  New results for scalar Casimir self-energies arising from interior modes are
presented for the three integrable tetrahedral cavities. Since the eigenmodes
are all known, the energies can be directly evaluated by mode summation, with a
point-splitting regulator, which amounts to evaluation of the cylinder kernel.
The correct Weyl divergences, depending on the volume, surface area, and the
edges, are obtained, which is strong evidence that the counting of modes is
correct. Because there is no curvature, the finite part of the quantum energy
may be unambiguously extracted. Cubic, rectangular parallelepipedal, triangular
prismatic, and spherical geometries are also revisited. Dirichlet and Neumann
boundary conditions are considered for all geometries. Systematic behavior of
the energy in terms of geometric invariants for these different cavities is
explored. Smooth interpolation between short and long prisms is further
demonstrated. When scaled by the ratio of the volume to the surface area, the
energies for the tetrahedra and the prisms of maximal isoareal quotient lie
very close to a universal curve. The physical significance of these results is
discussed.
