  Recent advances enable the creation of nanoscale building blocks with complex
geometries and interaction specificities for self-assembly. This nearly
boundless design space necessitates design principles for defining the mutual
interactions between multiple particle species to target a user-specified
complex structure or pattern. In this article, we develop a symmetry-based
method to generate the interaction matrices that specify the assembly of
two-dimensional tilings which we illustrate using equilateral triangles. By
exploiting the allowed 2D symmetries, we develop an algorithmic approach by
which any periodic 2D tiling can be generated from an arbitrarily large number
of subunit species, notably addressing an unmet challenge of engineering 2D
crystals with periodicities that can be arbitrarily larger than subunit size.
To demonstrate the utility of our design approach, we encode specific
interactions between triangular subunits synthesized by DNA origami and show
that we can guide their self-assembly into tilings with a wide variety of
symmetries, using up to 12 unique species of triangles. By conjugating specific
triangles with gold nanoparticles, we fabricate gold-nanoparticle supercrystals
whose lattice parameter spans up to 300 nm. Finally, to generate economical
design rules, we compare the design economy of various tilings. In particular,
we show that (1) higher symmetries allow assembly of larger unit cells with
fewer subunits and (2) linear supercrystals can be designed more economically
using linear primitive unit cells. This work provides a simple algorithmic
approach to designing periodic assemblies, which may open new doors to the
multiscale assembly of superlattices of nanostructured "metatoms" with
engineered plasmonic functions.
