"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.