  Multi-component aggregates are being intensively researched in various fields
because of their highly tunable properties and wide applications. Due to the
complex configurational space of these systems, research would greatly benefit
from a general theoretical framework for the prediction of stable structures,
which, however, is largely incomplete at present. Here we propose a general
theory for the construction of multi-component icosahedral structures by
assembling concentric shells of different chiral and achiral types, consisting
of particles of different sizes. By mapping shell sequences into paths in the
hexagonal lattice, we establish simple and general rules for building a wide
variety of magic icosahedral structures, and we evaluate the optimal
size-mismatch between particles in the different shells. Our predictions are
confirmed by numerical simulations for different systems.
