第十二章 蛇腹折

1Box2Pleatingne of the characteristics of many artistic endeav- orsas well as science and engineering, which also possess a significant artistic componentis the presence of creative bursts. Origami is no ex- ception. The progress of origami design throughthe 20th century was one of steady, incremental advance punc- tuated by occasional episodes of remarkable creativity. This is a universal phenomenon: It is as if some threshold is reached, that a truly new approach to design is discovered, then the technique or techniques are so rapidly explored and exploited that a jaw-dropping new field appears as if from thin air. Usu- ally after the fact, historians can tease out the antecedents of a particular revolution, but in the days and years leading up to the critical event, no one saw it coming. This phenomenon happens in many fields of endeavor: Quantum theory revo- lutionized physics in the early 20th century; Impressionism changed the world of painting forever. In origami, the most outstanding example of a creative burst was the mid-1960s appearance of Dr. Emmanuel Moosers Train, which ushered in an era of multiple subjects from a single sheet and of origami representing man-made articles, along with the collection of techniques that has come to be known as box pleating.12.1. Moosers TrainIn the small, loosely knit world of Western origami, Moosers Train, shown in Figure 12.1, was something of a bombshell. While many folders had grown comfortable with the notion of using multiple sheets of paper to realize a single subjecthead and forelegs from this square, hind legs and tail from thathere was the far opposite extreme: use of a single sheet of paper to 459 Figure 12.1.Moosers Train, folded by the author.realize many different objects, the engine and cars of a complete train! The result was so unbelievable that folders scrambled to see how it was done.Such a novel result was accomplished by an equally novel approach. What set Moosers Train apart from the vast major- ity of origami designs was the folding style and technique, as well as the complexity of the resulting model. The difference was immediately apparent to even a superficial examination of the crease pattern. In nearly all ancient and early modern origami, the major creases were predominantly radial. They emanated, star-like, from various points in the square: the center, the corners, the midpoints of the edges, as shown in Figure 12.2.Figure 12.2.Crease patterns of the Bird and Frog Bases, illustrating the radialpattern of creases.469Chapter 12: Box PleatingBut in Moosers design, things were different. First, he started from a long rectangle; that alone was not a novelty, as several traditional models begin with a rectangle. But in contrast to most origami, the creases in Moosers Train formed a grid of mostly evenly spaced parallel lines, occasionally bro- ken by diagonals running at 45 to the edges of the paper. The overall appearance of the crease pattern was wholly unlike the patterns of conventional origami.Fortunately for the curious, origami has by and large fomented a culture of sharing of both results and how-to, and it wasnt long before a hardy folder, Raymond K. McLain, had constructed and circulated an instruction sheet for the design. In lieu of formal publicationorigami books were few and far between in the 1960s and 1970sit was passed from person to person, photocopied, and recopied (this at a time when copi- ers were far from ubiquitous). Dauntingly, the instructions consisted of a single page containing the crease pattern, no step-by-step diagrams, and a smattering of tiny, handwritten verbal instructions wrapped around the edges of the pattern. Ive redrawn McLains instructions in Figure 12.3 if youd like to give it a try yourself; for the adventurous soul whod like to experience folding from the original instructions, they are reproduced in Figure 12.4.The challenging diagrams and their lack of widespread availability only added to the aura of mystery surrounding this model, and soon after its appearance it became one of the test pieces against which the origami-hopeful must apply his or her folding skills. And any folded Moosers Train instantly became a focal point for the origami gathering at which it appeared.Moosers Train fulfilled a valuable role: Its folding provided evidence that the folder had attained the pinnacle of the art. That was itself a worthy role. But Moosers Train was not the culmination of a new style; on the contrary, it was the road map, leading the way to an entirely new approach to origami design and a new class of origami subject matterthe man-made ob- ject. It would inspire a small group of origami designers through a decade of creative growth, of exploration, and of pushing the boundaries of what was possible within the one sheet/no cuts origami paradigm. Their innovations, in turn, by showing that truly anything was possible with folding alone, would lead to the near abandonment of multi-sheet, or composite, origami design. And their work would go on to inspire an entire generation of origami designers, including the author of this book.The revolution that was initiated by Moosers model began in earnest when its techniques were adopted and expanded by another innovative folder. By the mid-1960s Neal EliasMoosers Train Crease Pattern & Order of AttackWorked out by R. K. McLain, March 20, 1967Hindman, KY 41822AABegin with (2) x (1) square.Divide (2) into 32 squares.Divide (1) into 16 squares. Remove 4 squares the long way. You now have 32 x 12 squares.Mtn. fold under 1 square the long way on each side.Now make the crease pattern as indicated. Each box car requires 10 squares long and 12 squares wide. The locomotive requires 12 x 12.Now mould the model much as you would clay.Several things must give at once so that a firm crease pattern without extraneous creases is helpful. Be patient& gentle.When moulding is completed, squash & partially petal fold the wheels & turn under the end points a little.