## Recent Results in Computational Origami (2001)

Venue: | In Proceedings of the 3rd International Meeting of Origami Science, Math, and Education |

Citations: | 18 - 3 self |

### BibTeX

@INPROCEEDINGS{Demaine01recentresults,

author = {Erik D. Demaine and Martin L. Demaine},

title = {Recent Results in Computational Origami},

booktitle = {In Proceedings of the 3rd International Meeting of Origami Science, Math, and Education},

year = {2001},

pages = {3--16}

}

### OpenURL

### Abstract

Computational origami is a recent branch of computer science studying efficient algorithms for solving paper-folding problems. This field essentially began with Robert Lang's work on algorithmic origami design [25], starting around 1993. Since then, the field of computational origami has grown significantly. The purpose of this paper is to survey the work in the field, with a focus on recent results, and to present several open problems that remain. The survey cannot hope to be complete, but we attempt to cover most areas of interest.

### Citations

101 | A Combinatorial Approach to Planar Non-Colliding Robot Arm Motion Planning
- Streinu
- 2000
(Show Context)
Citation Context ... [6] recently proved that the answer is always yes. In addition, every polygon (polygonal loop) can be folded into a convex conguration. Some examples of these motions are shown in Figure 5. Streinu [=-=35] has -=-developed another motion based on [6]. Figure 5: Convexifying the \doubled tree" on the left. See http://db.uwaterloo.ca/ ~eddemain/linkage/ for more animations. 4.2 Folding and Unfolding Polyhed... |

78 | Straightening polygonal arcs and convexifying polygonal cycles, Discrete and Computational Geometry 30
- Connelly, Demaine, et al.
- 2003
(Show Context)
Citation Context ...blem is usually stated in the (equivalent) reverse direction: can every polygonal chain be straightened out without self-intersection and while preserving the bar lengths? Connelly, Demaine, and Rote =-=[6-=-] recently proved that the answer is always yes. In addition, every polygon (polygonal loop) can be folded into a convex conguration. Some examples of these motions are shown in Figure 5. Streinu [35]... |

56 | Folding and unfolding in computational geometry
- O’Rourke
- 1998
(Show Context)
Citation Context ...roblem of interest is how a particular object (e.g., a piece of paper, a linkage, or a polyhedron) can be folded subject to some natural constraints. For general surveys on folding and unfolding, see =-=[7, 30]-=-. 7 4.1 Linkage Folding: One-Dimensional Origami Consider a one-dimensional piece of paper, a line segment, marked at certain points with creases. It is most natural to fold such a piece of paper in t... |

38 |
A computational algorithm for origami design
- Lang
- 1996
(Show Context)
Citation Context ...Jun Maekawa, Fumiaki Kawahata, and Toshiyuki Meguro. This work has led to several successful designs, but a full survey is beyond the scope of this paper; see [24]. Here we concentrate on Lang's work =-=[23, 24, 25-=-]; over the past several years, he has developed the tree method to the point where an algorithm and computer program have been explicitly dened and implemented. The tree method allows one to design a... |

26 | When can a polygon fold to a polytope
- Lubiw, O’Rourke
- 1996
(Show Context)
Citation Context ...and, given a polygonal piece of paper, we might ask whether it can be folded and its edges can be glued together so as to form a convex polyhedron. This problem has been studied by Lubiw and O'Rourke =-=[26]-=-, and by Demaine, Demaine, Lubiw, and O'Rourke [12]. A particularly surprising discovery from this work [26] is that the well-known cross unfolding of the cube can be folded into exactlysve convex pol... |

19 | Folding and cutting paper
- Demaine, Demaine, et al.
- 1998
(Show Context)
Citation Context ...ght cut, and unfold the pieces. What shapes can result? This fold-and-cut problem wassrst formally stated by Martin Gardner in 1960 [16], but has a much longer history, going as far back as 1721; see =-=[9]-=-. More formally, given a planar graph drawn with straight edges on a piece of paper, can the paper be foldedsat so as to map the entire graph to a common line, and map nothing else to that line? The s... |

19 |
On the relation between mountain-creases and valley creases of a flat origami
- Kawasaki
- 1989
(Show Context)
Citation Context ...sat foldability. Without specied crease directions, a single-vertex crease pattern issat-foldable precisely if the alternate angles around the vertex sum to 180 . This is known as Kawasaki's theorem [=-=5, 18, 20, 21-=-]. When the angle condition is satised, a characterization of valid mountain-valley assignments andsat foldings can be found in linear time [5, 20], using Maekawa's theorem [5, 18, 20] and another the... |

