第十三章 单轴蛇腹

Uniax1ial BoxCircle packing, molecules, and tree theory had the potential to change the world of origami when they began to be widely used in the mid- 1990s and going forward, but they quickly ran up against a barrier: although one could,3Pleatingin principle, design arbitrarily complex tree-like structures of theoretically perfect efficiency, in practice, the resulting crease patterns were both highly irregular, making them difficult to precrease, and all of the folds in the pattern were highly coupled to one another, making them difficult to break down into a simple step-by-step folding sequence. The latter property is, unfortunately, often unavoidable. The vast major- ity of theoretically possible flat-foldable folding patterns are in this category; it is, in some sense, an accident of history that most published origami works have had step-by-step folding sequences, because they were discovered almost en- tirely through a step-by-step process of exploration. But the irregularity is not necessarily something that we, as origami designers, must live with.Circle/river packing often leads to irregularity, even if the circles themselves come in only one or a few sizes. My software tool, TreeMaker, constructs a circle-packed solution for any user-specified stick figure, but even when you have the complete crease pattern, the problem still remains: how do you transfer the pattern to the folded paper? With a computer program, you could perhaps print out the pattern, but then you have to contend with printed lines on the finished form. If you want to fold the pattern with no visible printed lines, then there may be tens, or even hundreds, of vertices with no easily constructible method. I developed another tool, ReferenceFinder, which can 561 find folding sequences to locate individual folds or lines, but this is an incredibly tedious process with a circle-packed design if there are tens of points and/or lines to be found.It is not surprising, then, that artists have found variations on circle packing that lead to far more tractable (and therefore foldable) designs. One of the most powerful and versatile is also, surprisingly, one of the oldest toolkits of technical design: box pleating, which we met in the previous chapter.The term “box pleating,” as it is used now, actually takes in two distinct sets of techniques. In one form of box pleating, one creates three-dimensional structures in which the walls meet at right angles to form boxes and partial boxes (hence the first part of the name). This form of box pleating is inherently three-dimensional; examples are to be found in Moosers Train and in the 3D works of Max Hulme and Neal Elias, such as the formers “Stephenson Rocket” and the latters “Dump Truck.” The other form of box pleating results in flat shapes with arbitrary flap combinations, often incorporating the design pattern we now call the “Elias stretch.” Both styles of folding have most major folds running at multiples of 90 and lying on a grid, with secondary creases at multiples of 45, and it is in fact difficult to distinguish from a crease pattern whether the model is 3D or flat without careful examination or even trial folding. For this reason, it has become common to call any fold in which most creases lie on a square grid a “box-pleated fold.” Many of the designs from the “golden age of box pleating” in the late 1960s and 1970s used both 3D and flat box-pleated formsas part of their structure.Since I have already written about box formation, Ill focus now on the subset of box pleating in which the major creases lie on a square grid, the secondary creases run at multiples of 45, and the base folds entirely flat. This subset of box pleating can be further subdivided into uniaxial basesbases in which all flaps lie along a common line and all hinges are perpendicular to the lineand, shall we say, everything else (which takes in a lot). Despite it being only a subset of the broader world of box pleating, the set of box-pleated structures that are also uniaxial bases is broad and useful. I call this subset uniaxial box pleating.Within the world of uniaxial box pleating, one can design bases using a process very much like circle packing, with one big difference: while a complex circle-packed design can be extremely irregular and practically impossible to construct without a computational device, even the most complex and ornate uniaxial box-pleated base can be constructed with no more tools than a pencil and square-grid paper. Because it is so569Chapter 13: Uniaxial Box Pleatingeasily implemented, uniaxial box pleating can be a powerful way to design extraordinarily complicated bases.Uniaxial box pleating, though it has historical roots that predate the development of circle packing, can be viewed as an extension and generalization of circle packing and works in essentially the same way.13.1. Limitations of Circle PackingCircle/river packing creates the most efficient uniaxial base for a given tree and sheet of paper, and it is guaranteed to give you every flap of the appropriate length that you specify. That makes it an extremely powerful tool in the origami designers arsenal. However, as with any tool, it is essential that one be aware of its limitations, of which there are several.First, there is no guarantee for the existence of a fold- ing sequence. Circle packing and many other origami design techniques create a valid folded form (“valid” meaning it is flat foldable without self-intersection), but in general, there may not be a sequential series of small steps that leads from the square to the finished shape. Traditionally, origami designs were discovered as the end result of a series of step-by-step explorations; not too surprisingly, then, such models could be constructed by a step-by-step sequence. But in the vast world of possible origami designs, step-by-step models are actually in the minority; most models cannot be broken up into a set of independent folds; they are “irreducibly complex .”