第九章 圆河分配

Circle9Packingn the last chapter, we saw how new bases can be constructed by assembling tiles composed of crease patterns in ways that allow the individual tiles to fold flat. By decorating the individual tiles with circles and rivers, we created matchingrules for the tiles; if two tiles mate so that their circles and riv- ers line up, then the union of the two tiles can fold flat without creating any new creases.Furthermore, we are able to use the pattern of circles and rivers to divine the structure of the resulting base: how many flaps there are, how long they are, how they are connected to one another. While a given polygon may give rise to tiles with different crease patterns that have different widths of flaps, any two tiles with the same pattern of circles and rivers necessarily gives rise to the same flap configuration. We can represent this common configuration by a stick figure, in which each segment corresponds to a distinct flap.When we have built a valid tiled crease pattern, the circu- lar arcs of mating tiles align, creating partial or full circles. A contiguous segment of a circle in the tiled pattern corresponds to a distinct flap in the folded base. Why, you might wonder, is this so? Why use circles? The choice of circles to create match- ing rules is not arbitrary; there is a deep geometric connection between flaps and circles. The use of circles to represent flaps is a powerful tool within origami, and so we shall investigate it a bit further.9.1. Three Types of FlapAs we have already seen, in origami there are three differenttypes of flaps: corner flaps, edge flaps, and middle flaps. These 291 types of flaps are named for the point where the tip of the flap falls on the square. A corner flap has its tip come from a cor- ner of the square, an edge flap has its tip lie somewhere along an edge, and a middle flap, as you would expect, comes from the middle of the paper. For example, the four large flaps on a Frog Base are corner flaps; the four stubby flaps are edge flaps; and the thick flap at the top is a middle flap. All three are illustrated in Figure 9.1.middle flapedge flap299Chapter 9: Circle PackingFigure 9.1.Flaps in the base have their tipsat unique points in the crease pattern.corner flapPaper, like people, can only be in one place at a time. Paper that goes into one flap cant be used for another. There- fore, every time you create a flap from the square, a portion of the paper gets consumed by that flap. The reason to make a distinction between the three different types of flapscorner, edge, middleis that for the same length flap, each of the three types of flaps consumes a different amount of paper.One way to see this difference is to fold corner, edge, and middle flaps of exactly the same size from three different squares. Figure 9.2 shows the folding of a corner flap. If you imagine (or fold) a boundary across the base of the flap, then that boundary divides the paper into two regions: The paper above the boundary is part of the flap, and the paper below the boundary is everything else. The paper that goes into the flap is for all intents and purposes consumed by the flap; any other flaps must come from the rest of the square.So, as Figure 9.2 shows, if you fold a flap of length L from a square so that the tip of the flap comes from the corner of the square, when you unfold the paper to the original square, you see that the region of the square that went into the flap is roughly a quarter of a circle; precisely, its a quarter of an octagon. Suppose we made the flap half the width, as shown in Figure 9.3, before we unfolded it; then the flap becomes a quarter of a hexadecagon. If we kept making the flap thinner and thinner (using infinitely thin paper!), the boundary of theflap would approach a quarter-circle. In all cases, the polygonal segment is inscribed by a circular segment, which represents the limiting case of an infinitely narrow flap. Thus, a circle is the minimum possible boundary of the region of the paper con- sumed by the flap. A corner flap of length L, therefore, requires a quarter-circle of paper, and the radius of the circle is L, the length of the flap.LLFigure 9.2.Folding a corner flap of length L from a square.LLLLFigure 9.3.Left: making a narrower flap makes the boundary a quarter of ahexadecagon.Right: the limit of the boundary as the flap becomes infinitely thinapproaches a quarter circle.Therefore, all of the paper that lies within the quarter- circle is consumed by the flap, and the paper remaining is ours to use to fold the rest of the model.Now, suppose we are making a flap from an edge. For ex- ample, if we fold the square in half, then the points where the crease hits the edge become corners, and we can fold a corner flap out of one of these new corners, as shown in Figure 9.4. If we fold and unfold across the flap to define a flap of lengthLLFigure 9.4.Folding an edge flap of length L from a square.L and then unfold to the original square, you see that an edge flap of length L consumes a half-circle of paper, and again, the radius of the circle is L, the length of the flap.Similarly, we can make a flap from some region in the interior of the paper (it doesnt have to be the very middle, of course). Figure 9.5 shows how such a flap is made. When you unfold the paper, you see that a middle flap requires a full circle of paper, and once again, the radius of the circle is the length of the flap.LLFigure 9.5.Folding a middle flap of length L from a square.The amount of paper consumed doesnt depend on the angle of the tip of the flap, only its length and location. So any flap in any model consumes a quarter, half, or full circle of paper, depending upon whether it is a corner flap, edge flap, or middle flap.This relationship doesnt depend on whether