Geometry and Kinematics of Cylindrical Waterbomb Tessellation

Abstract

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.