[Cos-top] binary polyhedral groups

[Jeff Weeks]

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