16 | Ununfoldable Polyhedra
- Bern, Demaine, et al.
- 1999
(Show Context)
Citation Context ... issat: a polygon in the plane, possibly with holes. Suppose instead we start with a piece of paper that is a polyhedral surface in space. How can polyhedra be folded? Specically, a natural question [=-=11]-=- is whether every polyhedron can besattened : folded into asat origami. Demaine, Demaine, and Lubiw [11] have shown that there aresattened states of several classes of polyhedra, including convex poly... |

13 | A disk-packing algorithm for an origami magic trick
- Bern, Demaine, et al.
- 2001
(Show Context)
Citation Context ...to somesexibility in the edge lengths. TreeMakersnds a crease pattern that results in the desired uniaxial base. Work on a related problem (see the next section) by Bern, Demaine, Eppstein, and Hayes =-=[4]-=- suggests a method forsnding an appropriate mountain-valley assignment for the crease pattern, and possibly also the resulting folded state. The reader is referred to [25] in this proceedings for more... |

11 | Enumerating foldings and unfoldings between polygons and polytopes
- Demaine, Demaine, et al.
(Show Context)
Citation Context ... whether it can be folded and its edges can be glued together so as to form a convex polyhedron. This problem has been studied by Lubiw and O'Rourke [26], and by Demaine, Demaine, Lubiw, and O'Rourke =-=[12]-=-. A particularly surprising discovery from this work [26] is that the well-known cross unfolding of the cube can be folded into exactlysve convex polyhedra by edge-to-edge 8 gluing: a doubly covered (... |

11 |
Towards a mathematical theory of origami
- Justin
- 1994
(Show Context)
Citation Context ...sat foldability. Without specied crease directions, a single-vertex crease pattern issat-foldable precisely if the alternate angles around the vertex sum to 180 . This is known as Kawasaki's theorem [=-=5, 18, 20, 21-=-]. When the angle condition is satised, a characterization of valid mountain-valley assignments andsat foldings can be found in linear time [5, 20], using Maekawa's theorem [5, 18, 20] and another the... |

10 | Computing extreme origami bases
- Demaine, Demaine
- 1997
(Show Context)
Citation Context ...proved formally, by Demaine and Mitchell [14], and so far only for rectangular pieces of paper. The only other paper of which we are aware that proves the existence of continuous folding processes is =-=[8]-=-. This paper proves that every convex polygon can be folded into a 6 uniaxial base via Lang's universal molecule [25] without gussets. Furthermore, unlike [14], no additional creases are introduced du... |

9 |
Paper cutting
- Gardner
(Show Context)
Citation Context ...aight Cut Take a piece of paper, fold itsat, make one complete straight cut, and unfold the pieces. What shapes can result? This fold-and-cut problem wassrst formally stated by Martin Gardner in 1960 =-=[16]-=-, but has a much longer history, going as far back as 1721; see [9]. More formally, given a planar graph drawn with straight edges on a piece of paper, can the paper be foldedsat so as to map the enti... |

8 | When can you fold a map
- Arkin, Bender, et al.
- 2001
(Show Context)
Citation Context ...ases, each marked mountain or valley, can it be foldedsat via a sequence of simple folds? Traditionally, map folding has been studied from a combinatorial point of view; see, e.g., [27]. Arkin et al. =-=[1]-=- have shown that deciding foldability of a map by simple folds can be solved in polynomial time. If the simple folds are required to fold all layers at once, the running time is at most O(n log n), an... |

8 |
The complexity of origami
- Bern, Hayes
- 1996
(Show Context)
Citation Context ...ple layers of paper. For what polyhedral shapes (shapes made up ofsat sides) is this possible? This problem is implicit throughout origami design, and wassrst formally posed by Bern and Hayes in 1996 =-=[5]. The-=- surprising answer is \always," as established by Demaine, Demaine, and Mitchell in 1999 [13]. The basic idea of the approach is to fold the piece of paper into a thin strip of paper, and then wr... |

7 |
Multi-dimensional map-folding
- Lunnon
- 1971
(Show Context)
Citation Context ...al and vertical creases, each marked mountain or valley, can it be foldedsat via a sequence of simple folds? Traditionally, map folding has been studied from a combinatorial point of view; see, e.g., =-=[27]-=-. Arkin et al. [1] have shown that deciding foldability of a map by simple folds can be solved in polynomial time. If the simple folds are required to fold all layers at once, the running time is at m... |

6 | Folding and unfolding linkages, paper and polyhedra”, pp. 113–124 in Discrete and computational geometry (JCDCG, Tokyo, 2000: Revised papers), edited by M
- Demaine
- 2001
(Show Context)
Citation Context ...roblem of interest is how a particular object (e.g., a piece of paper, a linkage, or a polyhedron) can be folded subject to some natural constraints. For general surveys on folding and unfolding, see =-=[7, 30]-=-. 7 4.1 Linkage Folding: One-Dimensional Origami Consider a one-dimensional piece of paper, a line segment, marked at certain points with creases. It is most natural to fold such a piece of paper in t... |

