  Wigner's little groups are subgroups of the Lorentz group dictating the
internal space-time symmetries of massive and massless particles. These little
groups are like O(3) and E(2) for massive and massless particles respectively.
While the geometry of the O(3) symmetry is familiar to us, the geometry of the
flat plane cannot explain the E(2)-like symmetry for massless particles.
However, the geometry of a circular cylinder can explain the symmetry with the
helicity and gauge degrees of freedom. It is shown further that the symmetry of
the massless particle can be obtained as a zero-mass limit of O(3)-like
symmetry for massive particles. It is shown further that the polarization of
massless neutrinos is a consequence of gauge invariance, while the symmetry of
massive neutrinos is still like O(3).
