Noncommutative Fractal Origami Structures

Abstract

Mechanical metamaterials have emerged as a powerful design paradigm in which geometry and structure, rather than material composition alone, govern mechanical response. Origami-based mechanical structures enable tunable stiffness, bistability, and coupling between distinct deformation modes, which enables systems to respond to both their current state and also its history of loading. This paper develops a theoretical and experimental framework for history-dependent, pathway-programmable mechanics in Kresling origami columns and multi-column assemblies. Using the Preisach formalism, we model each bistable Kresling unit cell as a mechanical hysteron characterized by two switching thresholds. For an n-cell column with ordered thresholds, we prove that the resulting state-transition diagrams are equivalent across all chirality arrangements and cell counts. We further demonstrate that these diagrams exhibit non-commutative behavior and possess a self-similar, fractal structure, which have a recursively embedded transition logic. This structure was utilized to develop a shortest-pathway algorithm that deterministically predicts the unique actuation sequence required to navigate between any two configurations. Our theoretical results are validated experimentally under both translational and twist loading, confirming that pathway predictability is a robust and programmable feature of this system we developed. Broadly, this work contributes to the growing understanding that physical matter can be designed to encode sequential information, perform logical operations, and respond to loading history in prescribed ways. The principles introduced here have implications across fields ranging from soft robotics and mechanical computing to deployable structures and adaptive architecture, pointing toward a future in which the mechanical response of a structure is not merely reactive but purposefully encoded.