  Self-organized complex structures in nature, e.g. viral capsids, hierarchical
biopolymers, and bacterial flagella, offer efficiency, adaptability,
robustness, and multi-functionality. Can we program the self-assembly of
three-dimensional (3D) complex structures with simple building blocks, and
reach similar or higher level of sophistication in engineered materials? Here
we present an analytic theory of tetrahedral nanoparticles (NPs)
self-assembling in 3D space, where unavoidable geometrical frustration combined
with competing attractive and repulsive inter-particle interactions lead to
controllable, high-yield, and enantiopure self-assembly of helicoidal ribbons.
This theory, based on crystal structures in non-Euclidean space, predicts
morphologies that exhibit qualitative agreement with experimental observations.
We expect that this theory will offer a general framework for the self-assembly
of simple polyhedral building blocks into complex morphologies with new
material capabilities such as tunable optical activity, essential for multiple
emerging technologies.
