Origami embedding of piecewise-linear two-manifolds

Abstract

We show that any compact, orientable, piecewise-linear two-manifold with Euclidean metric can be realized as a flat origami, meaning a set of non-crossing polygons in Euclidean 2-space "plus layers". This result implies a weak form of a theorem of Burago and Zalgaller: any orientable, piecewise-linear two-manifold can be embedded into Euclidean 3-space "nearly" isometrically. We also correct a mistake in our previously published construction for cutting any polygon out of a folded sheet of paper with one straight cut. © 2008 Springer-Verlag Berlin Heidelberg.