  We propose and investigate an extension of the Caspar-Klug symmetry
principles for viral capsid assembly to the programmable assembly of
size-controlled triply-periodic polyhedra, discrete variants of the Primitive,
Diamond, and Gyroid cubic minimal surfaces. Inspired by a recent class of
programmable DNA origami colloids, we demonstrate that the economy of design in
these crystalline assemblies -- in terms of the growth of the number of
distinct particle species required with the increased size-scale (e.g.
periodicity) -- is comparable to viral shells. We further test the role of
geometric specificity in these assemblies via dynamical assembly simulations,
which show that conditions for simultaneously efficient and high-fidelity
assembly require an intermediate degree of flexibility of local angles and
lengths in programmed assembly. Off-target misassembly occurs via incorporation
of a variant of disclination defects, generalized to the case of hyperbolic
crystals. The possibility of these topological defects is a direct consequence
of the very same symmetry principles that underlie the economical design,
exposing a basic tradeoff between design economy and fidelity of programmable,
size controlled assembly.
