"Origami Graphics"
Abstract
The goal of this project was to derive an algorithm for modeling origami on a computer. First, the nature of origami, or paper-folding, was studied. Then, an origami model, the Fortune Teller, was chosen as an example to be illustrated on the computer. The pattern made by the creases in an unfolded model was studied and labeled. From this point, the image was drawn into the computer and manipulated to simulate paper folds.
Narrative
About Origami
The word "origami" came from a Japanese term meaning "paper-folding." Origami might have originated in China, but the techniques and traditions of origami became more developed in Japan. Many schools of origami existed. Some strictly believed that only square papers should be used, while others accepted all shapes and sizes, but believed that cutting the paper should be strictly prohibited. Many forms of origami were accepted today.
Origami artists have made such a wide variety of objects from boxes and ornaments to fully functioning grandfather clocks. Surprisingly, the variety of objects that could be folded using a piece of paper was not random. In fact, most folds were symmetrical, meaning what was done on one side was usually done on another. Usually the first step included a fold that divided the area of the paper in half. This could either be a diagonal fold or a fold parallel to one of the edges. After this, there was something called a "base." A base was a set of folds that resembled an unfinished model. It allowed enough freedom so that the next step could make a difference in the finished model.
Project: Comuter Modeling of Origami
The purpose of this project was to construct a computer model of the "Fortune Teller," an origami object resembling a child's toy. (To see what the Fortune Teller is, or to learn how to make one,
go here, and
here.)
Although the goal was to reconstruct all the steps for making this figure (eventually ending up with the completed model), the goal was not achieved in the allowed time. Only the first four folds were completed.
To do origami by hand differed from illustrating origami using computer graphics because computer graphics involve writing detailed codes and more mathematics than simply taking a piece of paper and folding it by hand. In a computer model, the image was not a piece of paper, but a plane in space defined by a coordinate system.
Before making the computer model, a paper model of the Fortune Teller was made. To study its structures and constructions, the model was unfolded entirely. The creases on the unfolded paper generated a folding pattern that was defined here as a "sequence map." Although each object had their own characteristic sequence map, the pattern itself was not unique because different objects could generate the same pattern (if folds were done in different order/orientation). However, if both the order and orientation of folds were provided, then a sequence map would generate a path for reconstructing the object.
Another use of sequence maps allowed origami artists to search for patterns. For instance, if the sequence maps of six-legged insects were compared to each other, then certain resemblance, such as the location of all the legs, could be identified. Once the pattern was found, it could lead to new inventions or further development of the object.
The sequence map of a Fortune Teller divided the paper into thirty-two identical triangular sections. From this sequence map, three things were done. First of all, since all the vertices lined up in a grid, a coordinate system was sufficient to illustrate the map. If one corner was the origin a side was on the x- or y-axis, and the rest of the vertices could be labeled according to their positions on the x-y plane. Second, each small triangle was assigned a number. Third, the edges were thought of as line segments, and were also assigned a number. When the piece of square paper was drawn, it was actually a construction of multiple triangles whose positions were assigned according to their coordinates. Now, the paper not only had an appearance of an unfolded Fortune Teller, but could also be manipulated by specifying where the folds (line segments where two triangles met) should take place.
In order to do origami graphics, we looked for the "functions" that needed to be defined in the computer program. These functions include two types of folds:
1. mountain folds, in which a flap paper is folded OVER
2. valley folds, in which the flap is folded UNDER
Mountain and valley folds were actually the same folds oriented in opposite directions. In other words, the valley fold could almost always be done by doing a mountain fold after turning the paper over. To do this on the computer, a ¡§bend¡¨ function was defined. This function specified a line (an edge), and rotated the plane on one side of the line along this edge. The problem with this method was that once the flap moved 90º in one direction, it was no longer on the specified side. The consequence was a fold that bent halfway, and locked in place at 90º to the direction it rotated.
This program was originally called the IlliSkeleton. The IlliSkeleton was written by many people, and now modified to fit this model by my partner, Yana Malysheva.
Project: Suggestions for Improvement
In origami, not everything can be made with just folding, so some difficulties arose while modeling origami on a computer. For example, a plane in space was sufficient in illustrating a piece of paper. However, folding the paper required manipulation of this piece of paper in the 3-D space. The algorithm chosen here was to rotate part of the plane along a line on the plane. This method worked for the first few steps (see explanation above), but it would soon become too complicated because what appeared to be on the same line might have been different segments on the original paper due to overlapping.
A different algorithm was chosen for this project in the very beginning, but it was not used in the actual project. This could be experimented as the alternative of the method above. First, a model was made from paper. Then, the sequence map was drawn. In this map, each crease divided the paper into a smaller portion of the plane. The method was to name each crease, vertices, or plane, and then simply map a specified portion of the plane to the new location.
Realistically, many constructions require opening up a figure and then flatten it, or twisting an edge so that the figure remains "flat." These are all problems that should be solved in this project because they will allow a wider range of modeling, and possibly allow the computer to construct a step-by-step instruction for a final product.
A suggestion for future projects is to not look at the object, but only the sequence maps. Usually a sequence map appears to have many symmetrical patterns. Perhaps a way to understand general origami construction is to analyze the creases from various models and find their common parts. Another interesting idea would be to find out how and why only slight variations in the crease patterns allow the construction of a great variety of objects.
Instructions
The name of file was ori3. Once the file was retrieved, there were four buttons to animate the figure. Each button, "b," "c," "d," and "e" activated one of the four corners of the square paper. A summary of the instructions follows:
"b," "c," "d," and "e"
--These four keys operate the "bend" function on each of the four different corner flaps of the figure in clockwise order. The "b" key operates the upper left corner flap; the "c" key operates the upper right corner flap; the "d" key operates the bottom left corner flap; the "e" key operates the bottom right corner flap.
"f" and "m"
on the screen. (The buttons do the same operation, but operate by a different set of codes. This resulted because Yana and I both wrote a "freeze" function, but a different one. An advantage is that it can "lock" the figure in place, provided that both buttons are pressed.)
"z"
--"Zap," returns to default setting; sets the figure to the original position.
right click
--Right-clicking on the mouse turns the figure clockwise.
left click
--Left-clicking on the mouse turns the figure counter-clockwise.
Bibliography
Engel, P. Origami. New York: Dover Publications, Inc. (1989)
Origami Mathematics.
http://web.merrimack.edu/~thull/OrigamiMath.html
Origami Fortune Teller.
http://www.enchantedlearning.com/crafts/origami/fortuneteller/
Origami Fortune Teller.
http://www.tappi.org/paperu/art_class/fortuneTeller.htm
Wu, J. Joseph Wu's Origami Page.
http://www.origami.vancouver.bc.ca/