  We calculate the Casimir force between parallel plates for a massless scalar
field. When adding the energy of normal modes, we avoid infinities by using a
discrete spacetime lattice; however, this approach proves ineffective as long
as both space and time are kept discrete. Yet, when time is treated as
continuous while the scalar field forms a spatial periodic lattice, our method
succeeds, and we refer to this approach as Hamiltonian lattice theory. The
dispersion relation for both square and triangular lattices accurately
reproduces the subtle Casimir effect, providing evidence that the Casimir force
is independent of the type of lattice used. At low frequencies, both lattices
exhibit a high level of rotational symmetry. However, at high frequencies, they
lose this symmetry, even though the propagation of high-frequency waves becomes
limited as their group velocity approaches zero.
