第十四章 多边形分配

1Polygon4PackingAt this point, you have seen that uniaxial box pleating has all the versatility of circle/river packing but has the additional benefit of pro- ducing symmetric, easily precreaseable (if not necessarily easily collapsible) folding patterns.These patterns can be highly complex, and while the tech- nique may be used for all types of subject, it is particularly suited to insects and arthropods (many of which you will meet in this chapter), such as the Flying Walking Stick shown in Figure 14.1.This design contains all the elements of uniaxial box pleat- ing: rectilinear hinge polygons, ridge creases along the straight skeleton, and two elevation levels for axis-parallel folds: axial and axial+1. Although you might enjoy the challenge of figuring out a crease assignment from the contour map, a fully assigned crease pattern is given at the end of the chapter.If, however, you restrict your designs only to the basic ele- ments of box pleating described in the previous chapter, you will quickly bump up against one of the barriers of uniaxial box pleating, for this design approach carries with it several limitations. Fortunately, there are more specialized tech- niquesmodifications and variations of the basic ideathat let you creatively work around the limitations of uniaxial box pleating. Better yet, uniaxial box pleating is just a special case of a much broader, much more powerful concept, whose name I have already introduced: polygon packing. Polygon packing allows one to create complex designs while striking an aesthetic balance between efficiency, symmetry, precreasability, collaps- ibility, and, of course, the desired visual representation of the subject. In this chapter, we will delve deeply into the subtleties 625 Figure 14.1.Contour map and folded model of the Flying Walking Stick.of uniaxial box pleating and will, eventually, arrive at the full- up technique of generalized polygon packing.14.1. Level ShiftingOne drawback of box pleating relative to circle packing is the issue of widthor rather, lack of width. It is not uncommon for the axial creases to be separated by two or even only one grid square, which means that the resulting flaps will be only one or one-half grid square wide. This may not be a problem for insect legs, but it can definitely be a problem for the (typically wider) body. It would be nice to have a technique for selectively widening parts of the base in an elegant and straightforward way.A more serious issue can also arise: what happens if, in the process of bouncing, two contour lines at different elevations turn out to meet head-on, as shown in Figure 14.2?Now, ones first reaction might be that this cant happen. But we might have made decisions in several places about el- evation (for example, forcing axial contours along the symmetry line of the model) that would result in this situation somewhere else in the model.It cant really happen, of course; we cant possibly allow two contour lines of different elevations to run into one another.629Chapter 14: Polygon Packingaxial+1?axialFigure 14.2.Two colliding contours at differ-ent levels.Figuratively, we have a head-on train wreck. What we need is a way to get the two trains onto parallel tracks.And we will find a solution lurking within a very simple structure, shown in Figure 14.3. This is simple to fold: take a Waterbomb Base; sink the point; crease the result through all layers; then spread-sink two corners as you fold the near edge downward. Then closed-sink the flap into the interior of the model.Figure 14.3.A level-shifting test structure.What I would like to do is to compare the crease patterns of the first and last steps of this model, emphasizing the con- tour lines (where I have taken the bottom edge of the folded shape as the axis). First, we have the original shape, as shown on the left in Figure 14.4. It consists of a series of concentric contour lines, with the lowest elevation, axial (green) around the outside and in the center, axial+1 inside of that (brown), and the highest elevation, axial+2 inside of that (violet). Then, on the right, we have a contour map of the result.Figure 14.4.Top left: contour map of the test structure before sinking.Bottom left: the folded form.Top right: the contour map after sinking. Bottom right: the folded form.We have, of course, added some diagonal folds in red (which correspond to ridge creases). But the important thing to observe is that the second line down in the middle, which used to be axial+2, is now just plain axial. We have shifted the elevation of this crease.The folds that created the shifting were the creases along the diagonal ridge crease on each side of the former ridge. Lets focus on just one side of this structure. This pattern of creases, created by the spread-sink, when isolated, becomes a tool for shifting the elevation of an axis-parallel fold, as shown in Figure 14.5.001 12 0110001210Figure 14.5.01210Contour map of a level-shifting gadget. Left: prior to level shifting.Right: after level shifting. The numbers along each contour line indicate the elevation of the contour.Once one knows the contours, then one can work out a layer ordering and assign creases. Figure 14.6 shows one pos- sible crease assignment of the pattern with mountains and valleys but retaining the structural coloring.001 12 0110001210Figure 14.6.01210Crease-assigned contour map. Left: prior to level shifting.Right: after level shifting.I call an isolated pattern of creases like this a gadget. This particular gadget is a design pattern for level shifting. Whenever a contour crosses a ridge crease, as in Figures 14.4 and 14.5, we can use this gadget to shift the elevation on one side by an amount equal to twice the distance to