When can you fold a map?

  • Authors:

  • Esther M. Arkin;Michael A. Bender;Erik D. Demaine;Martin L. Demaine;Joseph S. B. Mitchell;Saurabh Sethia;Steven S. Skiena

  • Affiliations:

  • Department of Applied Mathematics and Statistics, SUNY, Stony Brook, NY;Department of Computer Science, SUNY, Stony Brook, NY;MIT Laboratory for Computer Science, 200 Technology Square, Cambridge, MA;MIT Laboratory for Computer Science, 200 Technology Square, Cambridge, MA;Department of Applied Mathematics and Statistics, SUNY, Stony Brook, NY;Department of Computer Science, SUNY, Stony Brook, NY;Computer Science, Oregon State University, 102 Dearborn Hall, Corvallis, OR

  • Venue:
  • Computational Geometry: Theory and Applications - Special issue on the 10th fall workshop on computational geometry
  • Year:
  • 2004

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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 fiat 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 1D crease patterns, and reconstruction of a sequence of simple folds. Indeed, we prove that a 1D 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.