Figure B-1. The four classic bases of origami.
Origami is the Japanese name for the art of paper-folding. The purest form of origami stipulates that you fold one sheet of paper, which must be square, and no cutting is allowed. These rules might seem restrictive, but over the hundreds of years that origami has been practiced, tens of thousands of origami designs have been developed for birds, flowers, animals, fish, cars, planes, and almost everything else under creation. Despite the age of the art, the vast majority of origami models have been designed within the last fifty years. The works of Japanese master Akira Yoshizawa in the early twentieth century led to a worldwide origami renaissance in the latter half of the century, but a contributing factor to the growth in number, quality, and sophistication of modern origami was the development of a body of systematic techniques for the design of origami models. Notable among these developments is the concept of a generalized base. Growing out of the four classic bases of origami --- the bird, fish, frog, and kite bases --- the concept of a base forms the foundation for nearly all sophisticated origami designs.
Figure B-1. The four classic bases of origami.
What, exactly, do we mean by "a base?" A base is a geometric shape that contains flaps corresponding to all of the appendages of the origami subject. For example, the base for a dog would have six flaps, corresponding to four legs, a head, and a tail. The base for a beetle would have nine flaps, corresponding to six legs, two antenna, and an abdomen. As origami subject matter has moved from relatively simple birds to complex insects with legs, wings, horns, and antennae, the advances that enabled this transition were the development of geometric and other mathematical techniques for the design of the underlying origami base.
All origami bases are general abstract, geometrical shapes. What makes many of them useful is that each base is a collection of more fundamental structures, called "flaps." A flap is loose bit of paper that can be transformed into an appendage of the subject: an arm, leg, or wing, for example. The Classic Bases of origami had different numbers and arrangements of flaps, which made each base particularly well-suited for certain types of subject. As origami design expanded worldwide through the 1960s and 1970s, origami artists designed and/or discovered new bases with new arrangements of flaps, or other features that made them suitable as building blocks for origami compositions.
The Classic Bases of origami are uniaxial bases, a property that they share with many other origami bases. All uniaxial bases share the following attributes:
In the figure above, the axis of each base runs vertically through the center line of the base.
In the early 1990s, several origami artists/scientists developed a body of techniques that allowed the systematic design of arbitrarily complex origami uniaxial bases. Two of us, Japanese biochemist Toshiyuki Meguro and myself, independently developed a particularly versatile theory known variously as the "circle/river/molecule method," or "tree theory." This theory allowed one to draw a simple stick figure of a desired shape, then construct a crease pattern that, when folded up, gave a uniaxial base whose projection (or shadow) was precisely the given stick figure. The advent of tree theory and its widespread adoption has led to an explosion of complexity and realism in origami.
While tree theory can be practiced with no more equipment than a pencil and paper, it can be described mathematically in a way that makes it suitable for computer implementation. Beginning in the early 1990s, I began writing a computer program that implemented the algorithms of origami tree theory. I called this program TreeMaker.
Fundamentally, TreeMaker is a program for designing uniaxial bases. It calculates the crease pattern for a base that has any number of flaps of arbitrary size and distribution. TreeMaker specializes in bases that can be represented by a tree graph, i.e., a simple (one piece) acyclic (no loops) graph (stick figure) in which each segment is characterized by a specified length. Each segment of the tree corresponds to a flap of the base, so the graph serves to describe the lengths and connections between the flap of the desired base. TreeMaker will compute a crease pattern showing how to fold a square (or rectangle, for non-purists) into a multi-pointed base whose distribution of flaps match the lengths and connections of the tree.
While I envisioned TreeMaker primarily as an intellectual exercise that gave me a vehicle to explore the underlying theory, over the years I regularly rewrote it to take advantage of advances in the algorithms as I discovered them (and advances in my own programming ability). This program is now version 5, and it offers several new capabilities compared to previous versions.
To use TreeMaker, you represent the subject as a tree, or stick figure, that defines all of its appendages and their relative lengths. You construct the tree using a simple graphical point-and-click interface to define the segments of the tree and to specify the lengths of the flaps. You can also include constraints that enforce symmetry requirements among the appendages --- for example, forcing the model to be bilaterally symmetric, or forcing particular points to come from a corner or edge of the square (to control thickness or to allow color-changes). After you have defined the stick figure, TreeMaker will find an optimally efficient arrangement of points on a square that correspond to the nodes of the tree and will have all the creases of the base, including the direction of folding (mountain versus valley fold). The crease pattern so computed is guaranteed to be foldable into a flat base with the proper proportions and is in fact a locally optimum solution for the base. "Locally optimum" means that for a given starting configuration, you get the largest possible base for a given square, but an entirely different starting configuration might give a different base. You can subsequently add points to the tree to simplify the crease pattern (I'll explain more about this later); when you have sufficiently simplified the pattern, a single command computes all of the creases. The full pattern can be printed (and cut out and folded) at arbitrary size or (if your OS supports it) printed to a file for further editing.
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