* This leads to a base construction procedure that could be described as, “precrease forever, then collapse,” at which point all of the folds of the base are brought together at once. Or, as my colleague Brian Chan once described one of his designs, “fold this model in three easy steps: precrease, collapse, shape.” (Each of the “three easy steps” took several hours.) Such “three-step mod- els” are becoming the norm in complex designed origami.Second, in circle/river packing, there is little control over flap width. Length, yes: length is specified in the algorithm. But when it comes to flap width, you get what you get; you dont get to specify flap width as part of the design process. Now, it is often possible to employ sink folds (and multiple nested sink folds) to make a wide flap narrower. But it is not always possibleand you cant easily have a narrow flap con- nected to a wide flap (at least, not without sacrificing some length). As it turns out, the universal molecule appears to give* The term “irreducible complexity” regrettably has another usage, in the pseudoscientific doctrine known as “Intelligent Design.” I hope it is clear that the terms usage in origami has nothing to do with such other usage.the widest possible flapand thus, the most width to work with. And even if a flap is not wide enough initially, techniques such as strip grafting can be used to make a flap wider.Third, the vertices are at arbitrary locations, and creases run at arbitrary angles. This is, perhaps, the most significant drawback of “pure” circle packing. There is no easy way to transfer an irregular crease pattern onto the paper to be folded. There is a strong aesthetic within origami to find all refer- ence points by folding alone (no measuring and marking), and there is a substantial field of origami-mathematics devoted to finding both exact and approximate methods for locating refer- ence points. But even with a tool such as my ReferenceFinder (which can give a pure folding sequence for any point or line in a small number of folds; see the References), a circle-packed design can be overwhelming, with tens or hundreds of points to be located. Even if you fall back on measure-and-mark, the process of transferring key vertices to the square is mind- numbingly tedious.Circle-river packing is not the only game in town, however. We have seen that with box pleating, all of the creases fall on regular grids and run at just a few angles. We can introduce ideas from box pleating into circle packing to realize techniques closely related to circle packing that produce much more easily foldable bases that are far more geometrically regular, with only a slight penalty in efficiency. These regular patterns are not only more easily folded; they are more easily designed, and in fact usually require nothing more than a pencil and grid paper to construct. Before going into them, however, I would like to work through a real example design problem, which will illustrate some of the problems associated with circle packing and will also introduce some concepts essential to their resolution.13.2. A Circle-Packed BeetleLet me start with a real problem, of the sort that inspired much of the development of circle packing: an insect. To be specific, I will design a beetlea rather generic beetle, with just the basic appendages: three sets of legs and antennae, spaced out along a three-segment body (head, thorax, abdomen). The first step in the creation of this beetle is to create the tree graph, the stick figure, and to assign lengths to all the flaps. These parts are shown in Figure 13.1.The absolute lengths that one assigns to a flap are entirely arbitrary; what matters is their length relative to one another. To make this simple, I have chosen all of the flap lengths to4 1Figure 13.1.Left: a generic beetle to design.Right: its stick figuree integral multiples of the smallest distance that appears in the stick figurewhich is assigned a length of 1 unit. The rest of the flaps follow: legs are graduated in length, with the back legs being longest at 8 units, followed by the center legs (6 units) and front legs (4 units).There are a few extra flaps in this stick figure: one at the top of the head and two along the body. These flaps serve to create “extra paper” at strategic places in the design. The 1-unit flap at the top will allow the design to be opened flat in plan view (viewed from above, as in the drawing); without it there would not be a complete hinge allowing the two sides to be spread apart. The other two 1-unit flaps along the body create excess paper that will allow a distinct line between the head, thorax, and abdomen, to be created.Next, we create the packing shapes, as shown in Fig- ure 13.2.It is fairly common that an origami model exhibits left/ right mirror symmetry. When that is the case, I commonly design only half of the model, as shown in Figure 13.2 (the left half). Flaps that lie on the line of symmetry of the subject must usually have their circles lie on the line of symmetry of the base, and this is the case in Figure 13.2.And now its time for the circle/river packing. With this many objects, finding an optimum packing by hand is fairly hard, even with the use of physical manipulatives (cardboard circles and spacers for the rivers). With this packing, its fairly easy to see that most of the circles and rivers will be arrayed around the outside of the square, and one can set up an algebraic set of equations for the coordinates of