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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bern, M., Mitchell, S., Ruppert, J.: Linear-size nonobtuse triangulation of polygons. Disc. Comput. Geom. 14, 411–428 (1995)
Bern, M., Hayes, B.: The complexity of flat origami. In: Proc. 7th ACM-SIAM Symp. Disc. Algorithms, pp. 175–183 (1996)
Bern, M., Demaine, E., Eppstein, D., Hayes, B.: A disk-packing algorithm for an origami magic trick. In: E. Lodi, L. Pagli, N. Santoro, (eds.) Preliminary version: Fun with Algorithms, pp. 32–42, Carleton Scientific (1999); Also: Hull, T., Peters, A.K. (ed.) Origami3, pp. 17–28 (2002)
Burago, Y.D., Zalgaller, V.A.: Polyhedral realizations of developments (Russian). Vestnik Leningrad. Univ. 15, 66–80 (1960)
Burago, Y.D., Zalgaller, V.A.: Isometric piecewise linear embedding of two-dimensional manifolds with a polyhedral metric in IR3. St. Petersburg Math. Journal 7, 369–385 (1996)
Connelly, R.: A flexible sphere. Math. Intelligencer 1, 130–131 (1978)
Connelly, R., Sabitov, I., Walz, A.: The bellows conjecture. Contributions to Algebra and Geometry 38, 1–10 (1997)
Demaine, E., Demaine, M., Lubiw, A.: Flattening polyhedra. (Manuscript 2001)
Demaine, E., O’Rourke, J.: Geometric Folding Algorithms: Linkages, Origami, and Polyhedra. Cambridge University Press, Cambridge (2007)
Erickson, J., Har-Peled, S.: Optimally cutting a surface into a disk. In: Symp. Comp. Geometry (2002)
Hull, T.: On the mathematics of flat origamis. Congressus Numerantium 100, 215–224 (1994)
Krat, S., Burago, Y.D., Petrunin, A.: Approximating short maps by PL-isometries and Arnold’s “Can you make your dollar bigger” problem. In: Fourth International Meeting of Origami Science, Mathematics, and Education, Pasadena (2006)
Kuiper, N.H.: On C 1-isometric imbeddings I. Proc. Nederl. Akad. Wetensch. Ser. A 58, 545–556 (1955)
Lang, R.J.: Origami Design Secrets: Mathematical Methods for an Ancient Art, A.K. Peters (2003)
Nash, J.F.: C 1-isometric imbeddings. Annals of Mathematics 60, 383–396 (1954)
Nash, J.F.: The imbedding problem for Riemannian manifolds. Annals of Mathematics 63, 20–63 (1956)
Pak, I.: Inflating polyhedral surfaces. Department of Mathematics. MIT Press, Cambridge (2006)
Zalgaller, V.A.: Isometric immersions of polyhedra. Dokladi Akademii, Nauk USSR, 123(4) (1958)
Zalgaller, V.A.: Some bendings of a long cylinder. J. Math. Soc. 100, 2228–2238 (2000)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bern, M., Hayes, B. (2008). Origami Embedding of Piecewise-Linear Two-Manifolds. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds) LATIN 2008: Theoretical Informatics. LATIN 2008. Lecture Notes in Computer Science, vol 4957. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78773-0_53
Download citation
DOI: https://doi.org/10.1007/978-3-540-78773-0_53
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-78772-3
Online ISBN: 978-3-540-78773-0
eBook Packages: Computer ScienceComputer Science (R0)