the base was constructed from tiles; it doesnt depend on whether the base is a uniaxial base. It is a property of any flap that comes to a point in any origami model whatsoever. This relationship gives us a new set of tools for designing origami bases that permits a more direct approach than the assembly of preexisting tiles; we can represent each flap by a circle and work from the pat- tern of circles directly.One of the goals of all scientific endeavors is the concept of unification: describing several disparate phenomena as differ- ent aspects of a single concept. Rather than thinking in terms of quarter-circles, half-circles, and full-circles for different kinds of flaps, we can unify our description of these different types of flaps by realizing that the quarter-circles, half-circles, and full circles are all formed by the overlap of a full circle with the square, as shown in Figure 9.6. The common property of all three types of flaps is that the paper for each can be represented by a circle with the center of the circle lying somewhere within the square. With middle flaps, the circle lies wholly within the square. However, with corner and edge flaps, part of the circle laps over the edge of the square. The center of the circle still has to lie within the square, though. Thus, any type of flap can be represented by a full circle whose center, which corresponds to the tip of the flap, lies somewhere within the square.Figure 9.6.All three types of flaps can be represented by a circle if we allow thecircle to overlap the edges of the square.9.2. OverlapsYou might have noticed an interesting feature of the circle pattern for any base; the circles corresponding to individual flaps touch, but do not overlap. In tiled crease patterns, this is, of course, by design; no circles overlapped within the tiles we started with, so no circles will overlap in the assembled crease pattern.However, in any base, if we represent distinct flaps by circles, they can never overlap, whether the crease pattern was constructed from a tiling or not. A moments reflection will reveal why this must be so. Each circle encloses the paper used in asingle flap. If two circles overlapped, the paper shared between the circles would be included in each of the two different flaps, which is obviously impossible. Thus, we can generalize:In any valid circle pattern corresponding to an origami base, the circles corresponding to two distinct flaps may not overlap.This condition must hold for any origami basenot just for the Classic Bases and not just for tiled bases. No matter how many flaps the base may have, the circles in the circle pat- tern for the base cannot overlap. Although this property seems pretty innocuous, it is in fact both restrictive and useful. Put one way, it is an interesting property of existing bases: Unfold any origami base and draw a circle for each flap of the base, each circle centered on the point that maps to the tip of the flap. No two circles will overlap. This relationship can become a useful tool for origami design, for the converse is also true: If you draw N nonoverlapping circles on a square, it is guaranteed that the square can be folded into a base with N flaps whose tips come from the centers of the circles.If you have a pattern of N circles on a square, its quite evident that its necessary that they not overlap to fold into a base with N flaps. Its not at all obvious that this condition is sufficient; but it is, and we will see why in later chapters. This is an incredibly powerful result. No matter how many flaps you want to achieve in your base, all you need to do is draw a set of nonoverlapping circles, and the centers of the circles map out the tips of each and every flap in the base.Using this fact, we can replace a somewhat abstract prob- lem (design an origami base with N flaps of a given length) by a simpler, geometric, more easily visualized and more easily solved problem (draw N nonoverlapping circles whose centers lie inside a square). By representing each flap in the model by a circle and placing all of the circles on the square, we ensure that we have allocated sufficient paper to construct each flap. The problem of placing circles so that they dont overlap re- sembles the packing of cylindrical cans into a box; we call such a pattern a circle packing.The circles need not be the same size. If we use different- sized circles, well get different-length flaps (recall that the length of the resulting flap is the radius of the corresponding circle). So merely by shuffling circles around on a square, you can construct an arrangement of points that can be folded into a base with the same number of flaps, no matter how complex. For that matter, by choosing different arrangements of circles, you can devisemany different folding sequences giving many different bases,even though they have the same number of flaps.While such a circle pattern is guaranteed to be foldable into a base, a guarantee is not the same as a blueprint. But here is where we can apply the tiles from the previous chapter. If the circle pattern can be cut into tiles so that the circles within each tile match up with known tiles, then we can assemble the complete crease pattern from these same tiles.9.3. Connections to TilesConsider, for the moment, those tiles we have seen that con- tain only circles (no rivers). They are of two types: triangles and square (the Waterbomb Base tile). In both types of tile, the inscribed circles touched each other along the tile edges, which, youll recall, were axial creases. In fact, the only places that circles touched each other were along axial creases. This is more than coincidence; it can be shown (and we will do so later) that there exist axial creases in any circle packing wherever any two circles