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

Frustrated Self-Assembly of Non-Euclidean Crystals of Nanoparticles