Proteins are linear chain molecules that play a central role in life and health. Protein native state folds are modular assemblies of space-filling building blocks of {\alpha}-helices, \{beta}-sheets and tight turns. Here we deduce the structures of a countable set of space-filling helical forms of a uniform discrete thick string from first principles with no additional input or adjustable parameters. These forms occur in correspondence with the natural numbers, loosely analogous to the energy levels in a Bohr atom. We find the remarkable result that one of these helical forms is an excellent candidate for an {\alpha}-helix through seemingly improbable quantum chemistry coincidences that fit the geometrical requirements. Our work suggests that geometry and chemistry are complementary ways of looking at proteins and suggests a route for developing a unified framework for understanding proteins.
@article{arxiv.2510.07011,
title = {Space-filling discrete helices},
author = {Jayanth R. Banavar and Achille Giacometti and Trinh X. Hoang and Amos Maritan and Tatjana Škrbić},
journal= {arXiv preprint arXiv:2510.07011},
year = {2025}
}