Geometry and Kinematics of Cylindrical Waterbomb Tessellation

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.

Automated summary

This paper proposes a kinematic model for the waterbomb tube and investigates its behavior by parameterizing the geometry and deriving a recurrence relation between the modules of the waterbomb tessellation. Through visualization, solutions are classified into cylinder, wave-like, and finite solutions. Analyzing the stability and bifurcation of the dynamical system, the changes in the waterbomb tube's behavior with changes in the crease pattern are examined. Furthermore, the existence of a wave-like solution around one of the cylinder solutions is proven.