  Single molecular species can self-assemble into Frank Kasper (FK) phases,
finite approximants of dodecagonal quasicrystals, defying intuitive notions
that thermodynamic ground states are maximally symmetric. FK phases are
speculated to emerge as the minimal-distortional packings of space-filling
spherical domains, but a precise quantitation of this distortion and how it
affects assembly thermodynamics remains ambiguous. We use two complementary
approaches to demonstrate that the principles driving FK lattice formation in
diblock copolymers emerge directly from the strong-stretching theory of
spherical domains, in which minimal inter-block area competes with minimal
stretching of space-filling chains. The relative stability of FK lattices is
studied first using a diblock foam model with unconstrained particle volumes
and shapes, which correctly predicts not only the equilibrium {\sigma} lattice,
but also the unequal volumes of the equilibrium domains. We then provide a
molecular interpretation for these results via self-consistent field theory,
illuminating how molecular stiffness regulates the coupling between
intra-domain chain configurations and the asymmetry of local packing. These
findings shed new light on the role of volume exchange on the formation of
distinct FK phases in copolymers, and suggest a paradigm for formation of FK
phases in soft matter systems in which unequal domain volumes are selected by
the thermodynamic competition between distinct measures of shape asymmetry.
