Some remarks on Origami and its limitations

From a mathematical point of view the Japanese art of Origami is an art of finding isometric injections of subsets of ℝ2 into ℝ3. Objects obtained in this manner axe developable surfaces and they are considered to be fully understood. Nevertheless, until now it was not known whether or not the local shape of the Origami model determines the maximum size and shape of the sheet of paper it can be made of. In the present paper we show that it does. We construct a set Ω ⊂ ℝ2 containing the point (0, 1/2) and an isometry F: Ω → ℝ3 such that for every neighborhood ω ⊆ Ω of the point (0, 1/2) and for every ϵ > 0 and δ > 0, F restricted to ω cannot be extended to an isometry of the set {-ϵ < x < ϵ,-δ < y < 1 + δ} into ℝ3. We also prove that all the singularities of an Origami model are of the same type - there can appear only cones.