Kinematic folding propagation in degree-4 origami strips

Degree-4 origami strips, one-DOF mechanisms constructed by sequentially connecting degree-4 origami vertices, have inspired origami-based engineering design. However, thorough kinematic analyses were limited to a special subset of degree-4 origami strips that exhibit uniform folding along the sequence. In this study, we show how folding propagates non uniformly, i.e., gets attenuated or amplified, in a general degree-4 origami strip. We introduce the concept of kinematic folding propagation and analyze it by studying discrete dynamical systems. Our results reveal that, despite its simple structure, the strip exhibits diverse folding propagation behaviors, strongly influenced by design parameters such as sector angles and topology of the crease patterns. We show that the propagation behavior is topologically linear when adjacent vertices are connected via opposite creases. We compute and visualize folding motions, including strips that transition from a flat-folded state to a helical shape through uniform or nonuniform (attenuated/amplified) propagation upon actuation of the boundary crease. Additionally, we demonstrate folding propagation in physical models using 3D-printed prototypes with thick panels. Furthermore, we show that topologically nonlinear propagation emerges when adjacent creases are used to connect adjacent vertices. We also discuss folding propagation in curved-crease origami that is achieved by taking a continuum limit of the strip. Our findings establish kinematic folding propagation as a core functionality enabled by nonuniform folding, thereby laying the foundation for programmable origami.