@Rahi09:PRD; @Kenneth08], and we employ the generalization of [@Rahi09:PRD]. Although this process is described in detail elsewhere [@Rahi09:PRD] and is reviewed for the specific geometries of this paper in the appendix, the basic idea of the calculation is as follows. The Casimir energy can be expressed via path integrals as an integral $\int_0^\infty \log \det A(\kappa)
d\kappa$ over imaginary frequencies $\kappa$, where $A$ is a “T-matrix” related to the scattering matrix of the system. In particular, one needs to compute the scattering matrices relating outgoing spherical waves from each sphere (or planewaves from each plate) being reflected into outgoing spherical waves (or planewaves) from every other sphere (or plate), which can be expressed semi-analytically (as infinite series) by “translation matrices” that re-express a spherical wave (or planewave) with one origin in terms of spherical waves (or planewaves) around the origin of the new object [@Rahi09:PRD]. This formalism is exact (no uncontrolled approximations) in the limit in which an infinite number of spherical/plane waves is considered. To obtain a finite matrix $A$, the number of spherical waves (or spherical harmonics $Y_{\ell m}$) is truncated to a finite order $\ell$. Because this expansion converges exponentially fast for spheres [@Canaguier09; @Rahi09:PRD], we find 