When Can You Fold a Map?

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, but 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.

[1]  Esther M. Arkin,et al.  An algorithmic study of manufacturing paperclips and other folded structures , 2002, Comput. Geom..

[2]  P. Di Francesco,et al.  Folding and coloring problems in mathematics and physics , 2000 .

[3]  Robert J. Lang,et al.  A computational algorithm for origami design , 1996, SCG '96.

[4]  Marshall W. Bern,et al.  The complexity of flat origami , 1996, SODA '96.

[5]  John E. Koehler.  Folding a strip of stamps , 1968 .

[6]  Richard M. Karp,et al.  Efficient Randomized Pattern-Matching Algorithms , 1987, IBM J. Res. Dev..

[7]  Robert E. Tarjan,et al.  Fast Algorithms for Finding Nearest Common Ancestors , 1984, SIAM J. Comput..

[8]  Joseph S. B. Mitchell,et al.  Folding flat silhouettes and wrapping polyhedral packages: New results in computational origami , 2000, Comput. Geom..

[9]  Mihalis Yannakakis,et al.  On the Complexity of Protein Folding , 1998, J. Comput. Biol..

[10]  Liang Lu,et al.  Folding cartons with fixtures: a motion planning approach , 1999, IEEE Trans. Robotics Autom..

[11]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[12]  Rajeev Motwani,et al.  Randomized algorithms , 1996, CSUR.

[13]  Uzi Vishkin,et al.  On Finding Lowest Common Ancestors: Simplification and Parallelization , 1988, AWOC.

[14]  W. F. Lunnon.  A map-folding problem , 1968 .

[15]  Joseph S. B. Mitchell,et al.  Reaching folded states of a rectangular piece of paper , 2001, Canadian Conference on Computational Geometry.

[16]  W. F. Lunnon.  Multi-Dimensional Map-Folding , 1971, Comput. J..

[17]  Mikkel Thorup,et al.  Faster deterministic sorting and priority queues in linear space , 1998, SODA '98.

[18]  M. Farach.  Optimal suffix tree construction with large alphabets , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[19]  Thomas C. Hull.  On the Mathematics of Flat Origamis , 1994 .

[20]  Mihalis Yannakakis,et al.  On the complexity of protein folding (abstract) , 1998, RECOMB '98.

[21]  Toshikazu Kawasaki.  On relation between mountain-creases and valley -creases of a flat origami , 1990 .