The major and minor axes of the polarization ellipses that surround singular
lines of circular polarization in three dimensional optical ellipse fields are
shown to be organized into Mobius strips. These strips can have either one or
three half-twists, and can be either right- or left-handed. The normals to the
surrounding ellipses generate cone-like structures. Two special projections,
one new geometrical, and seven new topological indices are developed to
characterize the rather complex structures of the Mobius strips and cones.
These eight indices, together with the two well-known indices used until now to
characterize singular lines of circular polarization, could, if independent,
generate 16,384 geometrically and topologically distinct lines. Geometric
constraints and 13 selection rules are discussed that reduce the number of
lines to 2,104, some 1,150 of which have been observed in practice; this number
of different C lines is ~ 350 times greater than the three types of lines
recognized previously. Statistical probabilities are presented for the most
important index combinations in random fields. It is argued that it is
presently feasible to perform experimental measurements of the Mobius strips
and cones described here theoretically