Intermetallic phases—solid state compounds that form upon alloying metallic elements together—comprise a realm of immense structural diversity:  their structures range from simple variants of the familiar fcc, bcc and hcp lattices, to the giant cubic unit cells of NaCd₂ (>1,000 atoms/cell) and Al_55.4Cu_5.4Ta_3.9 (23,134 atoms/cell), to quasicrystals such as YbCd_5.7 whose geometries defy description with 3-dimensional unit cells.  A limiting factor in realizing the broad technological applications promised by this diversity of atomic arrangements is our inability to understand, let alone control, the crystal structures of these compounds.  An emerging theme in the study of these phases is a link between structural complexity and the coexistence of mutually exclusive bonding or packing modes.  One focus of our group’s research has been pursuing this theme using an interaction of theory and experiment.  In this seminar, we will discuss some of our recent advances in this pursuit, including (1) the development of the DFT-chemical pressure analysis for creating graphical and intuitive representations of the tension between electronics and atomic size requirements, and (2) the revealing of an expanding range of structural features, including periodic interfaces and icosahedral clusters, as the products of chemical pressure release.