Opposing vertices are used for the poles of spin. This creates six great circles which subdivide each face of the spherical icosahedron. This marks on the sphere the spherical equivalent of the icosidodecahedron with 20 spherical triangles and 12 spherical pentagons.
Note: the icosidodecahedron is an Archimedean polyhedron which forms the interior part common to the compound of an icosahedron and dodecahedron; it is also the dual of the rhombic triacontahedron.
[Refs: ‘Geodesic Math And How To Use It’ p.50, fig 7.6; ‘Polyhderon Models’ p.73, by Magnus J. Wenninger; ‘Mathematical Models’ p. 108, by H.M. Cundy and A.P. Rollet]
Opposing mid-edge points are used for poles of spin. This creates fifteen great gircles which outline every icosahedral face and slice through each three ways. In so doing, these fifteen great circles create 120 right triangles that tessellate the sphere. These are arranged in 60 enantiomorphous (mirror image) pairs and are in fact icosahedral Schwarz triangles, named after the 19th Century mathematician who discovered them. They form the “essential particles” of icosahedral subdivision; Buckminster Fuller called them “alpha particles” and they are the smallest repeating element of spherical tessellation. Thus this family subdivides the icosahedron in exactly the same way as Basic Triacon 2V subdivision.
[Ref: ‘Geodesic Math And How To Use It’ p.50, fig 7.7]
Opposing mid-face points are used for poles of spin. This creates ten great circles; each one leaves the midpoint of a face edge to cross the next face edge at right angles.
[Ref: ‘Geodesic Math And How To Use It’ p.50-51, fig 7.8]
Families I & II
Families I & III
Families II & III
Families I, II & III