Unfolding Simple Folds from Crease Patterns

Traditionally, the folding process of origami was transmitted orally and visually by directly showing the folded paper. However, throughout history, attempts have been made to register instructions of particular origami models. Among all types of “written origami," only two are frequently used by contemporary origami artists. They are commonly referenced as origami diagrams and crease patterns. Origami diagramming was initially devised by Akira Yoshizawa in the 1950s and 1960s [Robinson 04]. It uses lines and arrows indicating the position of the folds and the movement of the paper, as shown in Figure 3 (top). Each step shows the current state of the paper and some indications on how to obtain the state shown in the next step. Usually, the diagrams show the unfolded paper in the first step and the final model in the last step. In this paper, we will show the paper with one side white and the other gray. Dot-dot-dash lines mark the locations of mountain folds, and dashed lines mark the locations of valley folds. On the other hand, crease patterns show only one picture, which is the unfolded paper containing the creases left by the folds that define the origami model. An example of a crease pattern can be seen in Figure 1. In this paper, we will show the crease pattern white face up with solid lines marking valley folds and dashed lines marking mountain folds. The importance of the crease pattern grew with the rise of mathematical origami and the new design techniques that allowed origami to reach an incredible level of complexity. The fact that the crease pattern only shows the developed/opened state of the paper makes it difficult for nonexperts to grasp any important information regarding the model. However, it indeed can be more illuminating about the origami structure than the image of the folded shape or even diagrams [Lang 04a]. For this reason, many techniques for origami design give a crease pattern as the output. Usually, when the designer produces a technical design, he or she ends up with a crease pattern but has no clue of how to actually fold the model. In fact, it is very hard to fold a model based on a crease pattern. Besides the fact that there is no apparent folding sequence expressed in it, some crease patterns might not even have a folding sequence at all [Lang 11]. Origami diagrams are composed of steps showing subsequent states of the paper. In general, what differs from one state to another is the execution of a fold. In this work, we consider a fold to be the bending of one or more layers of the paper localized in a finite number of straight line segments resulting in a dihedral angle of π or −π. Let us consider that a fold can be categorized as a simple fold or a complex fold. A simple fold is a fold along a single line that does not end in any internal point of the paper. A complex fold is