Paper by Martin L. Demaine

Reference:

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.

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 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.

Comments:

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

Updates:

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.

Length:

The paper is 24 pages.

Availability:

The paper is available in PostScript (608k), gzipped PostScript (226k), and PDF (281k).

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Related papers:

MapFoldingWADS2001 (When Can You Fold a Map?)

CGW2000 (When Can You Fold a Map?)