In origami bases, flaps can be categorized as corner, edge, or middle flaps, depending on whether the tip of the flap is located at a corner of the paper, along an edge, or somewhere in the interior. It is fairly easy to see that for a given length and width of flap, corner flaps have the fewest layers; edge flaps have an intermediate number of layers; and middle flaps have the most. Thus, corner flaps are the best choice for flaps that must be thinned a lot; edge flaps are a second choice; and middle flaps should be avoided.
There are other differences, too. After you have run several TreeMaker designs and folded them, you will start to notice that some crease patterns are easy to collapse into bases while others are incredibly difficult to fold up. Closer examination of this phenomenon reveals a common thread: the more middle points a base has, the harder it is to fold up. While edge points can almost always be separated from the rest of the base by simply being folded out to the side (perpendicular to the plane of the the base), middle points are often surrounded by layers of paper and can only be separated out by being reverse-folded up and out of the model (lying in the plane of the base).
The more flaps there are in a base, the more likely it is to have middle flaps, and the greater the relative proportion of middle flaps. With only 3 or 4 flaps in the tree, they will almost inevitably be edge flaps. However, even with as few as 5 flaps, you will find that one or more flaps will tend to move into the middle. For example, suppose we want a base with five equal-length flaps. Construct a tree as shown in Figure Ti-3-1.
Figure Ti-3-1.
Depending on the initial configuration of nodes, there are two locally optimum arrangements of nodes. If you start with the initial configuration above, then you will probably wind up with the solution shown in Figure Ti-3-2, which has a scale of 0.3236.
Figure Ti-3-2.
If you start with one of the nodes in the middle of the square and the others near the four corners, you will probably arrive at the solution in Figure Ti-3-3, which is significantly larger with a scale of 0.3536.
Figure Ti-3-3.
In general, for 5 or more flaps, the most efficient crease pattern will have at least one middle flap. In this example, we were able to force all of the flaps to be edge flaps simply by starting from an initial configuration in which all of the leaf nodes were on the edges. This is not always possible, however. For example, if you try the same problem for seven equal-length flaps, most initial configurations will place a node in the middle of the paper, creating a middle flap. A typical circle pattern (with a scale of 0.2657) is shown in Ti-3-4.
Figure Ti-3-4.
However, we can force nodes to lie on the edge of the paper, thereby turning the associated flaps into edge flaps, by setting conditions on them. Select all the nodes, either by shift-clicking on each in turn, or by choosing Edit->Select->All. Then choose Condition->Node(s) fixed to Paper Edge. This will set a position constrained condition on each of the leaf nodes in the selection (but not the branch nodes; such a condition would have no effect anyhow). The conditions are identified in the Design window as small flags on each node, as shown in Figure Ti-3-5.
Figure Ti-3-5.
This will force each leaf node to stay on the edge of the paper. Then re-run the optimizer. When you do, you will find that all the nodes now lie on the edge of the paper, essentially forming one giant polygon, as shown in Figure Ti-3-6.
Figure Ti-3-6.
Choose Action->Build Crease Pattern and View->Creases View. This node arrangement results in the crease pattern shown in Figure Ti-3-7.
Figure Ti-3-7.
This crease pattern gives the folded form shown in Figure Ti-3-8.
Figure Ti-3-8.
Ease of collapsing the crease pattern into a base is a good reason to force flaps to be edge flaps; another good reason is the number of layers. For a given width flap, edge flaps have only half the number of layers of middle flaps, and so if a flap is to be folded further, it may be desirable to make it be an edge flap so that it will be easier to fold.
Another reason to make a flap be an edge flap is for color-changing. The bases produced by TreeMaker are inherently single-color. If you are making a model that exploits the colors of both sides of the paper --- a bald eagle, for example, which would have white head, tail, and feet --- then you will need to make sure that the flaps to be color-changed are edge flaps, so that they may be turned inside-out around the raw edges of the flap. (It is extremely difficult to turn a middle flap inside-out!) Of course, color-changing individual flaps is only the most rudimentary way of exploiting two-tone models and is insufficient for models with stripes or spots, for example. For zebras, tigers, and giraffes, you'll have to use different techniques than those provided by TreeMaker.
All bases designed by TreeMaker are of a type known as uniaxial bases. A uniaxial base is one in which all the flaps lie along a single line. A base in which all of the flaps are edge flaps has the very useful property that the layers can be arranged so that with all of the layers in the same half-plane, no major flap is enclosed inside of another flap. I call a base with this latter property a simple uniaxial base. Simple bases can also be designed by Fumiaki Kawahata's "string-of-beads" algorithm. It is fairly easy to prove that the perimeter of an edge-flap-only base cannot exceed the perimeter of the square; conversely, one can also show that by allowing middle flaps, it is possible to construct a flat origami shape with arbitrarily large perimeter from a finite size sheet of paper!
It is tempting to suppose that any simple uniaxial base must be an edge-flap-only base, but it is often possible to add auxiliary nodes to a base with middle points to make them "quasi-edge-like". An example will make this clear.
An eight-pointed base in which all of the flaps are forced to be edge flaps is shown in Figure Ti-3-9. It has a scale of 0.2500.
Figure Ti-3-9.
If we remove the conditions and re-optimize, we'll find the arrangement shown in Figure Ti-3-10, in which four of the points have become middle flaps. This has a scale of 0.2588, just a bit larger than the all-edge-flap base.
Figure Ti-3-10.
Although four of the flaps are middle flaps and will be trapped inside layers, it's possible to squash-fold the raw-edged flaps attached to each node in such a way that a simple uniaxial base is obtained. And clearly, the perimeter of this base exceeds the perimeter of the square by the factor (sec(15°)), or by about 3.5%.
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