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TIPS AND TECHNIQUES

WHILE MANY modular origami projects can indeed be challenging, keeping a few important things in mind ahead of time will make everything go easier.

Before you begin, it is important to be prepared. You should have all the paper that you will need for the project, as well as any tools you might need. Be careful to choose an appropriate size for the starting papers, as they will determine the size of your final model. This is especially true for the Wire Frames—it is very natural for beginners to want to expand the size of the units, but if you aren’t careful, you can easily end up with a model several feet in diameter. Conversely, you might decide to make a model very small, which would then make it quite difficult to fold. Scale the paper proportions up or down to determine an appropriate size.

Paper proportions are the length-to-width ratio of the paper for Wire Frames. In some cases, you may want to change the dimensions listed. Fortunately, this is easy. You just have to convert them to their original 1:X ratio (if necessary), and then multiply both numbers by the desired width. Here’s an example: if the paper proportions are listed as 1.25:5, and you want the width of each unit to be decreased to .875 in order to make a smaller model, you first convert the 1.25:5 back to a 1:X ratio. To do this, divide both numbers by the width. 1.25/1.25=1, and 5/1.25=4, so the proportions have been converted to 1:4. You must now multiply both numbers by the new width to get the final proportion: 1x.875=.875, and 4x.875=3.5. Therefore, your final proportions are .875:3.5. These rectangles will have the same height-to-width ratio as the proportions listed in the directions, only scaled down. The same procedure can be used to increase the proportions; simply reverse the process to increase the proportions.

Once all of the preparations are made, cutting and folding the units is generally very straightforward, especially for the Wire Frames. However, if you have difficulty with a step, look ahead to the next step to see the result of the fold. The folding of the units can be tedious, but you can fold small “batches” of them and then assemble them later. And the more effort you put into the units, the more rewarding the finished model will be.

A batch of units for The Alphabet (page 89), just begun.

The complete collection of units for The Alphabet.

The Wire Frame units themselves are surprisingly simple, as they have a standard design pattern. The nature of their design is somewhat redundant, to the point that after having folded several different models, you will probably be able to infer an approximate folding sequence even before reading the diagrams. This will make variations and new concepts easier for you to explore on your own. Probably the most important thing to keep in mind when folding the units, aside from the proportions and pocket angles, is the dihedral angle of each unit. This is the interior angle between the two halves of a completed edge unit. This angle will determine how the units interact with each other: if it is near 180 degrees, the unit will be close to flat; if it is near 0 degrees, the unit will be narrow, and the two halves will be pressed against each other. Unless a special effect is desired, the optimal angle is around 90 degrees. (It is possible for edge units to have a dihedral angle greater than 180 degrees, but that is a subject for another volume.)

The assembly of the units is usually the most difficult—and the most exciting—part of making modular origami. Various types of locks may be used the hold the units together, but the standard method involves sliding a tab of paper into a pocket. Different paper types each have their own pros and cons in assembly. Thinner paper has the advantage of being more flexible, and will have fewer gaps where the units come together. It is useful in assembly where mobility is limited. Thicker paper is stronger, and is less likely to crumple, bend or rip during the assembly. It can make the completed model more rigid as well. The last units in a modular will be the most difficult to assemble—especially the solid, ball-like models that offer no way to manipulate the paper into place from underneath once they are near completion. Be patient and deliberate as you slowly ease the units into place.

"Neighborhoods" of weaving patterns

For Wire Frames, it is important to know where to place any frame-holding pieces to ease assembly. The areas of each unit that press against other units in the assembled model, and which hold the model together, are referred to as the limiting factors.

Limiting factors

These determine the proportions of the starting paper. If frame holders are used during assembly, it is important to put them in a place where they will maintain the model’s stability, but not interfere with the construction process. The limiting factors are usually the best areas to place any wire, string, etc. that you are planning to use as frame holders.

In addition, the weaving pattern has to be taken into account when assembling the units of the Wire Frames. Getting all of the units woven around each other in the proper pattern so that the model is symmetrical on all sides is a fun puzzle to figure out. To start, having an understanding of basic geometry, especially polyhedra, is absolutely critical to understand the weaving of a Wire Frame. This is because the weaving pattern for any given model will almost always follow the symmetry of a regular polyhedron, most likely one of the Platonic or Archimedean solids (the last model in the book represents the exception to this). For the first two models, I have specified a corresponding polyhedron, but I’ve left this information to be inferred in later projects. Once you identify what polyhedron symmetry any given model is based on, you can determine which of several methods to use to carry out the weaving and completing the assembly.

The first important key to understanding how Wire Frames are assembled is through their axes. These will be referred to in every Wire-Frame assembly diagram in this book. Strictly speaking, an axis is basically a line around which a figure can be rotated. The axes here will manifest as woven polygonal shapes that form on the model where each frame goes underneath or over another in a rotating manner. This repeats with several others in a circuitous fashion, and the resulting axes align with certain parts of the basic polyhedron on which the model is based. For example, the three-fold axes of a compound might align with the facial viewpoints of an icosahedron, which would equate to aligning with the vertices of a dodecahedron. They are often used to represent different viewpoints in completed models. They are referred to as an n-fold axis, n being the number of sides on the axis.

