Rinki Imada

Non-periodic folding of periodic crease patterns paves the way to novel nonlinear phenomena that cannot be feasible through periodic folding. This paper focuses on the non-periodic folding of recursive crease patterns generalized from Spidron. Although it is known that Spidron has a 1-DOF isotropic rigid folding motion, its general kinematics and dependence on the crease pattern remain unclear. Using the kinematics of a single unit cell of Spidron and the recursive construction of the folded state of multiple unit cells, we consider the folding of Spidron that is not necessarily isotropic. We found that as the number of unit cells increases, the non-periodic folding is restricted and the isotropic folding becomes dominant. Then, we analyze the three kinds of isotropic folding modes by constructing 1-dimensional dynamical systems governing each of them. We show that the dynamical system can possess different recursive natures depending on folding modes even in an identical crease pattern. Furthermore, we show their novel nonlinear nature, including the period-doubling cascade leading to the emergence of chaos.

<jats:title>Abstract</jats:title> <jats:p>Folded surfaces of origami tessellations have attracted much attention because they often exhibit nontrivial behaviors. It is known that cylindrical folded surfaces of waterbomb tessellation called waterbomb tube can transform into peculiar wave-like surfaces, but the theoretical reason why wave-like surfaces arise has been unclear. In this paper, we provide a kinematic model of waterbomb tube by parameterizing the geometry of a module of waterbomb tessellation and derive a recurrence relation between the modules. Through the visualization of the configurations of waterbomb tubes under the proposed kinematic model, we classify solutions into three classes: cylinder solution, wave-like solution, and finite solution. Through the stability and bifurcation analyses of the dynamical system, we investigate how the behavior of waterbomb tube changes when the crease pattern is changed. Furthermore, we prove the existence of a wave-like solution around one of the cylinder solutions.</jats:p>