  We first apply Connes' noncommutative geometry to a finite point space. The
explicit form of the action functional of U(1) gauge field on this n-point
space is obtained. We then consider the case when the n-point space is replaced
by {space-time}\times{n-point space}. This action is shown to relate the
Hamiltonian of the continuous-spin formulation of the Potts model. We argue
that U(1) gauge theory on the discrete space-time determines the geometric
origin of a class of phase transitions.