6 |
Folding silhouettes and wrapping polyhedral packages: New results in computational origami
- Demaine, Demaine, et al.
- 1999
(Show Context)
Citation Context ...is problem is implicit throughout origami design, and wassrst formally posed by Bern and Hayes in 1996 [5]. The surprising answer is \always," as established by Demaine, Demaine, and Mitchell in =-=1999 [13]-=-. The basic idea of the approach is to fold the piece of paper into a thin strip of paper, and then wrap this strip around the desired shape. This wrapping can be done particularly eciently using meth... |

5 |
Folding and one straight cut suce
- Demaine, Demaine, et al.
- 1999
(Show Context)
Citation Context ...is is always possible, for any collection of line segments in the plane, forming nonconvex polygons, adjoining polygons, nested polygons, etc. There are two solutions to the problem. Thesrst solution =-=[9, 10]-=- is based on a structure called the straight skeleton, which captures the symmetries of the graph, thereby exploiting a more global structure to the problem. The second solution [4] is based on disk p... |

5 | Reaching folded states of a rectangular piece of paper
- Demaine, Mitchell
- 2001
(Show Context)
Citation Context ... by appropriately exing the paper, any folded state can be reached by a continuous motion, so the two models should be equivalent. Only recently has this been proved formally, by Demaine and Mitchell =-=[14]-=-, and so far only for rectangular pieces of paper. The only other paper of which we are aware that proves the existence of continuous folding processes is [8]. This paper proves that every convex poly... |

5 |
Origami Inside-Out
- Montroll
- 1993
(Show Context)
Citation Context ...both sides, and the shape may be two-colored according to which side should be showing. In principle, this allows the design of two-color models similar to the models in Montroll's Origami Inside-Out =-=[29]-=-. An example is shown in Figure 2. Of course, because of the reliance on thin strips, none of these methods lead to practical foldings, except for small examples or when the initial piece of paper is ... |

5 |
Pureland Origami 1, 2, and 3
- Smith
- 1980
(Show Context)
Citation Context ...plex model of simple folds. A simple fold (or book fold) is a fold by 180 along a single line. Examples are shown in Figure 4. This model is closely related to \pureland origami" introduced by Sm=-=ith [33, 34]-=-. We can ask the same foldability questions for a sequence of simple folds. Given a crease pattern, can it be foldedsat via a sequence of simple folds? What if a particular mountainvalley assignment i... |

4 |
TreeMaker 4.0: A Program for Origami Design
- Lang
- 1998
(Show Context)
Citation Context ...ical origami community, in particular by Jun Maekawa, Fumiaki Kawahata, and Toshiyuki Meguro. This work has led to several successful designs, but a full survey is beyond the scope of this paper; see =-=[24-=-]. Here we concentrate on Lang's work [23, 24, 25]; over the past several years, he has developed the tree method to the point where an algorithm and computer program have been explicitly dened and im... |

3 |
Computational tools for origami tessellations
- Bateman
- 2001
(Show Context)
Citation Context ...specifying creases around these faces is based on shrinking the tiles and introducing the dual tiling in the resulting gaps, using the notion of a hinged primal-dual tiling (see, e.g., [37]). Bateman =-=[3]-=- has formalized this method to the point of a computer implementation, called Tess. See [3] in this proceedings for more details. Some of the key people working on origami tessellations include Alex B... |

3 | African Animals in Origami - Montroll - 1991 |

2 |
Sojo suru origami asobi no shotai (Invitation to creative origami playing). Asahi Culture
- Fujimoto
- 1982
(Show Context)
Citation Context ...omputer implementation, called Tess. See [3] in this proceedings for more details. Some of the key people working on origami tessellations include Alex Bateman [3], Paulo Barreto [2, 31], S. Fujimoto =-=[15]-=-, Thomas Hull (unpublished), Toshikazu Kawasaki [22], Robert Lang (unpublished), Chris Palmer [31, 32], and Helena Verrill [36]. Unfortunately, much of the work on origami tessellations has not been w... |

2 |
Crystallographic origamis
- Kawasaki, Yoshida
- 1989
(Show Context)
Citation Context ...lyhedra. 4 2.4 Origami Tessellations Roughly, an origami tessellation is asat folding of a piece of paper based on a tessellation or tiling of the plane [17]. One way to make this notion more precise =-=[22, 36-=-] is to consider the whole plane as the piece of paper and dene a symmetric origami tessellation to be asat folding of the plane whose symmetry group is one of the 17 crystallographic groups. To avoid... |

