What drives the stability of complex intermetallic compounds? Many, if not most, metals and alloys crystallize in one of the familiar body-centered cubic (bcc), face-centered cubic (fcc), or hexagonal close-packed (hcp) crystal structures. But what of Cu5 Zn8 with 54 atoms in its unit cell, or Ti2 Ni with its 96? Or larger still Fe11 Zn39 , with 408 atoms in the repeat unit? Chemically and physically straight-forward semi-empirical calculations can ¨ lend us insight, and we review one such technique-the extended Huckel method-with new examples. Along the way, we solve the crystal structures for the new family of ternary compounds Ir x Ru1[-] x Zn10 and see what clues the Ir site preferences can give us about the connection between atomic structure and electronic structure via the Mott-Jones method of constructing electronic wavefunctions from the intensity of Bragg peaks. Finally, we investigate the connection between intermetallic structures that possess pseudo five-fold diffraction patterns. We find these peaks corresponding to features of several different types of tetrahedral close-packing, which in turn can all be described as three-dimensional projections of a four-dimensional perfect packing of tetrahedra: the 600-cell. We further show that the Bragg peaks comprising the pseudo five-fold diffraction pattern are linked to structural stability, and hence the higher-dimensional geometry.