the two sur- rounding contour lines. So in the example above, the axial+2contour line is shifted by the gadget between the two axial+1 contours down to axial+0 (that is, plain old axial) elevation. And clearly, it is possible to use the same gadget to go the other direction as well.In box pleating, ridges can propagate at two angles with respect to incident axial contours: 45 and 90. There are level- shifting gadgets for both situations, and the two possibilities are shown in Figure 14.7.Figure 14.7.The two level-shifting gadgetsfor box pleating.As an illustration of this technique, Figure 14.8 shows a contour map for a Salt Creek Tiger Beetle that is somewhat similar to the generic beetle base of the previous chapter, but2 1 0 12 Figure 14.8.Contour map, base, and folded form for the Salt Creek Tiger Beetle.Note that the abdomen is widened by use of level shifters inserted into the body.631Chapter 14: Polygon Packinghas extra width in the abdomen. This extra width is obtained by inserting two of the 90-incidence level shifters into the ab- domen, which connects an axial contour to an axial+2 contour. (Can you find this connection in the contour map?)A fully assigned crease pattern is given at the end of the chapter.The symmetric gadgets of Figure 14.7 are not the only possible level-shifting gadgets; there are asymmetric versions as well. You can discover these by, for example, spread-sinking the corner in Figure 14.3 at some angle other than the sym- metric angle. You can also construct them graphically. The angle at the tip of the triangle is fixed: 45, in the 90 angle level shifter on the left in Figure 14.7. (In general, the angle at the tip is equal to the angle between the ridge crease and an incident axial). You can imagine keeping that angle fixed and swinging the two lines back and forth from side to side to orient the level shifter more closely toward one axial fold or the other, as illustrated in Figure 14.9.45Figure 14.9.Left: one can swivel the levelshifter back and forth about its tip so long as the angle between the two lines is fixed.Right: an asymmetric level shift- er whose vertices all lie on grid lines.One particularly interesting and useful level shifter is shown on the right in Figure 14.9, which is a pattern I learned from Japanese artist Satoshi Kamiya. A small perturbation in angle puts all four vertices of the level shifter on grid points, making it easy to construct in a grid-based box-pleated de- sign.Other versions of level shifter apply when a contour hits a junction of several ridge creases. Special cases can often be found simply by drawing just the region of paper around the junction with contour lines and the original ridge creases and then spread-sinking to make the contour lines wind up in the right place. And I should point out that just as the spread- sink has been around a very long time, the use of structures like this can be found in many crease patterns for advanced complex designs. Like box pleating, level shifting itself is notnew, but once we recognize the function of a structure, we can then use, adapt, modify, and improve it, and make it one more tool in our designers arsenal.14.2. Layer ManagementLevel shifting allows one to selectively widen (or narrow) flaps. That allows one to, for example, make a body wider than the legs, or distribute layers across the width of a flap, reducing (or at least, balancing) the overall thickness of the flap. This is a useful capability. When one is designing a complex base, even with thin paper, the paper thickness plays a non-negligible role in the finished figure. When used well, it can add needed three-dimensionality to the fold. It can also get in the way, though, driving upward the thickness of flaps that need to be thin (legs, antennae), or simply unbalancing thickness. If one folds an insect with six legs so that four of the legs come from the corners and the other two come from the edges, then those edge flaps will have roughly twice as many layers as the corner flaps. This can produce a notable imbalance in the apparent thickness of the legs.Paradoxically, the solution to such an imbalance, with some legs too thick, is to add layers to the legs that are too thin. If the thicknesses are balanced, it is much less noticeable. This selective addition of layers to flaps can be accomplished by enlarging the corresponding hinge polygons.Fine-grained layer control is an ability that polygon pack- ing offers that is not readily available in circle/river packing. In the latter, the individual flap polygons are defined late in the design process, and you “get what you get.” In polygon packing, we can tinker with the layers in individual polygons, giving much more control over the thickness of the correspond- ing flaps.The way we add layers to flaps in uniaxial box pleating is simple: we make the flap polygon larger than its minimum size. Since all of the paper within the polygon is going to go into the flap, making the polygon larger while maintaining the length and width of the flap insures that the average amount of paper in the flap increases.All leaf flaps (those with one free end) taper in their number of layers, with the fewest layers near the tip and the most near the base, where it joins the rest of the model. This relationship is evident in circle-packed bases, where flaps tend to be triangular; it is less evident, but no less true, in uniaxial box-pleated bases. The number of layers at the base tends to increase linearly with distance from the tip, and is, for evenly639Chapter 14: Polygon Packinglayered 1-unit-wide flaps, given by the perimeter of the flappolygon, expressed in those same units.So, for example, one can “fatten up” a corner flap by add- ing a few more grid