第八章 模块拼接

8Tilinghile there are many different approaches to origami design, the ones that Ive shown thus far can be arranged in a rough hierarchy of complexity. We started with some simple structuresthe traditionalbases. Then, we modified these bases by various distortions offsetting the crease pattern from the center of the square or distorting the entire pattern. Both types of modification leave the number of flaps unchanged; they only alter the lengths and widths of the flaps.Then we increased the number of flaps by subdividing some flaps into smaller flaps using the various point-splitting techniques. While in principle any number of flaps can be at- tained, point-splitting is inherently a process of reduction; the flaps you end up with are always smaller than what you started with. Thus, there are definite limits on what you can accomplish by point-splitting.We can escape those limitations by using grafting, by ef- fectively adding paper to an existing crease pattern in such a way that the paper remains square after the graft. Grafting allows you to add features to an origami base without taking anything away from features that are already present. The simplest grafts are border grafts, which consist of adding paper around one or more edges of the square, but this method, too, has its limits. You can only add paperand thus featuresto flaps that are made from a raw edge, i.e., corner and edge flaps. Another limitation is that when you are border grafting, edge flaps dont offer quite the same freedom of point creation that corner flaps did; a border graft that can create four points at a corner flap only creates two points of the same size at an edge flap. 241 Yet more variety in added features comes when we realize that the existing crease patterns are not indivisible; we can cut them up and insert strip grafts throughout their structure. Strip grafts create points and flaps along edges just as border grafts do, but they also create extra points in the interior of the paper without diminishing the size of adjacent flaps. As an expansion of strip grafts, we can graft in pleats to create extra edges running across a face, and weave cross- ing groups of pleats to create scales, bristles, and other tex- tural elements. Although they all start with an existing crease pattern, strip and pleat grafts are much more versatile than point-splitting and border grafts and come in many more varia- tions. Strip and pleat grafting possess this great versatility because they are based on dissected crease patterns, and there are usually many different ways to dissect a given pattern.Once weve taken the step to incorporate grafting into dissected crease patterns, an enormously richer variety of ori- gami structures becomes accessible. When grafting in strips of paper, we can vary the width, length, direction, and location of the strips; we can insert multiple strips; and we can create branching networks of strips, all to place additional points and/ or textural elements into the basic design.In the models to which weve applied graftingthe Song- bird, the Lizard, the Turtleour grafts have taken the form of fairly narrow strips. These are still relatively small perturbations to a preexisting model. The precursor to the songbird was still a bird; the lizard with toes began life as a lizard without toes; and the turtle with a patterned shell was still recognizably a turtle when its shell was smooth. But grafts can be made much larger and more complex and can be used to create new bases so dif- ferent from their predecessors that they hardly seem related at all. We will expand our palette of design techniques by exploring further the concept of dissection and reassembly. Thus far, we have treated bases and grafts as two distinctly different types of objects; we start with a base, then we add a graft. In this chapter we will learn to decompose both bases and grafts into the same underlying structures, which can be reassembled in an infinite variety of ways. We will also learn to distill origami bases down to simple stick figures; we will then use these stick figures as tools for the design of new bases.8.1. Uniaxial BasesLets look at several of the bases that Ive shown so far. First, we have the Classic Bases: Kite, Fish, Bird and Frog Bases; to these, we add two new bases, those used for the Lizard and the Turtle. All six are shown in Figure 8.1.249Chapter 8: TilingFigure 8.1.Six bases. Top: crease patterns.Middle: bases.Bottom: representative models.All six of these bases share two properties: First, all flaps either lie along or straddle a single vertical line; second: the hinge at the base of any flap (i.e., the line between two adjacent flaps) is perpendicular to this line. When several flaps lie along a line, that line is called an axis of the base. Any base that possesses a single axis along which all flaps lie is called a uniaxial base. The six bases of Figure 8.1 are all uniaxial; their axes are shown by dashed lines in Figure 8.2. Every flap in each base lies along the bases unique axis.Figure 8.2.The axes of six uniaxial bases.Uniaxial bases are very common in origami, and they have several properties that make them relatively easy to construct, dissect, graft, and manipulate. We will study them intently for the next several chapters.Not all origami bases are uniaxial, however, and before casting aside all other origami bases, its worth taking a few moments to look at some exceptions.Among the traditional bases, the Windmill Base is not uniaxial because its four flaps do not lie along a single line; instead, it has two crossed axes, and the hinge