Paper by Martin L. Demaine

Reference:

Zachary Abel, Robert Connelly, Erik D. Demaine, Martin Demaine, Thomas Hull, Anna Lubiw, and Tomohiro Tachi, “Rigid Flattening of Polyhedra with Slits”, in Origami⁶: Proceedings of the 6th International Meeting on Origami in Science, Mathematics and Education (OSME 2014), volume 1, Tokyo, Japan, August 10–13, 2014, pages 109–118, American Mathematical Society.

Abstract:

Cauchy showed that if the faces of a convex polyhedron are rigid then the whole polyhedron is rigid. Connelly showed that this is true even if finitely many extra creases are added. However, cutting the surface of the polyhedron destroys rigidity and may even allow the polyhedron to be flattened. We initiate the study of how much the surface of a convex polyhedron must be cut to allow continuous flattening with rigid faces. We show that a regular tetrahedron with side lengths 1 can be continuously flattened with rigid faces after cutting a slit of length .046 and adding a few extra creases.

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