第五章 分支

Splittin5g Pointshe currency with which the origami designer most often deals is the point (or flap). Subjects can be classified by the number of significant points that they have. A snake has two (a head and a tail); a standing bird has four (six, if the wings areoutstretched). A mammal has six (four legs, a head, and a tail), while a spider has eight. A lobster may have twelve; a centipede, one hundred. The number of points in the subject dictates the number of flaps needed in the base from which it is folded.The number of points assigned to the subject depends not only on the subject, but also upon what the designer defines as a “significant” point. Quite often, minor features such as ears may be derived from small amounts of excess paper in the model and may be safely ignored in the initial stages of design. However, the larger the point is, the more important it is to include it in the ground stages of design.Most peoples first designs are modifications of another model. Ones first structural design might very well be a modi- fication of an existing base, either a Classic Base or perhaps a more recent base from the origami literature. A prime reason to modify a base is to obtain one or more extra flaps. Short of redesigning the base entirely, it is often possible to convert one flap into two, three, or more flaps by folding alone, a process I call point-splitting. The ability to split a pointwithout cutting is a useful tactic to have in the designers arsenal, and it also provides tangible evidence of the mutability of origami bases.5.1. The Yoshizawa SplitOf course, there is always one way to split a pointcut it in two. Traditional Japanese designs quite often did split points in 93 this way (the custom of one-piece, no-cut folding is a relatively modern restriction) and many of the designers of the 1950s and 1960s had no compunctions about cutting a point into two or more pieces to make ears, antlers, wings, or antennae. Isao Hondawho for many in the West defined 1960s origami design via his English-language publicationsused cuts as a matter of routine. Yoshizawa, the man from whom Honda derived many of his designs, also on occasion used cuts, but even in the 1950s had developed what is to my knowledge the first technique of splitting a point into two by folding alone. This procedure is illustrated in Figure 5.1, on a Kite Base flap. (I encourage you to fold a Kite Base and try it out.)99Chapter 5: Splitting Points1. Fold the top of the flap down and unfold.2. Sink the tip on the existing creases.3. Mountain-fold a portion of the flap behind. The exact amount isnt critical. Turn the paper over.4. Fold the flaps up and spread-sink the corners.Figure 5.1.5. Finished split flap. The dashed line shows the outline of the original flap.Sequence for splitting a Kite Base flap into two smaller points.Where once there was one flap, now there are two flaps, albeit considerably shorter than the one we started with (which is indicated by the dashed line). This maneuver is particularly nice for turning one long flap into a short pair of ears, which is precisely what Yoshizawa used it for.Of course, the resulting points are smaller than the one we started with; nothing comes for free. Our folding sequence was not particularly directed at making them long, which leads to the question: How long could they get?That question actually raises a more fundamental one: How does one quantitatively define the length of a flap? Intu- ition suggests that we define the length of a flap as the distance from the baseline to the farthest point of the flap, but this only pushes the question down a level: What would we mean by the baseline of the flap? Let us define the baseline of a flap as an imaginary line drawn across the flap, above which the flap can freely move. While this still permits some wiggle room in the definition, it allows us to define a baseline for the two flaps in the split point above as the obvious horizontal crease; in this case, we can clearly define the length of the two flaps as shown in Figure 5.2.A6. The new flaps are shorter than the original we started with.The two new flaps are less than 1/4 of the length of the originalnot a very good tradeoff, it would seem. That gets us back to our question: How long can we make the two flaps? If you have folded an example to play with, you can answer this experimentally, by taking flap A and gently pulling it lower down while massaging the spread-sunk triangles and allowing them to expand toward each other, as shown in Figure 5.3. (It is actually easier to do this before having ever pressed the two triangles flat.)As Yoshizawa pointed out in his opus, Origami Dokuhon I, the optimum length is attained when the distance from the top edge down to the valley fold is equal to half the width of the top edge.Clearly, these are the longest flaps we can make, at least, by this technique. Note that the baseline of the flaps moved downward in the process. Quite often, we have the baseline of the flap already defined and wed like to make the longestFigure 5.2.Comparison of the original andnewly created flaps.AFigure 5.3.Construction of the maximum-length pair.Figure 5.4.First fold for the optimum-length pair.7. Grasp flap A, and pull it lower while expanding the spread-sunk corners. Flatten when the two spread-sunk triangles meet in the middle of the paper.8. The maximum length pair of points.points possible extending from that baseline. Examination of the geometry of the point pair shows that, with a few precreases, we can go straight to the optimum-length fold, as shown in Figure 5.5.Even in the optimum-length case, the two flaps you end up with are much shorter than the original flap you started with. The ratio of their lengths can be worked out using a bit of trigonometry.short flap = tan 33.75 0.277.