  We examine the Casimir energy of 5D electromagnetism in the recent
standpoint. The bulk geometry is flat. Z$_2$ symmetry and the periodic
property, for the extra coordinate, are taken into account. After confirming
the consistency with the past result, we do new things based on a {\it new
regularization}. In the treatment of the divergences, we introduce IR and UV
cut-offs and {\it restrict} the (4D momentum, extra coordinate)-integral
region. The regularized configuration is the {\it sphere lattice}, in the 4D
continuum space, which changes along the extra coordinate. The change
(renormalization flow) is specified by the {\it minimal area principle}, hence
this regularization configuration is string-like. We do the analysis not in the
Kaluza-Klein expanded form but in a {\it closed} form. We do {\it not} use any
perturbation. The formalism is based on the heat-kernel approach using the {\it
position/momentum propagator}. Interesting relations between the heat-kernels
and the P/M propagators are obtained, where we introduce the {\it generalized}
P/M propagators. A useful expression of the Casimir energy, in terms of the P/M
propagator, is obtained. The restricted-region approach is replaced by the
weight-function approach in the latter-half description. Its meaning, in
relation to the {\it space-time quantization}, is argued. {\it Finite} Casimir
energy is numerically obtained. The compactification-size parameter
(periodicity) suffers from the renormalization effect. Numerical evaluation is
exploited. Especially the minimal surface lines in the 5D flat space are
obtained both numerically using the Runge-Kutta method and analytically using
the general solution.
