Paper by Erik D. 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?”, in Proceedings of the 7th Workshop on Algorithms and Data Structures (WADS 2001), Lecture Notes in Computer Science, volume 2125, Providence, Rhode Island, August 8–10, 2001, pages 401–413.

BibTeX

@InProceedings{MapFoldingWADS2001,
  AUTHOR        = {Esther M. Arkin and Michael A. Bender and Erik D. Demaine
                   and Martin L. Demaine and Joseph S. B. Mitchell and
                   Saurabh Sethia and Steven S. Skiena},
  TITLE         = {When Can You Fold a Map?},
  BOOKTITLE     = {Proceedings of the 7th Workshop on Algorithms and Data
                   Structures (WADS 2001)},
  BOOKURL       = {http://www.wads.org/},
  xEDITOR       = {Frank Dehne and J\"org-R\"udiger Sack and Roberto Tamassia},
  SERIES        = {Lecture Notes in Computer Science},
  SERIESURL     = {http://www.springer.de/comp/lncs/},
  VOLUME        = 2125,
  ADDRESS       = {Providence, Rhode Island},
  MONTH         = {August 8--10},
  YEAR          = 2001,
  PAGES         = {401--413},

  COPYRIGHT     = {The paper is \copyright Springer-Verlag.},
  LENGTH        = {12 pages},
  PAPERS        = {MapFolding_CGW2014; MapFolding; CGW2000},
  doi           = {https://dx.doi.org/10.1007/3-540-44634-6_37},
  dblp          = {https://dblp.org/rec/conf/wads/ArkinBDDMSS01},
  COMMENTS      = {This paper is also available from <A HREF="https://doi.org/10.1007/3-540-44634-6_37">SpringerLink</A>, and as <A HREF="https://arXiv.org/abs/cs/0011026">arXiv:cs/0011026</A>.},
  UPDATES       = {Ivars Peterson wrote an article describing these results,
                   &ldquo;Proof clarifies a map-folding problem&rdquo;,
                   <I><A HREF="http://www.sciencenews.org/">Science
                   News</A></I> 158(26-27):406, December 23-30, 2002.
                   <P>
                   Helen Pearson also wrote an article describing these
                   results,
                   &ldquo;<A HREF="http://www.nature.com/nsu/020218/020218-1.html">Origami
                   solves road map riddle</A>&rdquo;,
                   <I><A HREF="http://www.nature.com/nsu/">Nature Science
                   Update</A></I>, February 18, 2002.},
}

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 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.

Comments:

This paper is also available from SpringerLink, and as arXiv:cs/0011026.

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.

Copyright:

The paper is \copyright Springer-Verlag.

Length:

The paper is 12 pages.

Availability:

The paper is available in PostScript (145k) and gzipped PostScript (47k).

See information on file formats.

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Related papers:

MapFolding_CGW2014 (Simple Folding is Strongly NP-Complete)

MapFolding (When Can You Fold a Map?)

CGW2000 (When Can You Fold a Map?)