The Sareh twist: A hidden geometric principle in origami tessellations

Origami techniques originate from the ancient paper-folding traditions of China and Japan. Since the mid-20th century, origami engineering has developed into an independent technology, transforming concepts of foldable and deployable structures in aerospace. More recently, new application areas have emerged, including the design of robots, biomedical devices, antennas, shock absorbers, and metamaterials. This study highlights a range of novel aspects of a twist that underlies both the folding of two-dimensional flat-foldable units and their periodic tilings, as well as three-dimensional structures, where it manifests as an internal, chiral movement enabling the transition between flat-folded and deployed configurations. The principle of twist was first explicitly identified by Pooya Sareh in crystallographic origami tessellations derived from the Miura-ori and is referred to here as the “Sareh twist.” Building on this concept, the present study extends the role of twist to four domains: degree-4 vertex units in single and docked tiles; conic transformations with the vertex as a pivot point; the twist-inducing diagonal fold in rectangles, which was crucial for defining industrial paper format ratios by Wilhelm Ostwald around 1911 and is adapted here to parallelograms; and finally, the internal twist of the cylindrical Kresling pattern developed by the author. The key innovation of this work is the development of a general approach that transforms even the most complex geometries into Sareh twist units defined by their specific symmetries. This approach not only advances the rational design of folded structures but also enhances our understanding of biological patterns.