Despite significant progress in the structural characterization of the quasicrystalline state, the chemical origins of long- and short-range icosahedral order remain mysterious and a subject of debate. In this Article, we present the crystal structure of a new complex intermetallic phase, Ca10Cd27Cu2 (mC234.24), whose geometrical features offer clues to the driving forces underlying the icosahedral clusters that occur in Bergman-type quasicrystals. Ca10Cd27Cu2 adopts a C-centered monoclinic superstructure of the 1/1 Bergman approximant structure, in which [110] layers of Bergman clusters in the 1/1 structure are separated through the insertion of additional atoms (accompanied by substantial positional disorder). An examination of the coordination environments of Ca and Cu (in the ordered regions) reveals that the structure can be viewed as a combination of coordination polyhedra present in the nearest binary phases in the Ca-Cd-Cu compositional space. A notable feature is the separation of Ca-Cd and Cu-Cd interactions, with Bergman clusters emerging as Ca-Cd Friauf polyhedra (derived from the MgZn2-type CaCd2 phase) encapsulate a Cu-Cd icosahedron similar to those appearing in Cu2Cd5. DFT chemical pressure calculations on nearby binary phases point to the importance of this segregation of Ca-Cd and Cu-Cd interactions. The mismatch in atomic size between Cu and Cd leads to an inability to satisfy Ca Cu and Ca Cd interactions simultaneously in the Friauf polyhedra of the nearby Laves phase CaCd2. The relegation of the Cu atoms to icosahedra prevents this frustration while nucleating the formation of Bergman clusters.