The ability of DNA to store and encode information arises from base pairing of the four-letter nucleobase code to form a double helix. Expanding this DNA 'alphabet' by synthetic incorporation of new bases can introduce new functionalities and enable the formation of novel nucleic acid structures. However, reprogramming the self-assembly of existing nucleobases presents an alternative route to expand the structural space and functionality of nucleic acids. Here we report the discovery that a small molecule, cyanuric acid, with three thymine-like faces, reprogrammes the assembly of unmodified poly(adenine) (poly(A)) into stable, long and abundant fibres with a unique internal structure. Poly(A) DNA, RNA and peptide nucleic acid (PNA) all form these assemblies. Our studies are consistent with the association of adenine and cyanuric acid units into a hexameric rosette, which brings together poly(A) triplexes with a subsequent cooperative polymerization. Fundamentally, this study shows that small hydrogen-bonding molecules can be used to induce the assembly of nucleic acids in water, which leads to new structures from inexpensive and readily available materials.