Symmetric Self-folding of N-Gon Hypar Origami

The hypar origami is folded from concentric pleated polygons, whose folded configuration is characterized by a negative Gaussian curvature. Experiments show that the hypar origami with N-sides exhibit multiple symmetric states. When N=4, the folded square hypar pattern is proven to converge to a standard hyperbolic paraboloid shape, with two symmetric stable states. When N=6, the hexagon hypar pattern demonstrates two distinct configurations that follows different symmetry groups. In this research, we systematically look into the bifurcation configurations of N-gon hypar origami by means of self-folding. The simulation is performed based on a modified version of the MERLIN software that models nonlinear deformation of origami structures. We discover that the folded shapes of N-gon hypar origami is strongly related to the symmetry type of the initial perturbation. However, as N increases, configurations with higher order of symmetry become energetically unstable, and the least symmetric configuration approaches a mechanism with floppy mode. This work paves the way to develop a unified mechanics model for N-gon hypar origami, and shed light on the possible applications of the hypar origami as shape morphing metasurfaces.