In the same year Conyers Herring was working on his solution to the problem of plane-wave convergence in crystals, John C. Slater, professor of physics at MIT, tackled the same general problem — calculating electron energy bands in real solids — from a completely different geometric angle. During a sabbatical semester at the Institute for Advanced Study in Princeton in 1937, Slater proposed dividing the crystal's space into two regions of very different nature: small spheres centered on each atom — the so-called "muffin-tin" approximation, where the potential is treated as spherically symmetric, as in an isolated atom — and the interstitial region between those spheres, where the potential is approximately constant and the wave function can simply be represented as an ordinary plane wave. The central piece of Slater's proposal consists of "augmenting" each plane wave in the interstitial region by matching it continuously, at each sphere's surface, with the appropriate solution of the Schrödinger equation inside that sphere. The result is the so-called augmented plane waves (APW), a basis of functions that accurately captures both the nearly free behavior of the electron in the space between atoms and its strongly bound behavior near the nucleus, with no need for the explicit orthogonalization trick used in Herring's method. The APW method proved more computationally expensive than OPW — its energy dependence forces recalculating the basis for each energy level sought — which delayed its practical adoption until the 1950s, but its linearized descendant (LAPW, developed by Andersen in the 1970s) remains today, together with the pseudopotential methods derived from Herring, one of the two great computational lineages on which all first-principles materials physics is built.