Issue 3 March-April, 2011

Crease Pattern: Hex-Pleated Scarab Beetle

Posted by robert.lang

Last issue, I described the design of a Cicada Nymph in which I used the design technique of uniaxial box pleating, which is a version of
box pleating suitable for uniaxial bases. Box pleating itself is, of course, not new at all. The technique has been a part of origami since
the 1960s and many origami artists, notably Neal Elias and Max Hulme, created many beautiful and complex figures using box pleating
(and figures that went well beyond the limitations of uniaxial bases). Nevertheless, by focusing ones attention on the subset of uniaxial
design that is uniaxial box pleating, it is possible to create bases with all the versatility of circle/river packed forms, but with the relative
ease of precreasing of classical box-pleating.

I have used uniaxial box pleating for many designs, primarily insects and their kin. Once you've designed a few figures using uniaxial box
pleating, though, their designs begin to take on a certain boring sameness. Sure, if what you're primarily interested in is the finished
result, you can, perhaps live with a boring design. But I rather like it when both the finished figure and the design embodied in the crease
pattern are interesting to look at, having enough symmetry to be beautiful, but enough diversity to be stimulating.

So, this issue, I would like to show off another design technique that is, I believe, entirely new (at least, I have not seen any examples of
its usage outside of my own work). I call it hex pleating, and to show it off, I will use it to design a relatively simple scarab beetle. Hex
pleating is a technique for designing uniaxial bases, and it is best understood by analogy with uniaxial box pleating, so I will jump back
and forth between the two design families to emphasize differences and similarities.

In uniaxial box pleating, one designs the crease pattern on a regular square grid. All of the important vertices of the crease pattern lie on
the grid. The axial and axis-parallel creases (those that run parallel to the axis in the folded form) run along grid lines, as do the hinge
creases (those folds that run perpendicular to the axis). The ridge creases run at multiples of 45°; this property is a necessary condition
because all ridge creases turn out to be bisectors between pairs of hinge creases. To design a uniaxial box-pleated design, we choose
a set of hinge polygons and hinge rivers, each shape corresponding to a flap of the base; we pack them onto the grid, and then
construct ridge, axial, and nonzero-elevation axis-parallel creases that give rise to the base.

(And if the preceding paragraph sounds like gibberish, as it likely does, you might take a look at last issue's description of the uniaxial
box-pleated Cicada Nymph design, which describes all of these concepts. And if that doesn't help, uniaxial box pleating, hex pleating,
and more will get a lengthy treatment in a new edition of Origami Design Secrets that will be published this fall.)

Hex pleating works the same way as box pleating, but uses a different grid: one composed of equilateral triangles, as shown in Figure 1.
Like uniaxial box-pleating, we create polygons and polygonal rivers for each of the flaps (leaf flaps and branch flaps, respectively) of the
desired base, polygons and polygonal rivers whose edges run along the lines of our desired grid. For box-pleating, the grid is square,
and so those polygons are squares, rectangles, and generally, rectilinear polygons. For hex-pleating, the grid is triangular, and so
fundamental polygons are hexagonal and polygonal rivers that contain bends at angles defined by the grid: 60°, 120°, and 180°.

Figure 1. The triangular grid of polygon packing and several hinge polygons and hinge rivers. Each polygon creates a flap at least as
long as its enclosed circle. Light blue lines indicate the minimum-sized polygon for the given circle.

Both uniaxial box pleating and uniaxial hex pleating are examples of uniaxial polygon packing, which is a generalization of circle packing
in which we choose nice, well-behaved polygons, rather than circles, to represent the various flaps in the base. There is a close
relationship between polygon packing and circle/river packing: in the latter, each polygon must fully enclose the corresponding circle
(whose radius is the length of the desired flap). In the same vein, the width of polygonal rivers must be equal to the length of their
corresponding flaps.

So, for a hex-pleated design, we will choose hexagonal hinge polygons and hinge rivers, rather than rectilinear ones as in box pleating.
But from there, the design procedure is the same as for uniaxial box pleating: fill in ridge creases, follow with forced axials, then finish up
with all axis-parallel creases of nonzero elevation.

Let's see how this technique plays out. Figure 2 shows the stick figure for my desired scarab beetle. It's a very simple design, quite
generic, with six legs emanating from a single point, a head, and two antenna. We'll keep this simple: we won't add any extra claws, or
flaps spaced along the body to create body divisions or width variation (though I will leave a little extra length in the thorax and abdomen
that will allow some 3-D shaping).
 Figure 2. The desired stick figure with dimensions.

As one typically does with grid-based polygon packing, the flap lengths are expressed in integer multiples of a basic unit. For box
pleating, that unit is, of course, the edge length of the square. For hex pleating, the length unit is something else. It is not, as you might
have guessed, the side length of the equilateral triangle. It is, instead, the height of one of these triangles, since that is what defines the
size of a circle enclosed by a hexagonal hinge polygon.

Those flap lengths specify the minimum sizes of the packing hinge polygons (and the width of the sole polygonal river in this design,
which lies between the legs and antennae). As with circle/river packing and uniaxial box pleating, we must pack polygons so that their
circle centers lie within the confines of the square paper and polygons may touch but may not overlap.

