Closed-form solution for semi-symmetric folding of waterbomb tessellations

Abstract

Rigid-origami folding has versatile applications in the fields of architecture, engineering, medicine and aerospace. Given the design parameters of a single waterbomb cell, the global geometry of the origami tessellated shell at an intermediate state of folding is directly determined. Even though existing numerical methods of rigid-origami can be used to describe geometric forms, algebraic solutions of the waterbomb folding kinematics and tessellations reduce the algorithmic complexity and determine the range of admissible configurations exactly. The existing analytical solution for a symmetrically folded waterbomb cell and tessellation is extended to the case of a cell folded semi-symmetrically, i.e. a cell with a 180° rotation symmetry. The algebraic derivation provides the solution for the entire folding process. The resulting functional relationship between the design parameters and the 3D cell geometry of the folded waterbomb cell enables efficient studies of achievable shell geometries. This development facilitates a deeper understanding of the correspondence between the geometric parameters of a single waterbomb cell and the statically relevant geometric properties, i.e. the shell curvature and the effective cross-sectional height. The systematic description of this correspondence provides the basis for innovative design concepts using modern cementitious composites, e.g. textile-reinforced concrete, to produce thin-walled, high-performance, lightweight shells.