpose of this work and its sequels is to obtain
similar results for the quotients of the three--dimensional sphere. Various
properties of the group $SU(2)$ and of its representations are used to obtain
explicit expressions for coordinates and the relations they satisfy.