A five-fold axis

Once the concepts of axes and axial weaving are understood, they can be expanded to represent entire “neighborhoods” of the models’ weaving pattern. The most common “neighborhoods” in an icosahedral/dodecahedral model, for instance, are the five-fold, three-fold, and two-fold axis views. Each axis, and the surrounding units in its vicinity, represent a “neighborhood” on the surface of the compound; adjacent “neighborhoods” will integrate seamlessly into each other. See the bottom left illustration on the opposite page.

Note that axial weaving alone is not sufficient for more complicated models. These sometimes have double overlapping sections, which can result in illusory axes—areas that have a circuitous whorl in similar frames, but do not exactly represent any polygonal faces of polyhedra.

Another important factor in figuring out the weaving of a complex Wire Frame is identifying if there are any clear relationships between individual frames. One of the most commonly referenced relationships is in-and-out weaving. This is an interlocking pattern in which one frame weaves outside of a second frame on one side of the model. On the other side, the frame that was outside now weaves inside the frame that was the inside frame on the other side. Basically, opposite sides of the model are mirror images of one another. See the bottom left illustration on the following page.

Another commonly referenced pattern is envelope weaving, where one complete frame is entirely inside of another frame, but entirely outside of a third frame. One example of this is the famous Borromean weaving pattern, where three links are held together through weaving, but any two frames are not interlocked. See the top right illustration on the following page.

When folding Wire Frames, your initial impulse may be to use the pictures of the assembly in the text to exactly follow the pattern. This will work to a limited degree, but in complex constructions, your view will be obstructed by other parts of the model. In these cases you will have to instead focus on understanding the pattern of each axial area so as to intuit obscured areas based on geometrical patterns, rather than visual images.

Knowing how to weave a complex model is only useful if you are able to physically assemble the units. I have used three different assembly methods; the most practical one will depend on the model you are attempting. The most commonly used method, which was, up until the last few years, the only practiced method, is frame-at-a-time weaving. Essentially, any Wire Frame compound is composed of a certain number of identical interlocking polygons, or polyhedra, which are not actually connected to each other, but which interlock around each other in a symmetrical pattern to hold together. In the frame-at-atime method, you simply assemble one complete frame around another, then add another to the first two, and another, and so on, until the model is completed. This method dates back to the first Wire Frames, including Tom Hull’s FIT.

The second method, tailored for the assembly of complex models where most of the units are near the outside surface of the completed piece, is referred to as bottom-up weaving. Pieces consisting of perhaps five or ten units of all frames are added simultaneously, so that all frames are assembled with the same progress. The model thus becomes fully completed on the bottom, and more pieces are added to all the frames in a sequentially upward fashion until the compound is complete. One of the advantages of this assembly method is that it makes it easier to weave a Wire Frame in the correct pattern, and it is helpful in understanding the weaving. It is also a particularly useful method if you are experimenting with a new compound idea. The first known use of this method was by Daniel Kwan, with the construction of his compound of Six Irregular Dodecahedra.

The third method, referred to as scaffolding, is a hybrid of the previous two; it is generally only used for the most complicated models. It is best used for models that are too complex to be woven with just the frame-at-a-time method, but whose units reach too deeply into the center of the model for stable bottom-up weaving. With this method, as many complete frames as possible are assembled, and then the remaining frames are “bottom-up” woven over the existing “scaffolding,” which makes the half-assembled model more stable. I had not seen anyone specifically using this method before I tried it.

An example of "in-and-out" weaving

Of the various assembly techniques listed here, frame-at-a-time weaving will likely be used the most, followed by bottom-up weaving. Scaffolding weaving will be used the least. The photos of the assemblies in this book show the method I would use for each specific model. In the end, however, the methods you use are up to you.

Another subject that often comes up is coloring guidelines. These aren’t specifically mentioned in the instructions themselves. The decorative models can have a variety of coloring patterns depending on the type of assembly. The number of colors should be divisible by the total number of units; i.e., a five-color pattern for a thirty-unit model would require six units per color. I will leave it as an exercise to the reader to figure out the assembly order of the colors. Generally, Wire Frame coloring is basic, especially since all of the Wire Frames in this book are woven compounds. Every frame should get its own color for best results. Alternately, all frames can be colored the same for a special effect, but that will make the separation and interaction between the frames less visible.

A Borromean weaving pattern

If you don’t succeed at your first attempt, simply keep trying. Remember that, like puzzles, these models are supposed to be difficult to figure out. Experience and practice will increase your confidence, and eventually even exceedingly difficult models will seem more reasonable. Follow the path of every frame, and carefully observe how it interacts with every other frame. When the assembly looks impossibly complex, break it down into more manageable parts. Understanding complex models is a matter of accumulating experience. In the end, keep in mind that this book is meant to be an introduction to modular origami, not a complete extrapolation. Wire Frames, and modular origami in general, are too complex to explain fully here.

When you do complete a model, check to make sure everything is assembled, and, if it is a Wire Frame, that the weaving is correct. Be careful during this step—it is very easy to miss a small mistake in the weaving process. When you are sure that it is correct, sit back and take a minute to appreciate your work! I think you will find that it is much easier to fold and appreciate these pieces than it is to find a place for them, especially as you begin to make more.