For our scarab design, then, we will have a very small number of polygons to pack. As I did with last issue's Cicada Nymph, I will work
inside of a half square, with the right edge of the half-square being the center line (and line of symmetry) of the full crease pattern.

So, we start with the minimum-size features for all of the polygons. For uniaxial box-pleated figures, the minimum-size polygons are
squares; for hex-pleating, they are hexagons. Figure 3 shows the minimum-size polygons and a polygonal river packed into a
half-square which is itself superimposed upon the triangular grid. Those flaps that should end up on the line of symmetry are centered
on the symmetry line of the crease pattern, which is the right side of the half-square.

Figure 3. Packing of minimum-size polygons into a half-square. The right side of the figure is the vertical center line of the crease
pattern.

Now, as you can see, there is a lot of extra space in the half-square that doesn't beloing to any polygon. In polygon-packed designs,
though, every bit of paper must go into some hinge polygon or hinge river. We can deal with this by expanding some of the polygons
and adding a few more to absorb some of the excess paper without lengthening the flaps we've already created. This gives a new
packing, shown in Figure 4, and a stick figure modified to show the new extra flaps. The dark blue hinge lines are now the actual
boundaries between connected flaps.

Figure 4. A full packing of hinge creases (blue). Every part of the square now belongs to some flap. We have added two new flaps
along the center line, one of length 1, one of length 2.
Here's a little bit of rationale for these choices. I wanted to keep each of the length flaps as thin as possible, so I tried to add as little
extra paper to the leg polygons as I could; most of the extra paper is absorbed by widening the abdomen flap (because we don't care if
we have extra paper in the abdomen). I also added an extra flap near the head because I could; there was paper that needed to be
consumed, so I might as well use it fruitfully. And a new flap near the bottom absorbed the remaining extra paper; it will end up tucked
into the abdomen, again, where it can do no harm.

These polygons, hinge polygons, are outlined by hinge creases. In other words, these are the actual regions of paper that will go into
each flap.

Now, it's time to construct the ridge creases. As with uniaxial box pleating, the ridge creases are the straight skeleton of their
corresponding polygons, created by constructing angle bisectors between pairs of sides that propagate inward from the corners. The
network of ridge creases is shown in Figure 5.

Figure 5. Ridge creases (red) for the polygons of the scarab beetle (again, in a half-square whose right edge is the center line of the full
crease pattern).

Once the ridges are in place, it's time to construct the axial contour lines (which will be axial creases). Axial lines always run
perpendicularly to the hinge lines, so while the hinge lines always followed grid lines, axial lines will, in general, cut across grid lines but
will still pass through grid points. Some axial contours are forced by their origination along the center line of the crease pattern (always
an axial line for a plan view base); others may be chosen so as to give an even spacing between axial contours, which will result in the
exterior edges of the base being neatly aligned in the folded form. An axial contour set is illustrated in Figure 6.

Figure 6. Network of axial contours (green). Note that when an axial contour hits a ridge crease, it reflects across the crease.

The third big step, after the hinges, ridges, and axials, is to construct the nonzero-elevation axis-parallel creases. (For a discussion of
elevation, see the Cicada Nymph from the last issue.) This design is very simple: the axial contours are all evenly spaced, and so we can
simply place axial+1 contours halfway between them, as shown in Figure 7.
 Figure 7. The crease pattern with axial+1 contours (orange) added.

This would complete the crease pattern, giving a base of uniform width, and a rather narrow width at that. Hex pleated designs have a
natural width unit, but that unit is not the same as the length unit; in fact, the natural width unit in hex pleating is only about 58% of the
length unit, so for a given length flap, the hex-pleated version will be narrower than that of its box-pleated counterpart.

For insect legs, we don't really care about that (in fact, narrowness is downright desirable), but for insect bodies, a little width would be
nice. We can widen the body by the introduction of design gadgets I call level shifters. I used box pleated level shifters in the Cicada
Nymph to widen the wings; I can use comparable structures in this design to widen the body, as shown in Figure 8. The shifted folds are
highlighted, and they allow me to shift a portion of a contour from axial to axial+2 — two units away from the axis of the model.

Figure 8. Altered contour map. The highlighted crease has been shifted to elevation axial+2.

With that, the contour map is completed. One can now assign creases to the contour lines, mountain or valley, as appropriate. A
complete, crease-assigned pattern in shown in Figure 9 (which is also downloadable as a vector PDF file if you click on the image).
 Figure 9. The assigned crease pattern. Note that I have retained the structural color information. Mountain folds are solid, valleys are
dashed. (Click on the image to download a vector PDF version.)

The collapse into the base and final shaping of the model will be left as an exercise for the reader. The finished model is shown in
Figure 10.

Figure 10. Finished Scarab HP.

Hex pleating was certainly not necessary to create this simple beetle, but it's a nice design technique that gives rise to elegant,
symmetric crease patterns and fun, surprising base collapses. I've used it for several designs of my own, and I encourage readers to
give it a try.

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