(51)long flaptan 67.5Each of the short flaps is 28% of the length of the long flap; in other words, weve given up almost a factor of four in length. This seems unnecessarily wasteful. One might think that the length of a long flap could somehow be divided up when we split the flap; one might think we should be able to divide a long flap of length 1 into two flaps of length 1/2, or three of length 1/3, and so forth.And in fact, we can do better than the Yoshizawa split. This procedure is quick and (relatively) simple, and its1. Fold and unfold to define the desired location of the base of the two flaps.2. Fold and unfold along angle bisectors.3. Fold and unfold along a horizontal crease that passes through the intersection of the two bisectors.4. Sink the point on the crease you just made.Figure 5.5.5. Mountain-fold the far edge behind while spread-sinking the corners. Turn the model over.6. Finished pair of flaps.Alternate folding sequence for the optimum-length pair of points.certainly good enough to generate a short pair of ears, but it would be nice to do betterfor example, to take a three-legged Bird Base animal and give it that elusive fourth leg.5.2. The Ideal SplitThe key to making two longer flaps is to ignore the foreground and examine the background; that is, turn our attention away from the flaps themselves and instead, look at the space around the flaps. The thing that makes two flaps two instead of one is not the paper making up the flaps; its the space weve created between the flaps that defines the pair. What limits the length of the flap is the length of the gap. And that is significant because the gaps as well as the flaps consume paperand so we must allocate paper for both.A small thought experiment will bring this out. Supposeyou wished to travel from the tip of one flap to the tip of theother, but you could only travel along the paperyou couldnt jump across the gap. Imagine a microscopic bookworm that travels within a sheet of paperthat is, he crawls between the two sides of the sheet but never ventures to either surface. (He is a very shy bookworm.) How far must he crawl to get from the tip of one flap to the tip of the other? Even without knowing anything about the folded structure of the two flaps, we can say for certain that the bookworm must travel from the tip of one flap down to its baseline, then back out to the tip of the other flap (at a minimum) because theres no shorter path that doesnt require the bookworm to leave the paper, as shown in Figure 5.6. So the bookworm must travel the sum of the lengths of both flaps. (And since it may not be possible to go from the baseline of one flap directly to the other flap via the interior of the paper, the journey could be even longer.)Figure 5.6.Path followed by an origamibookworm.Suppose, for the moment, that our bookworm were further restricted to traveling only along folded or raw edges. Then there is only a small number of paths he could travel along. The two shortest paths, labeled A and B, are shown in Figure5.7 by dashed lines (in some cases, he is traveling along hidden layers of paper). A third path, labeled C, is shown that does not follow existing folds.It helps to distinguish the different paths by simultaneous- ly examining the crease pattern and the model with the paths drawn on each. These are shown together in Figure 5.7.Of course, paths A and B are only two of the possible paths the bookworm could take, but these are the two shortest paths that travel along folded or raw edges of the paper. Neither, however, is the shortest possible path from the bookworms point of view, which is the same whether the paper is folded or unfolded. That shortest path is easy to draw on the crease pattern; its a straight line. Its a bit harder to work out what it is on the folded model, lying as it does in hidden layers of paper, but it is shown as path C in Figure 5.7.X YX YX YBACBXYXYXYCAFigure 5.7.Upper row: path in the folded form.Bottom row: path traveling along the surface of the unfolded paper.Now, whats interesting is that although path C is the short- est path from tip to tip in the unfolded paper, its clearly not the shortest possible path in the folded model. As can be seen in the figure, the bookworm backtracks a bit and actually travels somewhat below the baseline of the two points. This means that weve devoted more paper to the gap than we really needed to paper that could have been used to make longer points.In the most efficient possible structure, the amount of paper that is used to create a gap between two points would be as close as possible to the minimum required. In other words, if we compared the tip-to-tip path in the folded model and the crease pattern, they would look something like Figure 5.8.Of course, we dont know what the rest of the crease pattern looks like or even what the folded model looks like. But weve identified several salient features of both. We know where the tips of the two points are (indicated by the black dots), and we know how deep the gap is in the folded model (half the distance between the point tips on the crease pattern). Knowing the depth of the gap, we also know where the baseline of the two flaps must be, and we can make cor- responding creases on the paper.Figure 5.8.The optimum