touch. The newly created axial creases divide the square into axial polygons; if we are fortunate enough that the axial polygons are recognizable as known tiles, we can fill them in with the creases associated with each tile and use the resulting creases to fold the shape flat.Thus, the six simple bases we used to illustrate tiling could have been derived directly from circle packings based on their desired number of flaps. We represent each desired flap by a circle; pack the circles into the square, and then construct axial creases that outline the tiles.The circle diagrams also allow us to address the problem of folding efficiency. By representing each flap of a model by the appropriately sized circle and drawing the circles on a square, you can easily find the arrangement of points on the square that gives the most efficient base containing those flaps and, as often as not, it will be an elegant base as well.And since the length of a flap is equal to the radius of the corresponding circle, if we design the base by laying out points on a square and represent each flap by a circle, the most efficient base will come from that layout in which the circles representing the flaps are as large as possible. In accordance with what weve shown above, the circles are to be placed ac- cording to the following rules: Each flap in the model is represented by a circle. The radius of each circle is equal to the length of the corresponding flap. The center of each circle lies within the square. The circles may overlap the edges of the square,but not each other.We have now assembled the necessary building blocks to carry out origami design from the ground up. Throughout the history of origami, most designs were modifications, and the techniques weve learned so faroffsetting, distortion, point- splitting, grafting, tilinghave implicitly assumed that we started with something reasonably close to our final objective. But proximity is no longer needed; we can proceed directly from the desired subject to a base that contains all the structure nec- essary to realize our subject. Here, therefore, is an algorithm for origami design, called the circle method: Count up the number of appendages in the subjectand note their lengths. Represent each flap of the desired base by a circle whose radius is the length of the flap. Position the circles on a square such that no two overlap and the center of each circle lies within the square. Connect the centers of touching circles to one an- other with axial creases, dividing the square into axial polygons. Identify tiles whose circles match up with thecircles in the axial polygons. Fill in the axial polygons with tile creases.The resulting pattern can be folded into a base with thenumber and dimension of flaps with which you started.9.4. Scale of a Circle PatternOne aspect of the circle method of design that we have already seen is that corner flaps consume less paper than edge flaps, which consume less paper than middle flaps. Turn this property around, and you find that for a given size square, you can fold a larger model (with fewer layers of paper) if you use corner flaps rather than edge flaps, and edge flaps rather than middle flaps. Seen in the light of the circle method, the traditional Craneand the Bird Base from which it comesis an extremely efficient design, since all four flaps are corner flaps, and almostall of the paper goes into one of the four flaps. However, add one or two more flaps, and you are forced to use edge flaps. Once you start mixing edge flaps and middle flaps, you begin to run into tradeoffs in efficiency. Sometimes, it is even better not to use the corners for flaps if there are additional flaps to be placed on the square.So, for example, suppose we want to fold a base with five equal-length flaps. A little doodling with a pencil and paper (alternatively, you can cut out some cardboard circles and shuffle them around) will reveal two particularly efficient ar- rangements of circles, as shown in Figure 9.7.r = 0.354r = 0.324Figure 9.7.Two circle patterns corresponding to bases with five equal-length flaps.Now we have two possible circle patterns. Which one is better? Is there any way to quantify the quality of a crease pattern?As mentioned earlier, one measure of the quality of a model is its efficiency, that is, the relationship between the size of the folded model and the square from which it is folded. A quantitative measure of efficiency is to compare the size of some standard feature of the basesuch as the length of a flapto the size of the original square. The most efficient base is the largest possible base for a given size square. Since the flaps of the base are represented by circles whose radius is equal to the length of the flaps, the most efficient base corresponds to the most efficient circle packing, i.e., the packing with the largest circles.To facilitate this comparison, lets assume our square is one unit on a side. (If youre using standard origami paper, aunit is equivalent to 10 inches or 25 cm.) For the crease pattern on the left in Figure 9.7, if all of the circles are the same size, it is fairly easy to work out that the radius of each circle, and thus the length of each of the five flaps in the base, is301Chapter 9: Circle Packingr = 12 0.354 .(91)For the pattern on the right, it requires some algebra tocalculate, but you will find that the circle radius isr = 1 + 1 21 + 2 0.324 .2(92)Since 0.324 is smaller than 0.354, the radius of each circle in the second pattern is about 10% smaller than in the one on the left. Since the radius of the circle is equal to the length of the flap, the flaps in a base folded from the pattern on the right are about 10% shorter than flaps folded from the pattern on the left. Thus, a five-flap base made from the patter