all theFigure 13.2.Packing shapes on the squareand stick figure. The rivers are shown connected to their respective segments on the tree graph.circle centers and the size of the enclosing square. This is a bit tedious, but it is worth going through as an illustration of how to solve for a packing pattern with minimal computa- tional tools.Figure 13.3 shows the packing of circles and rivers into the left half of a square whose side has length s. The most elegant arrangement would have circles packed neatly into the corners of the square, but one is rarely so lucky as to achieve this condition; more often, the situation is as shown in the figure, where no circle lies precisely in a corner.There are five unknowns in this figure: the square side s, and the four distances marked w, x, y, and z. In order to solve for all distances, we need five equations. Three of them come from adding up distances along the sides of the rectangle. Along the top edge, left edge, and bottom edge, we have, respectively,w + 4 + 1 = s /2 ,(131)x + 4 + 1 + 1 + 6 + y = s,(132)z + 8 = s /2 .(133)wx41 14s/24414 11114 111611682s46424y88zs/2Figure 13.3.Circle packing for the genericbeetle in a square of side s.And then at the two corners, the Pythagorean theorem gives the remaining two equations:x 2 + w2 = (4 + 1 + 1 + 4)2,(134)y 2 + z2 = (6 + 2 + 8)2 .(135)These equations can be solved exactly (with complex results), but all we really need are numerical values for the solution with all real positive values, which are readily found to be the values shown in Table 13.1.DistanceValuew9.63x2.70y14.56z6.63s29.26Table 13.1.Distance values for the genericbeetle circle packing.This packing is not complete, however. The packing of the three circles in the interior is not rigid; there is room for the circles (and the rivers around them) to “rattle around” in the interior. The way we deal with this situation (which occurssurprisingly often) is to “soak up” the extra space by enlarging one or more of the interior circles. With this design, the length-4 abdomen circle is an obvious candidate for enlargement; we can either turn the excess paper underneath, hiding it, or perhaps use the extra paper to create additional lines or features of the model. A similar analysis to the above, letting the size of the abdomen square now become an unknown variable, gives the circle/river packing shown in Figure 13.4, where we now show the full packing in both halves of the square.wx41 1s/244141 1141111111266.2s66.22yFigure 13.4.Expanding the abdomen circlemakes the packing rigid.888zs/2Note that the positions of the circles are fixed (“pinned”) in place, as are the rivers where they cross the axial paths be- tween the circle centers. Elsewhere, the positions of the rivers are not necessarily fixed; I have drawn them where they are only for convenience.For this packing, the abdomen circle has been increased in length by 55%, to a total length of 6.2 units. This means that there will be a fair amount of excess length to be hidden. But that extra paper was going to have to be hidden somewhere, and in a beetle, the abdomen is one of the fattest parts of the model; better to hide excess paper in the abdomen (or thorax) than in the antennae, for example.This design now consists of four axial polygons: a triangle at the bottom, a quadrilateral at the top (it looks like a triangle, but its actually a quadrilateral with one straight vertex), and the two heptagons that make up most of the model.(Heptagons? Surely I mean hexagons, right? No, these are heptagons from the standpoint of filling with molecules; each has one vertex along the midline of the base that is straight, i.e., with a vertex angle of 180. Each polygon takes in seven circles around its outside, ergo, it is a heptagon.)Once we have a rigid packing, we can fill in the creas- es using our favorite system of molecules. As we saw in Chapter 10, it is possible to break up large polygons into triangles and quadrilaterals by adding additional circles that create extra flaps in the open spaces, but for this example I am going to use the universal molecule (using TreeMaker to compute the positions of the vertices and creases). Figure 13.5 shows the resulting crease pattern using the generic form introduced in Chapter 10, with all creases colored according to their structural role.Figure 13.5.Crease pattern for the genericbeetle with structural coloring.As a reminder, with structural coloring, axial creases are green, ridges are red, hinges are blue, and gussets are gray. This coloring (which gives the orientation of the creases in the base) and the hints on folding direction provided by the generic form are enough to collapse such a crease pattern in practice using the approximate rules given in Chapter 10,Section 10.8, but TreeMaker (or a bit of manual manipulation of the folded base) can also find the full crease assignment, which is shown in Figure 13.6.Figure 13.6.Full crease assignment for theTreeMaker version of the generic beetle.TreeMaker also provides a picture of the folded form of the base, given as an “x-ray view” so that all creases are visible. This base is shown in Figure 13.7. The coloring of the creases in the folded form matches the structural coloring in the crease pattern, so you can see explicitly that all axial creases (green) do indeed coincide along the axis; the ridges (red) propagate toward and away from the vertical axis; the hinges (blue) are all perpendicular to the axis; and the gussets (gray) are parallel to the axis, but are removed from it at some distance.Figure 13.7.X-ray view of the base for thegeneric beetle with creases colored according to their type.571Chapter 13: Uniaxial Box PleatingIt is perhaps not so obvious that this base has all