**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?)

See also other papers by Martin Demaine.

Last updated November 17, 2022 by Martin Demaine.