(Make catcher with A & A.) Dent inwards the platform between cars, lock the end of the last car by valley folding inwards the platform part, lock the underside by folding inward the extra material between & behind the wheels.Bend the locomotives snout upwards, penetrate (with a cut) it inwards into the boiler & bring it back outwards (with another cut) (and a valley fold) as a smoke stack. If you succeed, you get the prize for diligence! Ill take one too! This surely is a clever model& points the way to future 3D origami. Perhaps the crease pattern could bescratched onto paper (making valleyfolds only on both sides of the paper) with a knife denting but not cutting through.Figure 12.3.Folding instructions for Moosers Train.Figure 12.4.Raymond K. McLains original instructions for Moosers Train.was already one of Americas most inventive folders and had diagrammed hundreds of his own new designs. Elias displayed an amazing ingenuity with the traditional origami bases. The classic Bird Basewhich some folders felt had already been played outin Eliass hands blossomed into new shapes. Most notably, Elias had a flair for multisubject creations, for example, a birdhouse with two birds peering out, from a single Bird Base. When he saw Moosers Train, he immediately saw its vast potential.To understand what this model signifies, we have to recall the state of origami design in the 1950s and 1960s. Origami designers typically picked a subject, then chose one of several bases that had varying numbers of flaps to work with. By choos- ing a base with the same number of flaps as the desired subject, and hopefully with the flaps arranged in roughly the same positions as the features of the subject, the budding designer could, with further shaping folds, massage the base into some semblance of the desired subject. The designers of the 1950s and 1960s in both Japan and the West had systematically identified a dozen or so known bases. They had combined pieces of two bases to make hybrid bases. A fewnotably American folder (and friendly rival of Elias) Fred Rohmhad devised new bases of their own.But a three-car train bears no resemblance to any known origami base, uniaxial or not. Such a train combines big, boxy shapes with the need for fourteen identical flaps to form the wheels, appropriately distributed along the bottoms of the three cars (six on the locomotive, four on each of the boxcars). This is pretty specific. No one was ever going to fold a train from a con- ventional base. Even though throughout the 1950s and 1960s new bases were continuously being discovered by trial and er- ror, the odds of a given base having the right number and size of flaps in just the right place to make a train were millions to one. Even fast-forwarding to the 1990s, the techniques of uniaxial basescircles, rivers, molecules, and treescould handle the flaps but were not going to produce the solid elements. What Mooser had found, and displayed brilliantly in his Train, was a set of techniques for apparently making three-dimensional boxes and flaps at will.How was this possible? What is it about the crease pat- tern of the Train that bestows this incredible versatility? The answer is not immediately obvious. The most distinctive aspect of the crease pattern of Moosers Train is the fact that most of the creases run up-and-down or left-to-right. A smaller number run at 45. This is to be contrasted with other origami bases in which the creases appear, at first perusal, to run every whichway at many different angles and directions. Which pattern shows greater flexibility: the constrained, uptown/downtown/ crosstown pattern of the Train, or the many-different-direction pattern of conventional origami? Clearly, the rules by which the Train was constructed were more restrictive than the rules of conventional origami. How could it be that a more restrictive set of rules leads to a less restrictive, more flexible result?Paradoxically, it is the very tightness of the constraints of box pleating that makes it possible to fold such complex designs. The reason it has always been difficult to develop new origami bases is that a base is a gestalt, an inseparable whole; all parts of the pattern interact with other parts, so that it is very dif- ficult to make a substantial change in one part of the pattern without having to change all other parts. The resemblance of a crease pattern to a spiders web is an apt analogy; pluck a single strand and it reverberates throughout the web. Perhaps a better analogy is a stack of apples: Move the wrong apple and the heap collapses. Move one circle in a circle-packing and the entire packing might need to rearrange. Change a single vertex in a crease pattern, and its effects propagate throughout the entire pattern.And those effects may very well precipitate a descent into unfoldability. Lets take a simple example: the Frog Base, shown in Figure 12.5. Suppose that for some reason we wished to move the vertex that corresponds to the central point. Move that vertex the tiniest amount away from the center, changing nothing else, and the crease pattern becomes unfoldable (or rather, un-flat-foldable; it can no longer be pressed flat without creating wrinkles). It is possible, however, to move other verti- ces to return the base to flat foldability, as shown on the