[Cos-top] Jesper tells why

[Jeff Weeks]

P.P.S.  I like Jesper's response because it hints
at *why* the various levels contain the number of dodecahedra
that they do.  In effect, Jesper is showing the relation
between the binary icosahedral group (which is the holonomy
group of the Poincaré dodecahedral space) and the ordinary
dodecahedral group (the symmetries of the original dodecahedron
at layer 1).  The relationship, which Jesper's table clearly shows,
is that if you travel in the direction of a symmetry axis
(of the original layer 1 dodecahedron) of order n, you'll
find 2n translates of the dodecahedron spaced pi/n radians apart.

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