Origami embedding of piecewise-linear two-manifolds

Abstract

Answering a question of Erik Demaine, we prove that the surface of any bounded polyhedron in Euclidean 3-space can be folded flat. That is, by adding extra vertices and edges (creases), we can re-embed the polyhedral surface as a flat origami, meaning a set of non-crossing polygons in Euclidean 2-space ‘plus layers’. More generally, we show that any compact, orientable, piecewise-linear 2-manifold with Euclidean metric can be embedded in Euclidean 2-space plus layers. This result gives a piecewise-linear analogue to the Nash embedding theorems for Riemannian manifolds.