right in Figure 12.5; but to do so requires that we shift the location of all the other interior vertices, resulting in moving nearly every crease in the pattern.One seemingly innocuous change in the pattern forces changes throughout the design. And this was the result of an attempt to shift the location of a single point. We have not even added any points. In the early days of origami, design was in- cremental, a change at a time. But if such a tiny change forces a complete redesign of the crease pattern, what hope has the designer of incrementally creating a fourteen-wheeled, three- vehicled conveyance such as a train? How would a designer of a real steel-and-wood train fare if the most minor changesay, moving a door handleforced an unpredictable change in every dimension of every part of the structure?But in Moosers Train, some changes dont cause so much trouble. In the Train, the creases dont run every which way.Figure 12.5.Left: the crease pattern for a Frog Base. Suppose we move the centervertex upward.Right: the new flat-foldable crease pattern. Note that every otherinterior vertex has also moved.In fact, they only extend in one of four different orientations: up/down, left/right, diagonally upward, diagonally downward. And the creases dont fall just anywhere: There is an underly- ing grid, so that up/down and left/right creases run solely along grid lines, while diagonal creases always connect diagonal grid points. So the crease pattern is quite tightly constrained.The constraint of the grid brings order to the crease pattern: It winnows the unimaginably vast space of possible patterns down to a manageable set. And most importantly, it limits the ways that different parts of the pattern can interact with each other. The problem with an old-style base like the Frog Base is not just that the central point interacts with the surrounding points: Its that it interacts with each surrounding point in a different way. So one type of change creates several types of changes in its surroundings, which then create more changes in theirs, and so forth. This means that the complexity induced by a change quickly cascades as the change propagates away from the original perturbation. But in a box-pleated pattern, by contrast, where different parts of the crease pattern correspond to different parts of the model, all interact in the same basic way. And so, a fairly small tool kit of basic techniques can be combined and built up into quite complex structures.The basic elements of this tool kit are visible in Moosers Train, the archetype for all the box-pleated models that fol-lowed. Those two elements are a technique for building and linking boxes (used for the bodies of the engine and the two cars), and techniques for creating flaps (used in the wheels and, especially, the smokestack). Both boxes and flaps grow out of the same rectilinear grid of creases, which allows arbitrary combinations of boxes and flaps to be created and combined at will.12.2 Box FoldingThe techniques to create box-like structures have their anteced- ents in well-known traditional models that include (perhaps not surprisingly) a simple box, known for decades, if not hundreds of years. The box displays the underlying mechanism that enables box pleating as a style and that makes up the overall structure of Moosers Train. Box pleating as a style was sitting there all along, waiting to be discovered, but the most common folding sequence for the traditional box (given in Figure 12.6) and the diagonal orientation of the model obscure the underly- ing structure and its relationship to the train.This is a fairly common occurrence in origami: the pub- lished folding sequence is usually constructed for ease of fold- ability, or in some cases, for elegance of presentation (with a surprise move at the end). In either situation, the choice of folding sequence may well conceal, rather than illuminate, the underlying structure of the model.Superficially, what we have here is simply a box with two handles. But lets look at it as a collection of forms. We have a linear series of forms: a flat form (the handle), a transition from a flat form to a three-dimensionalform, the three-dimensional form (the box itself), another transition from the three-dimensional form to a flat form, and finally another flat form (the opposite han- dle).How does this combination of two- and three-dimensional forms arise from the flat sheet? The best way to find out is to take the model back to the flat sheet, keeping track of which parts came from where. If we label the features of the boxbase, side, front, rear, handleand note where each region comes1/81. Begin with a square. Fold and unfold along the diagonals.2. Fold the four corners to the center of the paper.3. Rotate the paper 1/8 turn clockwise.4. Fold the sides in to the center line.5. Mountain-fold the top half of the model behind.6. Fold one flap up to the top edge in front; repeat behind.7. Pull the corners out to the sides as far as possible and flatten the model.8. Pull the raw corners out completely in front and behind.9. Fold the corners in to the center on existing creases. Repeat behind.10. Fold the top corner down; fold the resulting flap down again. Repeat behind.11. Grasp the two white flaps and pull them in opposite directions, opening out the model.12. Finished Box.Figure 12.6.Folding sequence for the traditional box.from in the unfolded sheet, we can establish a correspondence between the folded and unfolded forms of the model, as shown in Figure 12.7.Figure 12.7.Correspondence between theparts of the folded model and the crease pattern.If we examine the crease pattern by itself, w