2000/11/20 by Esther M. Arkin, Arkin, Esther M., Michael A. Bender +11 · 1 citation
Computer Science · #Computational Geometry (cs.CG) #Discrete Mathematics (cs.DM) #F.2.2 #FOS: Computer and information sciences #G.2.1 #cs.CG #cs.DM
paper · pdf · doi:10.48550/arxiv.cs/0011026
24 pages, 19 figures. Version 3 includes several improvements thanks to referees, including formal definitions of simple folds, more figures, table summarizing results, new open problems, and additional references
arxiv created 2003/08/31 · arxiv updated 2009/11/30
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 degrees. 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.