Paper by Martin L. Demaine

Esther M. Arkin, Michael A. Bender, Erik D. Demaine, Martin L. Demaine, Joseph S. B. Mitchell, Saurabh Sethia, and Steven S. Skiena, “When Can You Fold a Map?”, Computational Geometry: Theory and Applications, volume 29, number 1, September 2004, pages 23–46. Special issue of selected papers from the 10th Annual Fall Workshop on Computational Geometry, 2000.

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 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.

This paper is also available from ScienceDirect, and as arXiv:cs.CG/0011026 of the Computing Research Repository (CoRR).

Ivars Peterson wrote an article describing these results, “Proof clarifies a map-folding problem”, Science News 158(26-27):406, December 23-30, 2002.

Helen Pearson also wrote an article describing these results, “Origami solves road map riddle”, Nature Science Update, February 18, 2002.

The paper is 24 pages.

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Related papers:
MapFoldingWADS2001 (When Can You Fold a Map?)
CGW2000 (When Can You Fold a Map?)

See also other papers by Martin Demaine.
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Last updated by Martin Demaine.