squares to the bite it takes out of the corner of the crease pattern, as illustrated in Figure 14.10.Figure 14.10.Left: a minimum-size corner flap.Right: the flap fattened by adding two more units of width.A similar technique can be used to fatten up an edge or middle flap (although the circumstances where you would want to fatten a middle flap are rare indeed).A side benefit to flap thickening comes when folding insects. The addition of one or two units to the width of the flap has the effect of squaring off the end, as shown in Figure 14.10. This squared-off end can then be easily point- splitcreating, for example, the pair of claws at the end of many insects feet.14.3. Whole vs. Half-Integer WidthsIn theory, the exact grid that one uses to make a uniaxial box- pleated base is not that important: if there are three sets of legs, theyll have the same relative proportions whether they are 1, 2, and 3 units long; 2, 4, and 6; or 3, 6, and 9. What willvary is their width relative to their length. For many designs, even this is not too important: a fat flap can be narrowed, par- ticularly easily if it has been turned at a right angle relative to the axis during the shaping folds.So the primary motivation for picking the basic unit is to establish a sort of minimum feature size. This becomes particularly important when the desired subject has a fairly wide regionthe main body, for example. One can use level shifters fairly easily to double the width of a portion of a model, but higher multiples are trickier: one must use multiple level shifters, or more complex level shifters, and the shifting itself consumes paper that might have been desired for other pur- poses.Once we have established a grid, we very often would like to keep all of the creases on the gridideally, without using level shifters at all. That means that in every region of the paper, we would like our contours to alternate axial, axial+1, axial, axial+1, and so forth.This goal may not be possible, though. In fact, it is possible to choose hinge polygons that make this choice impossible. A situation that arises not infrequently is to have a middle flap positioned along the center line of the base, which is usually an axial fold (so that the base can be opened out in plan view). When this situation occurs, the contour down the center is axial; the contour one unit away is axial+1; and then they alternate from there, as shown in Figure 14.11, as one moves around the outside of the polygon.Figure 14.11.A hinge polygon centered on anaxial contour.Now, if we start with an axial contour in the middle and, as shown on the left in Figure 14.11, start working our way around the polygon, we find that when we get to the middle of the left side, there are two axial+1 contours one unit apart. That means there must be a folded contour halfway betweentheman “axial+1/2” contour. And so that part of the fold, and anything that that folded contour connects to, will be half a unit wide, a potentially undesirable outcome.Note that while we chose the elevation assignments of the contour lines along the top and bottom edges of the polygon, the contour lines along the sides were forced by their bouncing off of the ridge creases inside the polygon. So one might consider that, perhaps with a different shaped polygonone with a dif- ferent pattern of ridge creases inside itthe bouncing might work out the way we want. And things might well work for a different hinge polygon, but wouldnt it be nice to know how to pick one? Or, at least, how not to pick the wrong one?Figure 14.11 is a good example of a wrong polygon. With- out even working out the ridge creases and bouncing creases inside the polygon, we could have determined we were in trouble from a simple parity argument. There must be some contour line at every grid point on the boundary of the hinge polygon that is perpendicular to the polygon boundary (be- cause the contours are axis-parallel, the polygon boundaries are hinges, and axis-parallel creases are perpendicular to hinges). So we could just move around the outside, alternating in parity. We are forced to have two consecutive contours of the same parity somewhere along the way because the semiper- imeter of the polygon is an odd number of units in length.And so, the first rule of packing is clear: if we want to avoid folded contours that create half-unit-width flaps, we should make sure that the distance between any two points on the polygon boundary that are required to be axial is an even number of units, measuring around the boundary of the polygon.This even/odd condition on the boundary is necessary; alas, it is not sufficient. It is possible to satisfy the even- semiperimeter condition for a specific polygon but to still have bouncing in the interior cause problems with the desired alternation of contours. Figure 14.12 shows a slightly more complex polygon whose straight skeleton induces a collision of contours.Now, you might notice a feature in each of these poly- gons that is a bit out of the ordinary: in both of them, the straight skeleton contains vertices that dont lie on the grid. And those non-grid vertices are, in fact, the nasty beasties of both of these patterns. At vertices where ridge creases come together, some additional axis-parallel creases must arise. If we force those points to lie on grid points, then we can insure that those additional creases are well behaved on-grid axis- parallel creases.Figure 14.12.A hinge polygon with an evensemiperimeter but that thwarts alternating contours because of the bouncing pattern.We can insure that all straight skeleton vertices lie on grid points by insuring that every side of the hinge polygon is an even number of units in length. That is often relatively e