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Geometry and Kinematics of Cylindrical Waterbomb Tessellation
Abstract
Folded surfaces of origami tessellations have attracted much attention because they sometimes exhibit non-trivial behaviors. It is known that cylindrical folded surfaces of waterbomb tessellation called waterbomb tube can transform into wave-like surfaces, which is a unique phenomenon not observed on other tessellations. However, 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. Furthermore, we give proof of the existence of a wave-like solution around one of the cylinder solutions by applying the knowledge of the discrete dynamical system to the recurrence relation.
... Tubular waterbomb tessellation called waterbomb tube has the non-trivial behavior that their folded states can approximate wave-like surface [2,8]. In our previous work [3], we clarified the mathematics behind the wave-like surface; the oscillating configuration is generated by the quasi-periodic solutions of the discrete dynamical system that comes from the geometric constraints. The study left two unsolved problems: (1) whether the oscillating configuration is peculiar in waterbomb tube or the universal phenomena observed in other origami tessellation; and (2) whether such oscillating origami tessellations form conservative systems analogous to an oscillating pendulum. ...
... We consider the rigid folding of waterbomb tube obtained by stitching both the left and right end of the crease-pattern of waterbomb tessellation (see Fig. 1). Following the previous research [3,4], we assume that the folded state of waterbomb tube has the symmetry of C NV , the pyramidal group of order 2N . This symmetry assumption implies that the N modules belonging to the same "ring" (yellow part in Fig. 1) are in the same configuration that is mirror-symmetric through the plane spanned by the two mountain creases. ...
... Nevertheless, it is not always true that the origami tube has the oscillating configuration. As discussed in [3], oscillating configurations of waterbomb tube are generated by the quasi-periodic solution around the elliptic fixed point of the system. Based on this, the necessary conditions for the existence of oscillating configuration of origami tube dominated by M is that the system M p have quasi-periodic solutions around the elliptic fixed point which is a p-periodic orbit of M. Furthermore, these quasi-periodic orbits must be bounded in (0, l) × (−π, π) that is the parameter space of the zigzag. ...
Folded surfaces of origami tessellations sometimes exhibit non-trivial behaviors, which have attracted much attention. The oscillation of tubular waterbomb tessellation is one example. Recently, the authors reported that the kinematics of waterbomb tube depends on the discrete dynamical system that arises from the geometric constraints between modules and quasi-periodic solutions of the dynamical system generate oscillating configurations. Although the quasi-periodic behavior is the characteristic of conservative systems, whether the system is conservative has been unknown. In this paper, we decompose the dynamical system of waterbomb tube into three steps and represent the one-step using the two kinds of mappings between zigzag polygonal linkages. By changing parameters of the mappings and composite them, we generalize the dynamical system of waterbomb tube to that of various tubular origami tessellations and show their oscillating configurations. Furthermore, by analyzing the mapping, we give proof of the conservation of the dynamical system.
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