tip-to-tip path.Top: path in the folded model. Bottom: path through the paper. The creases (and the exact shape of the folded model) are not yet specified.Now, we have two points with a gap between them, and the shortest bookworm path on the crease pattern is also the shortest bookworm path on the folded model. This strikes precisely the right balance between paper devoted to the flaps and paper devoted to the gap. The paper saved from the gap can go into making longer flaps. And indeed, a comparison of the folded and original flap in Figure 5.10 shows that the two flaps are indeed longer than in the simpler split.The ratio of the lengths of the new and original flaps is101Chapter 5: Splitting Pointsshort flap =long flaptan 45tan 67.5 0.414,(52)which is almost 50% longer than that obtained by simply sink- ing and spread-sinking the corners. This is, in fact, the longest possible pair of flaps that can be made from a standard Bird Base corner flap, and so I call it an ideal split.One quibble you may have: In this form, the two flaps over- lap each other while the Yoshizawa-split flaps have daylight between them. So the structures are not perfectly comparable. It is possible to further sink and squash the ideal split to put a gap between the two flaps (at the cost of a slight reduction of gap depth). But if you fold the two pairs in half (as one might, for example, in making a pair of ears), then the two arrange- ments can be compared directly, as shown in Figure 5.11.BaseBaseBase1. For an optimally efficient point split, the point where the base crease hits the edge is separated from the desired point tip by a distance equal to half the separation between the point tips.2. Now well refold the model. The top point isnt used; fold it down.3. Fold the sides in.Base4. Make creases that connect the edges of the base with the points that will become the tips of the two flaps. Turn the model over.5. Squeeze the sides in so that the extra paper swings downward, using the creases on the far layers that you made in the previous step.6. Fold the excess paper from side to side. Flatten firmly.Base7. Reverse-fold the edge inside. Observe that the edge of the reverse fold (the black dot) lines up with the previously defined base.8. The finished split point. Now the shortest path on the crease pattern is also the shortest path on the folded model.9. Crease pattern.Figure 5.9.Folding sequence for the ideal split.Figure 5.10. The finished point pair compared to the original flap length.Figure 5.11.The ideal split is about 50%longer than the Yoshizawa splitfor the same starting flap.The ideal split takes more folds to perform if you fold it as shown in Figure 5.9, but that sequence was designed to illustrate the connection between the paths and crease pat- tern. Thats not necessarily the most straightforward way to fold. Once youve worked out the crease pattern for a model or technique, its worthwhile going back and experimenting with different ways of folding. There are many ways of performing an ideal split on a standard flap. The sequence in Figure 5.12, which was developed by John Montroll, is one of the most elegant.There are numerous variations, both in arrangements of layers (note that this sequence has a slightly different arrange- ment of the layers) and in the folding sequence that gets you to the finish.Point-splitting can be used to breathe new life into old structures. For example, few shapes are as picked-over as the venerable Bird Base, possessed of four large flaps, correspond- ing to head, tail, and two wings. But by splitting the tail point, we can create two legs instead of a tail; by splitting the head point, we can create a head with an open beak, a head with a crest, or quite another flying beast altogether: a Pteranodon.You will find folding instructions for this figure at the endof this chapter. It includes both ideal and Yoshizawa splits.109Chapter 5: Splitting Points1. Fold the top of the flap down.2. Fold the flap up so that its rightedge is aligned with the layer underneath.3. Pull out the loose edge as far as possible.4. Squash-fold the edge and swing the flap over to the left on a vertical fold.5. Pull up the loose edge as far as possible, releasing the trapped paper under the flap.6. Outside-reverse-fold the white point.7. Reverse-fold the flap through the model. Turn the model over.Figure 5.12.8. Reverse-fold the flap back to the center line.9. Finished ideal split.Folding sequence for the ideal split, after Montroll.Splits can be used for more than point multiplication; as an auxiliary benefit, by splitting the large central point with a Yoshizawa split, we can reduce its height while preserving theFigure 5.13.Crease pattern, base, and folded model for a Pteranodon.1. Top: crease pattern for a corner flap. Bottom: the folded flap.2. Top: crease pattern for an edge flap. Bottom: the folded flap.Figure 5.14.3. Top: crease pattern for a middle flap. Bottom: the folded flap.Crease pattern (upper row) and folded flap (lower row) for three typesof points: (left) corner flap, (middle) edge flap, (right) middle flap.flapping action, so when you pull its head and legs like the traditional flapping bird, this Pteranodon flaps its wings.5.3. Splitting Edge and Middle FlapsThe folding sequence shown in Figure 5.12 works for a corner flap, a flap formed from a corner of the square, which describes the main flaps of the four Classic Bases. In addition to four corner flaps, the Frog Base possesses a different type of flap: Its central point comes from the middle of the paper