Abstract
We explore the following problem: given a collection of creases on a piece of paper, each assigned a folding direction of mountain or valley, is there a flat folding by a sequence of simple folds? There are several models of simple folds; the simplest one-layer simple fold rotates a portion of paper about a crease in the paper by ±180°. We first consider the analogous questions in one dimension lower—bending a segment into a flat object—which lead to interesting problems on strings. We develop efficient algorithms for the recognition of simply foldable 1-D crease patterns, and reconstruction of a sequence of simple folds. Indeed, we prove that a 1-D crease pattern is flat-foldable by any means precisely if it is by a sequence of one-layer simple folds.
Next we explore simple foldability in two dimensions, and find a surprising contrast: “map” folding and variants are polynomial, b ut slight generalizations are NP-complete. Specifically, we develop a linear-time algorithm for deciding foldability of an orthogonal crease pattern on a rectangular piece of paper, and prove that it is (weakly) NP-complete to decide foldability of (1) an orthogonal crease pattern on a orthogonal piece of paper, (2) a crease pattern of axis-parallel and diagonal (45-degree) creases on a square piece of paper, and (3) crease patterns without a mountain/valley assignment.
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
E. M. Arkin, S. P. Fekete, J. S. B. Mitchell, and S. S. Skiena. On the manufacturability of paperclips and sheet metal structures. In Proc. of the 17th European Workshop on Computational Geometry, pages 187–190, 2001.
M. Bern and B. Hayes. The complexity of flat origami. In Proc. of the 7th ACM-SIAM Symposium on Discrete Algorithms, pages 175–183, 1996.
P. Crescenzi, D. Goldman, C. Papadimitriou, A. Piccolboni, and M. Yannakakis. On the complexity of protein folding. J. of Computational Biology, 5(3), 1998.
E. D. Demaine, M. L. Demaine, and J. S. B. Mitchell. Folding flat silhouettes and wrapping polyhedral packages: New results in computational origami. In Proc. of the 15th ACM Symposium on Computational Geometry, 1999.
M. Farach. Optimal suffix tree construction with large alphabets. In Proc. of the 38th Symp. on Foundations of Computer Science, pages 137–143, 1997.
M. Gardner. The combinatorics of paper folding. In Wheels, Life and Other Mathematical Amusements, Chapter 7, pp. 60–73. W. H. Freeman and Company, 1983.
M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., 1979.
D. Harel and R. E. Tarjan. Fast algorithms for finding nearest common ancestors. SIAM J. on Computing, 13(2):338–355, 1984.
T. Hull. On the mathematics of flat origamis. Congressum Numerantium, 100:215–224, 1994.
J. Justin. Towards a mathematical theory of origami. In Koryo Miura, editor, Proc. of the 2nd International Meeting of Origami Science and Scientific Origami, pages 15–29, 1994.
R. M. Karp and M. O. Rabin. Efficient randomized pattern-matching algorithms. IBM Journal of Research and Development, 31(2):249–260, 1987.
T. Kawasaki. On the relation between mountain-creases and valley-creases of a flat origami. In H. Huzita, editor, Proc. of the 1st International Meeting of Origami Science and Technology, pages 229–237, Ferrara, Italy, December 1989. An unabridged Japanese version appeared in Sasebo College of Technology Report, 27:153–157, 1990.
R. J. Lang. A computational algorithm for origami design. In Proc. of the 12th ACM Symposium on Computational Geometry, pages 98–105, 1996.
L. Lu and S. Akella. Folding cartons with fixtures: A motion planning approach. IEEE Trans. on Robotics and Automation, 16(4):346–356, 2000.
W. F. Lunnon. Multi-dimensional map-folding. The Computer Journal, 14(1):75–80, 1971.
R. Motwani and P. Raghavan. Randomized Algorithms, Chapter 8.4, pages 213–221. Cambridge University Press, 1995.
B. Schieber and U. Vishkin. On finding lowest common ancestors: Simplification and parallelization. SIAM J. on Computing, 17(6):1253–1262, 1988.
J. S. Smith. Origami profiles. British Origami, 58, 1976.
J. S. Smith. Pureland Origami 1, 2, and 3. British Origami Society. Booklets 14, 29, and 43, 1980, 1988, and 1993.
M. Thorup. Faster deterministic sorting and priority queues in linear space. In Proc. of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 50–555, 1998.
C-H. Wang. Manufacturability-driven decomposition of sheet metal. PhD thesis, Carnegie Mellon University 1997. Technical report CMU-RI-TR-97-35.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Arkin, E.M. et al. (2001). When Can You Fold a Map?. In: Dehne, F., Sack, JR., Tamassia, R. (eds) Algorithms and Data Structures. WADS 2001. Lecture Notes in Computer Science, vol 2125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44634-6_37
Download citation
DOI: https://doi.org/10.1007/3-540-44634-6_37
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42423-9
Online ISBN: 978-3-540-44634-7
eBook Packages: Springer Book Archive