Diabolical configuration spaces in simple origami

Self-folding origami, flat sheets that use mechanical instabilities to fold into three-dimensional structures, have been actuated using swelling polymer gels, surface tension, and more. A universal problem with these structures is that there can be multiple fold configurations for one origami pattern, resulting in the possibility that the structure folds into a configuration other than the one desired. Using an energy model for the structures, we show that the configuration space of a generic origami structure takes the form of nested cones, and that the number of conical branches increases with the number of internal vertices. We will use this geometry to investigate misfolding, with a focus on origami with 5 external vertices.