Dear Cos-Top,

Jesper pointed out that the final sentence of my earlier message, namely

This same relation holds, of course, for the other single-action spherical manifolds.

implies something that is not correct. To make that statement more precise (and correct!) please note that in the construction

if you travel in the direction of a symmetry axis [of Polyhedron A] of order n, you'll find 2n translates of the fundamental domain [Polyhedron B] spaced pi/n radians apart

the polyhedra should be interpreted as follows:

Polyhedron A is the polyhedron for which the group is named (that is, a tetrahedron for the binary tetrahedral group, an octahedron for the binary octahedral group, or an icosahedrdon for the binary icosahedral group). Polyhedron B is the fundamental domain itself (respectively an octahedron, a truncated cube or a dodecahedron in the preceding three examples).

In the case of the binary icosahedral group, the symmetries of the icosahedron are exactly the symmetries of the dodecahedron, so no confusion is possible. In the case of the binary octahedral group, the symmetries of the octahedron are exactly the symmetries of the truncated cube, so again no confusion is possible. But... in the case of the binary tetrahedral group, the symmetries of the tetrahedron are a proper subset of the symmetries of the fundamental domain (the octahedron). In other words, the fundamental domain (the octahedron) acquires some "accidental" symmetries that are not a priori forced upon it by the symmetries of the basic tetrahedron. In this case, to enumerate the elements of the binary tetrahedral group, it's crucial that we start with the basic tetrahedron, and not with the octahedral fundamental domain.

(The reason the binary tetrahedral group's fundamental domain acquires extra symmetry is that the tetrahedron is self-dual. In effect the octahedral fundamental domain is the intersection of the basic tetrahedron and its dual. Thus the doubling of the symmetry.)

For an elementary explanation of binary polyhedral groups, see Section 3 (pages 5161-5163) of

"Topological lensing in spherical spaces", Classical and Quantum Gravity 18 (2001) 5155-5186 online at www.arxiv.org/gr-qc/0106033

(Someday I plan to write up a different explanation of these groups, but for now I hope the cited reference will serve well enough.)

In any case, many thanks to Jesper for pointing out the error in the earlier e-mail.

Best wishes to all, Jeff www.geometrygames.org/contact.html