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.
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Bern, M., Hayes, B. Origami Embedding of Piecewise-Linear Two-Manifolds. Algorithmica 59, 3–15 (2011). https://doi.org/10.1007/s00453-010-9399-8
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DOI: https://doi.org/10.1007/s00453-010-9399-8