Folding Equilateral Plane Graphs
We consider two types of folding applied to equilateral plane graph linkages. First, under continuous folding motions, we show how to reconfigure any linear equilateral tree (lying on a line) into a canonical configuration. By contrast, such reconfiguration is known to be impossible for linear (nonequilateral) trees and for (nonlinear) equilateral trees. Second, under instantaneous folding motions, we show that an equilateral plane graph has a noncrossing linear folded state if and only if it is bipartite. Not only is the equilateral constraint necessary for this result, but we show that it is strongly NP-complete to decide whether a (nonequilateral) plane graph has a linear folded state. Equivalently, we show strong NP-completeness of deciding whether an abstract metric polyhedral complex with one central vertex has a noncrossing flat folded state with a specified “outside region”. By contrast, the analogous problem for a polyhedral manifold with one central vertex (single-vertex origami) is only weakly NP-complete.
- 3.Connelly, R., Demaine, E.D., Rote, G.: Infinitesimally locked self-touching linkages with applications to locked trees. In: Calvo, J., Millett, K., Rawdon, E. (eds.) Physical Knots: Knotting, Linking, and Folding of Geometric Objects in R3, pp. 287–311. American Mathematical Society (2002)Google Scholar
- 5.Demaine, E.D., Devadoss, S.L., Mitchell, J.S.B., O’Rourke, J.: Continuous foldability of polygonal paper. In: Proceedings of the 16th Canadian Conference on Computational Geometry, Montréal, Canada, pp. 64–67 (August 2004)Google Scholar
- 6.Demaine, E.D., O’Rourke, J.: Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge University Press (2007)Google Scholar
- 8.Kawasaki, T.: On the relation between mountain-creases and valley-creases of a flat origami. In: Huzita, H. (ed.) Proceedings of the 1st International Meeting of Origami Science and Technology, pp. 229–237, Ferrara, Italy (December 1989); An unabridged Japanese version appeared in Sasebo College of Technology Report 27, 153–157 (1990)Google Scholar