Frankly, if someone had told me a year ago that I would have anything but contempt for modular folds, I would have laughed in their face. “Modular origami? That’s an oxymoron!” I would have scoffed. “Modular folding is for those who would rather make 1 fold in 90 sheets of paper than make 90 folds in one sheet of paper. It requires no special ability – simply a dulled mental state that adapts itself to menial drudgery of the sort encountered when sorting screws, counting blades of grass or folding and gluing 24 bird bases together. It is a sort of pastime designed for those persons who need to fold something 144 times before they get it right.”
Fortunately for all concerned, no one told me such a thing: consequently, I never made the preceding response and quite probably saved myself from a certain amount of embarrassment upon changing my tune, as well as a chance of bodily harm, depending upon who I responded to. Meanwhile, I have received several quite interesting modular folds in recent months (and invented one or two myself) which have convinced me that there may very well be some redeeming quality possessed by the genre.
Let’s first look at what we gain by using many sheets of paper. We get edges. Cut a square into 4 smaller squares and you double the amount of “edge” available for the same amount of area. And, as readers of this column are by now doubtless aware, the more available edge there is, the more efficiently one can make appendage-like shapes. So, for example, 3-D stars (or to get technical, stellated polyhedra) are natural subjects which exploit this peculiar attribute of modular folds.
Is modular folding necessary for stellated polyhedra? Probably not. There are those (myself included) who derive a perverse delight from making stellated thingamagigs from single sheets of paper. The grandaddy of them all is the “Omega Star” which seems to have been invented by everyone at one time or another (and which, by the way, is generically called a stellated rhombic dodecahedron). Then ther is my own entry into the field of pointedness, a stellated cuboctahedron (a.k.a the Stellated Cubocta of BOS Spring Convention ’85, sort of a truncated octahedron with points), Pat Crawford’s stellated octahedron, the Jackstone (a stellated hexahedron) and more. These efforts at pointliness pale, however, beside the “sweetgum balls” of John Montroll. Conceptually, they are quite simple. You simply sink each of the points of the “Omega Star” twice, and then repeat the process ad infinitum to generate stars of 24, 48 and (God help us) 96 points. The process could be continued (John had only taken it to 96), I suppose, but the idea has been demonstrated; carrying it further seems rather, er, pointless. Heh heh.
In all of these stellar one-piece folds, however, as the pointliness increases, the model accumulates ever increasing amounts of excess paper that must be hidden away, excess paper that has a disturbing tendency to reappear at undesirable times giving one’s star the appearance (to use another’s words) of a burst concertina. Or in the case of the 96- pointed Omega Star, 96 burst concertinas. the larger the points, the smaller the amount of paper that contributes to the visible portion of the model. Whereas, if we had used lots of individual sheets, the design could be far more efficient – half (or more) of the paper can actually contribute to the visible surface of a modular fold, and this efficiency is what makes modularity appealing.
The whole idea of modularity rests upon some sort of basic unit that is repeated over and over again. And while many module designs must be fitted together in a specific way to make a single star, it is tempting to make a basic unit that can serve as the basic unit for several different stars – sort of a “configuration independent” module. For example, a unit (courtesy of Dave Brill) which can be assembled into a variety of shapes is shown in figure 1.
One of the beauties of origami is its ability to produce figures that belie their means of construction. Fo a multi-piece construction, one of the most desirable effects to acheive is monolithic solidity, or in less verbose terms, a rigid construction, without resorting to glue or tape. An example of a module which is remarkable for the rigidity of the finished star is shown in figure 2 (by Roy Roberts). It is also remarkable for the fiendish difficulty involved in assembling the last few units. The module is from an American dollar — a number of them can be assembled to make any stellated uniform polyhedron, by replacing each edge of the polyhedron with the crease joining the two long triangles’ bases together. (Definition: ‘uniform polyhedra’ have faces composed of one or more types of regular polygon. Actually, there are more possibilities for stars than the uniform polyhedra (e.g. Archimedean duals, prisms and antiprisms), but a complete enumeration will have to wait for another column). A stellated octahedron takes 12 dollars; a stellated dodecahedron takes 30 dollars; a stellated snub dodecahedron takes 150. At this rate one could easily go bankrupt.
What else does a high edge-to-area ratio net us? Colour changes! Quite a few modulars are made with rather mundane shapes but rely on detailed and intricate patterns formed from the colour reversals. Even the lowly cube offers several unique possibilities. The simple (6-crease) shape shown in figure 3 (from a 3×4 rectangle, by Lewis Simon) can be woven with 11 others to make the two-tone woven cube shown. The woven appearance is appealing in itself, but that twisted hole in each face also adds to the effect (and, as we have mentioned, there is something aesthetic about a hole in an apparently uncut piece of paper). This model is also remarkable for the sturdiness of the mechanism which locks it together. While in the stellated shapes, we used the same basic shape to make different polyhedra, we can take the opposite tack with this basic structure, and make differently patterned basic units which are woven together to make different patterns on the same cube-with-a-twisted-hole.
As an example of this, and as a final problem for the reader, observe the shape in figure 4 (by me). 12 of these get woven together in the same way as the Simon cube to produce the cube in figure 4 with the woven chain-link pattern running around its edges. I’m rather fond of it because of the large number of “islands” of one color in a sea of another. The problem of folding it from a 14×3 rectangle is left as an exercise for the reader.