creases are not perpendicular to the axis.A base of a more recent vintageJohn Montrolls Dog Base, variations of which he has used for a score of diverse figuresis also not a uniaxial base, having two distinct par- allel axes. Montrolls base is remarkable for its efficiency in use of paper (and for my money, stands as the most elegant base in all of origami). So while uniaxial bases will prove to beFigure 8.3.Two bases that are not uniaxial.Left: the Windmill Base has two crossed axes.Right: Montrolls Dog Base has two parallel axes.remarkably versatile, they are not the magic solution for all origami problems.Montrolls Dog Base, in particular, highlights a limitation of uniaxial bases; for a given model, they may not provide the most efficient structure. However, uniaxial bases are read- ily constructed and quite versatile, and we will explore them thoroughly.It should also be noted that whether or not a base is uniaxi- al may depend on the orientation of the base. In the six example bases Ive shown, the axis lies along a line of mirror symmetry. This is usually, but not always, the case. For example, in the Waterbomb Base, if we attempt to draw the axis along the lineFigure 8.4.Top: crease pattern for theWaterbomb Base.Lower left: the Waterbomb Base is not uniaxial with respect to an axis along the symmetry line.Lower right: it is, however, uniaxial if we draw the axis along the raw edges of the base.of symmetry, we find that the raw edges of the flaps dont lie along the axis and the hinges arent perpendicular, so its not a uniaxial base. However, if we rotate the base by 90, we can re-draw the axis along the raw edges, the hinges are perpen- dicular to the axis, and it is thereby revealed to be a uniaxial base in this new orientation, as shown in Figure 8.4.Uniaxial bases lend themselves to strip grafting because the alignment of many folded edges along the axis of an exist- ing base makes the creases along those edges natural candi- dates for cutting to insert strip grafts into the crease pattern. The creases that lie along the axis in the base form a special set; they are called the axial creases in the crease pattern. In Figure 8.5 I have colored the axial creases green (whether mountain, valley, or unfolded) in the crease patterns for the six bases. I have also similarly colored those portions of the raw edge of the paper that lie along the axis.Figure 8.5.The axial creases and axial portions of the paper edge in the six uni-axial base crease patterns.Axial creases are natural candidates for cutting and insert- ing strip grafts, in part because in uniaxial bases every flap has at least one axial crease (or an axial raw edge) running to itstip. Consequently, we can always split any flap along an axialcrease to insert a strip graft.Observe that the network of axial creases divides the crease pattern into a collection of distinct polygons whose boundaries are entirely composed of either axial creases or the raw edge of the paper. We will call these polygons axial polygons.8.2. Splitting Along AxesThe axial polygons of the crease pattern have an interesting property in their own right: in the folded base, the entire pe- rimeter of each polygon comes together to lie along a common linethe axis of the model. You can observe this property by taking a base and cutting it along its axis. If you remove a slight bit of paper from either side of the axis so that the cut severs folded edges that lie along the axis, both the base andFigure 8.6.Dissected crease patterns for the Fish, Bird, and Frog Bases.the crease pattern will fall apart into distinct pieces, as shown in Figure 8.6 for the Fish, Bird, and Frog Bases.One or more strips can be inserted along any of the gapsto split or multiply flaps. Lets look at an example.1. Cut the square in half along the diagonal.2. Insert a strip of paper along the cut edges.The exact width of the strip isnt critical.3. Mountain-fold the corner underneath.4. Dent the top of the model and push the sides together so that the edges of the strip align.5. Reverse-fold the bottom corner. Repeat behind.6. Finished grafted shape.Figure 8.7.Folding sequence and creasepattern to form a strip graft within a Bird Base.7. Crease pattern.Figure 8.7 illustrates the process of inserting a strip into the middle of a Bird Base. We cut the base down the middle, then insert a strip into the gap. The resulting shape has paired points at the middle of the top and bottom where the original base had only single points.Now, lets look at what weve accomplished. The Bird Base that we started from had five flaps: four long ones pointing down and one short one pointing upward. Two of the long flaps at the bottom and the shorter flap at the top have now been split into a pair partway along their length. This is not entirely obvious from the final step in Figure 8.7, but if we rotate the layers so that the inserted strip stands out from the rest of the base, the gap becomes visible as shown in Figure 8.8.Figure 8.8.Strip-grafted Bird Base withflaps oriented so that the gapis visible.The interesting thing here is that after the inserted strip, we still have a uniaxial base. And it is instructive to highlight the axial creases of the new base and axial raw edges, as Ive done in Figure 8.9.Figure 8.9.Left: the crease pattern of the original Bird Base. Axial creases areshown in green.Right: the crease pattern of the strip-grafted version.Note that in the process of adding a vertical strip, we also created new horizontal axial creases. The Bird Base was com- posed of four axial polygons, which are four identical triangles. But our inserted strip