A Computational Algorithm for Origami Design Robert J. Lang 7580 Olive Drive Pleasanton, CA 94588 [email protected] 1. Introduction right number of flaps, it is relatively easy to transform it into a recognizable representation of the subject, so for complex 1.1 Background subjects, at least, most effort is focused on folding the base to Origami is the Japanese name for the centuries-old art of folding begin with. paper into representations of birds, insects, animals, plants, human I begin in section 2 by establishing a mathematical definition of a figures, inanimate objects, and abstract shapes. In the purest form base and several useful measures of its properties, including the of origami, the figure is folded from a single uncut square of tree graph that forms an abstraction of the base and that serves to paper. In the past 50 years, the complexity of origami figures has define the target design. In section 3 I establish a set of necessary increased dramatically, evolving from simple, stylized shapes to conditions relating key vertices of the crease pattern to properties highly detailed representations of crustacea [1], insects [2, 3], of the desired base. These necessary conditions help identify dinosaurs [4, 5], and other multi-legged, multi-winged, and “active paths” — primary valley creases that subdivide the crease generally multi-appendaged creatures. As the range of origami pattern into convex polygons, which correspond to distinct subject matter has expanded in complexity, the problem of folding portions of the base. Sections 4 and 5 introduce the “universal a base — a geometric shape with flaps for each of the appendages molecule,” an algorithm for filling in the creases in each polygon. of the subject — has assumed greater and greater importance. Section 6 describes the mathematical and computer Indeed, the problem of folding a multi-pointed base constitutes the implementation of the algorithm and gives an example of its usage bulk of that branch of the art known as origami sekkei, or in the construction of an origami base. “technical folding.” 2. Sufficient Conditions Throughout the history of origami, most origami design has been carried out by a combination of trial and error and/or heuristic 2.1 What is a Base? techniques based on the folder’s intuition. In this paper, I present To start, I will establish some definitions: for the first time a complete algorithm for the design of an A crease pattern is a division of the unit square into a finite set of arbitrary origami figure, specifically, for the solution of a crease polygonal regions by a set of straight line segments. Each segment pattern that folds flat into a base with any desired number of flaps is called a crease. Each polygon, which is bounded by a of arbitrary length, which become the arms, legs, wings, etc., of combination of creases and the edge of the square, is called a the origami figure. The algorithm is based on a set of facet of the crease pattern. mathematical conditions on the mapping between the crease A base is a non-stretching transformation of the unit square into pattern and a tree graph representing the base. In this fashion, the 3-space such that the facets remain flat, i.e., all folding occurs problem is transformed into a nonlinear constrained optimization, only on creases. A base can therefore be fully defined by the which as it turns out, is closely related to existing circle packing locations of the creases, their angles, and the orientation and and triangulation algorithms. I have implemented the algorithm in location of a facet. a computer program written in C++ that is available on the Internet; with it one can compute the crease pattern for origami A flat-foldable crease pattern is any crease pattern that can be designs of unprecedented complexity and sophistication. folded into a base so that all layers of the base lie in a common plane. 1.2 Previous Work A base may have several properties: The problem of designing an origami shape based on the number of appendages has been recognized for years [6] and several 1. Projectability. A projectable base can be oriented in 3-space so workers have addressed various aspects of origami design [7–9], that all facets of the base are perpendicular to the xy plane. The albeit typically at a conceptual rather than an algorithmic level. In projection of any base or element of a base is the projection of recent years, several authors, notably Meguro [10], Maekawa [11], that element into the xy plane. That is, the projection of a point Kawahata [12], and Kawasaki [13] have described useful (x,y,z) is the point (x,y). Since the facets of a projectable base are geometric algorithms for the design of a base in which the target perpendicular to the xy plane, the projection of any facet of a shape is represented as a stick figure, and flaps of the base are projectable base is a line segment. represented by nonoverlapping circles and/or circular contours. I y have also described a geometric algorithm for origami design in (0,1) (1,1) [14]. The present work builds upon and generalizes concepts presented in and implicit in [10–14], mathematically proves the underlying theory, and completes the algorithm begun in [14] in a form suitable for computer implementation. 1.3 Outline of the Approach The basis of many complex origami designs is a geometric shape called a base, which is a folded configuration of the original square that has the same number of flaps of the same length as the x appendages of the subject. Once the folder has a base with the (0,0) (1,0) Figure 1. Left: a crease pattern within the unit square. Right: the crease pattern embedded in 3-space and transformed into a base. The projection of the base lies in the xy plane. The projections of facets that meet along a crease are line segments that touch at a point or overlap along a line segment. Since the facets of a crease pattern are connected, the set of all 2.2 Comment line segments formed by projecting individual facets into the xy Throughout the history of origami, most but not all origami bases plane is connected as well and forms a planar embedding of a tree have been uniaxial bases. The four classic bases of traditional graph. The line segments thus formed are the edges of the graph; Japanese origami — the Kite, Fish, Bird, and Frog bases — may points where two edges come together are nodes of the graph. be configured to be uniaxial, as can many more complicated and Nodes that have exactly one edge connected to them are terminal sophisticated bases exploited by modern technical folders [3,4,6– nodes; the remaining nodes are internal nodes. 9]. It should be noted, however, that there are many bases that are Definition: a flap is a group of facets in a base that project to a not uniaxial (notably Montroll’s Dog Base [2] and its derivatives common edge of the tree graph such that: [5]) as well as curved and/or three-dimensional structures, all of 1. Every facet in the flap projects to the same edge; which require different design strategies. 2. Any facet in the base that projects to the edge and is connected 3. Necessary Conditions to a facet of the flap is itself a part of the flap. 3.1 Tree Conditions 3. The boundary of the facets that make up the flap projects to one The transformation from the two-dimensional crease pattern into or two nodes that are the endpoint(s) of the edge. the three-dimensional base can be expressed by a mapping Therefore each flap in a base is associated with an edge of the tree F(P)→Q, where P is a point in the unit square and Q is the graph. In many cases, the flaps of a projectable base can be corresponding point in the 3-space representation of the base. positioned at different angles, in 3-space giving projected tree According to the conventions of origami transformations (no graphs that differ in the arrangement of their edges. Note that two stretching, no self-intersection) [15–16], F is one-to-one and I will side-by-side flaps may project to overlapping line segments in the call F the transformation operator. xy plane; we still distinguish different flaps by associating them Similarly, a mapping G(Q)→R expresses the projection of the with distinct edges of the tree graph. To avoid ambiguity, a tree point Q onto the tree graph. G, which I will call the projection graph will always be drawn with nonoverlapping edges, but it operator, is obviously not one-to-one, since all points with the should be understood that such a graph could represent a base same x and y coordinates map onto the same point on the tree with side-by-side flaps or even one flap wrapped around another, graph. If S is the crease pattern in the unit square, the as shown in figure 2, whose projections include overlapping transformation of S into a base B and thence into a tree graph T edges. can be written T=G(F(S)) when F is known. The design of a uniaxial base boils down to the inversion of this problem: given T, find the mapping F and crease pattern S that transform and project S into T. As I will show, knowledge of T places some restrictive B conditions on the crease pattern S that are key to the solution of the problem.