Origami tessellations are origami obtained by tiling translational copies of a modular crease pattern. Origami tessellations can be folded from flat sheets of paper and transform into various surfaces, which can approximate both synclastic and anticlastic surfaces. Such macroscopic surfaces exhibit recently attracted much attention from scientists and engineers.In this paper, we study the kinematics of waterbomb tube, i.e., the cylindrical form obtained from waterbomb tessellation. Existing studies reported &nbsp;that waterbomb tubes could form wave-like surfaces, a unique phenomenon not observed in the folded surfaces of other tessellations. However, the theoretical reason why wave-like surfaces arise has been unclear.Our objective is to know “why” waterbomb tubes produce wave-like surfaces, i.e., to clarify the mathematics behind the behavior. In this paper, we first provide the kinematic model of each module of waterbomb tube and the relation with adjacent modules to obtain the recurrence relation dominating the folded states of modules. Then, we visualized the folded states of waterbomb tube as the sequence of points in the phase space using the recurrence relation. &nbsp;We observe that the solutions fall into three types: cylinder solution, wave-like solution, and finite solution. Finally, by applying theorems of discrete dynamical systems to the recurrence relation, we prove the existence of quasi-periodic orbits around one of the cylinder solutions, which is the mathematical structure producing wave-like surfaces.

Geometry and Kinematics of Cylindrical Waterbomb Tessellation

Paper Proceedings: https://asmedigitalcollection.asme.org/IDETC-CIE/IDETC-CIE2021/volume/85369

ASME 2021 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference (IDETC/CIE2021)

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