2 |
Area Rapid Folders Newsletter. Issues
- Bay
- 1994
(Show Context)
Citation Context ...d to the point of a computer implementation, called Tess. See [3] in this proceedings for more details. Some of the key people working on origami tessellations include Alex Bateman [3], Paulo Barreto =-=[2, 31]-=-, S. Fujimoto [15], Thomas Hull (unpublished), Toshikazu Kawasaki [22], Robert Lang (unpublished), Chris Palmer [31, 32], and Helena Verrill [36]. Unfortunately, much of the work on origami tessellati... |

2 |
Extruding and tesselating polygons from a plane
- Palmer
- 1994
(Show Context)
Citation Context ... people working on origami tessellations include Alex Bateman [3], Paulo Barreto [2, 31], S. Fujimoto [15], Thomas Hull (unpublished), Toshikazu Kawasaki [22], Robert Lang (unpublished), Chris Palmer =-=[31, 32]-=-, and Helena Verrill [36]. Unfortunately, much of the work on origami tessellations has not been written formally, so the exact computational results are unclear. Certainly a wide range of origami tes... |

2 |
Origami pro
- Smith
- 1976
(Show Context)
Citation Context ...plex model of simple folds. A simple fold (or book fold) is a fold by 180 along a single line. Examples are shown in Figure 4. This model is closely related to \pureland origami" introduced by Sm=-=ith [33, 34]-=-. We can ask the same foldability questions for a sequence of simple folds. Given a crease pattern, can it be foldedsat via a sequence of simple folds? What if a particular mountainvalley assignment i... |

2 |
Origami tessellations
- Verrill
- 1998
(Show Context)
Citation Context ...lyhedra. 4 2.4 Origami Tessellations Roughly, an origami tessellation is asat folding of a piece of paper based on a tessellation or tiling of the plane [17]. One way to make this notion more precise =-=[22, 36-=-] is to consider the whole plane as the piece of paper and dene a symmetric origami tessellation to be asat folding of the plane whose symmetry group is one of the 17 crystallographic groups. To avoid... |

2 |
Hinged tessellations
- Wells
- 1991
(Show Context)
Citation Context ...One method for specifying creases around these faces is based on shrinking the tiles and introducing the dual tiling in the resulting gaps, using the notion of a hinged primal-dual tiling (see, e.g., =-=[37]-=-). Bateman [3] has formalized this method to the point of a computer implementation, called Tess. See [3] in this proceedings for more details. Some of the key people working on origami tessellations ... |

1 |
Lines meeting on a surface: The \Mars" paperfolding
- Barreto
- 1994
(Show Context)
Citation Context ...d to the point of a computer implementation, called Tess. See [3] in this proceedings for more details. Some of the key people working on origami tessellations include Alex Bateman [3], Paulo Barreto =-=[2, 31]-=-, S. Fujimoto [15], Thomas Hull (unpublished), Toshikazu Kawasaki [22], Robert Lang (unpublished), Chris Palmer [31, 32], and Helena Verrill [36]. Unfortunately, much of the work on origami tessellati... |

1 |
On the mathematics of origamis. Congressum Numerantium
- Hull
- 1994
(Show Context)
Citation Context ...sat foldability. Without specied crease directions, a single-vertex crease pattern issat-foldable precisely if the alternate angles around the vertex sum to 180 . This is known as Kawasaki's theorem [=-=5, 18, 20, 21-=-]. When the angle condition is satised, a characterization of valid mountain-valley assignments andsat foldings can be found in linear time [5, 20], using Maekawa's theorem [5, 18, 20] and another the... |

1 |
Counting mountain-valley assignments for folds
- Hull
- 2001
(Show Context)
Citation Context ...em of Kawasaki [5, 18, 21] about constraints on mountains and valleys. In particular, Hull has shown that the number of distinct mountain-valley assignments of a vertex can be computed in linear time =-=[19]-=-. A crease pattern is called locally foldable if there is a mountain-valley assignment so that each vertex locally foldssat, i.e., a small disk around each vertex foldssat. Testing local foldability i... |

1 |
Trees and circles: an ecient algorithm for origami design
- Lang
- 2001
(Show Context)
Citation Context ...tational origami is a recent branch of computer science studying ecient algorithms for solving paper-folding problems. Thisseld essentially began with Robert Lang's work on algorithmic origami design =-=[25-=-], starting around 1993. Since then, theseld of computational origami has grown signicantly. The purpose of this paper is to survey the work in theseld, with a focus on recent results, and to present ... |