graft is similarly composed of polygons whose boundaries are axial creases (or the raw edge of the paper): In addition to the four triangles of the Bird Base, we have added two rectangles and two triangles.We can now view grafts in a new light. While we have previ- ously distinguished between the original base and the strip or border graft that weve added to the pattern, they are really not so different. Both the base and the graft are composed of the same fundamental elements, which are the axial polygons. The cre- ation of a graft simply divides the initial crease patternitself a collection of axial polygonsalong its axial folds, then inserts additional axial polygons into the opening as the graft.This unification allows us to approach design in a new way. In the past, we have almost always started with a base and then wrought variations upon it. But since bases are all composed of axial polygons, we can dispense with the idea of starting from a base and adding grafts; instead, we can actually build a base from scratchmaybe grafted, maybe notsimply by assembling axial polygons into a crease pattern. If we think of each axial polygon as a tile of creases, then the problem of design becomes a problem in fitting tiles together in such a way that we obtain all the desired flaps in our base, and the tiles fit together to make a square.8.3. Tiles of CreasesWe have already encountered several possible tiles in the Clas- sic Bases and the grafted variants seen so far. Lets enumerate them.(a) (b)(c)Figure 8.10.Crease pattern and folded form for three orientations of the triangulartile that makes up the Classic Bases.251Chapter 8: TilingFirst of all, there is the triangular tile that makes up the four Classic Bases. It comes in three distinct forms, depending on the orientation of the flaps (see Figure 8.10).These three forms are only distinguished from one another by the location of the mountain fold in the crease pattern and the positions of the flaps in the folded form. In the crease pattern within each triangle there are four folds one mountain fold and three valley foldsextending from the crease intersection to the corners and edges. Note that in all three cases, all edges of the triangle lie along a single line; the polygonal tile is uniaxial.The Lizard and Turtle bases are also composed of triangles, but different ones: an isosceles triangle from the Lizard, and an equilateral triangle from the Turtle, as shown in Figure 8.11. These, too, are uniaxial.Figure 8.11.Left: the triangle tile from the Lizard base, crease pattern and foldedform.Right: the equilateral triangle tile from the Turtle base.Every such triangular tile has three possible folded forms, just like the isosceles right triangle tile shown in Figure 8.10. The creases within each tile are the three angle bisectors from each corner (which always meet at a common point) as valley folds, and a mountain fold that extends from the intersection point perpendicularly to one of the three edges. Since there are three edges, there are three possible choices for the mountain fold. When we enumerate tiles, its not necessary to show all three forms for every triangle; you should keep in mind that for any triangle, all three flap arrangements are possible. The three tiles shown here are not the only possible triangular tiles, either. In fact, it can be shown that every triangle can be turned into such a tile by constructing the three angle bisectors as valley folds and dropping a perpendicular mountain fold from their intersection to an adjacent edge.Are the only such tiles triangles? Clearly not; look again at the grafted crease pattern in Figure 8.9. The strip graft is(a) (b)(c)Figure 8.12.(a) The rectangle tile from the strip graft.(b) A wider rectangle.(c) The limiting case of an equilateral rectangle, i.e., a posed of rectangles and triangles. The triangles are famil- iar; the rectangles are new. Rectangles, too, can be used as tiles from which crease patterns may be assembled. Figure 8.12 shows the rectangular tile from the strip graft; it, too, can be folded so that its perimeter lies along a common line. Thus, a rectangle can also serve as an axial polygon.Just as we saw that creases can be constructed inside of any triangle to make an axial polygon, so too can creases be constructed within any rectangle, no matter what its aspect ratio. Figure 8.12 shows creases for three different aspect ratios, including the limiting case of a squarewhich gives rise to the uniaxial orientation of the Waterbomb Base as its folded form.As we saw for the triangle, it is possible to orient the flapsof a tile in several different ways. Figure 8.13 shows severalFigure 8.13.Three different crease patterns and arrangements of flaps for arectangular tile.259Chapter 8: TilingFigure 8.14.Generic form of the rectangulartile and one possible arrangementof flaps.possible orientations for the flaps for one of the rectangular tiles. The variation arises in the creases that run perpendicular to an edge. We can recognize and treat the essential similar- ity among all such variations by simply drawing the tiles in a generic form, with undifferentiated creases perpendicular to all edges as in Figure 8.14. When the tiles are assembled into full crease patterns, some of those creases will get turned into mountain and/or valley folds, but we canand willdefer that assignment until a later time.Are there more possible tiles than these? Uncountably more, as it turns out. In addition to triangles and rectangles there are tiles from pentagons, hexagons, and octagons, both regular and irregular. In later chapters, we will lear