Once you have completed an optimization, the simplest crease pattern that collapses into a base is produced by choosing Action->Build Crease Pattern, which creates all creases using the "universal molecule" algorithm. You can also break up high-order active polygons into lower-order active polygons by adding new edges to existing nodes. When you create a new edge and scale it using Action->Scale Selection, it will be expanded until it forms new active paths; typically 3 new paths are formed, although if the polygon is mirror-symmetric, sometimes 4 new paths will be formed.
This process will break any polygon of order 5 or higher into polygons of lower order. In general, the lower the order of the active polygon, the simpler it is to fold, so this is often a desirable process. However, if you try to break up a polygon of order 4 (a quadrilateral), unless it happens to have a line of symmetry, you will form two triangles and another quadrilateral. Thus, you cannot in general reduce a crease pattern to all triangles simply by adding new edges to existing nodes.
As an example, construct the tree shown in Figure Ti-4-1, which has four edges of all different lengths.
Figure Ti-4-1.
When you optimize this tree, you will get an irregular quadrilateral, as shown in Figure Ti-4-2.
Figure Ti-4-2.
Building creases gives you a "gusset quadrilateral" as shown in Figure Ti-4-3.
Figure Ti-4-3.
Supposed we want to break up this quadrilateral into triangles. Kill the crease pattern and add a new edge to node 2; then choose Action->Scale Selection to maximize the length of the new edge. You should get something like the polygon pattern shown in Figure Ti-4-4. Indeed, the quad has been broken into two triangles --- and another quad.
Figure Ti-4-4.
The process can be repeated essentially infinitely; you can make the quad smaller and smaller, but you'll never get rid of it! Figure Ti-4-5 continues the process, but there is still one quadrilateral.
Figure Ti-4-5.
There is a fundamental reason for this limitation: breaking up an arbitrary quad into four triangles requires the formation of four active paths. But when you add a new edge to an existing node, you only have three degrees of freedom: the x and y coordinates of the new node and the length of the edge. To create four active paths, you need a fourth degree of freedom.
If, instead of adding the new edge to an existing node, we add the new edge to a newly-created node formed by splitting an edge, we have a fourth degree of freedom: the location of the split along the edge. An edge that creates four active paths with the corners of its enclosing polygon is called a stub, and TreeMaker has an entire submenu devoted to their generation: the Edit->Stub submenu. These commands let you break up one or more polygons by splitting one of the edges of the subtree spanned by the nodes at the corners of the polygon.
The simplest way to break up a polygon is when the crease pattern has been built: click near the border of the (yellow) polygon to which the stub is to be added. To see this in our example, delete the fifth node and edge we added and choose Action->Build Crease Pattern. Then click near the boundary of the big yellow polygon. The polygon will become selected, as shown in Figure Ti-4-6.
Figure Ti-4-6.
Fully-constructed polygons can contain smaller polygons; this polygon contains two smaller triangles, which are not selected. Only "outermost" polygons --- called tree polygons --- can be broken up by the addition of a stub. Now that the polygon is selected, choose Edit->Stub->Pick For Poly.... This brings up a dialog, shown in Figure Ti-4-7.
Figure Ti-4-7.
This dialog lists information about the possible stub(s) that can be added: the edge that would be split; the lengths of the two parts of the edge; the nodes to which active paths will be formed; and the length of the new edge. In this case, there is only one possible stub, but in general, there can be multiple possibilities. Click OK.
The edge is split, and a new node and edge added, which forms 4 active paths with surrounding nodes. A stub will always form at least 4 active paths, but it is sometimes possible that one forms 5 or more. This stub has now broken the quadrilateral up into four triangles, each of which can be filled with the rabbit ear molecule.
Figure Ti-4-8.
The resulting crease pattern looks like that of a distorted Bird Base, as shown in Figure Ti-4-9.
Figure Ti-4-9.
And the folded form looks something like a distorted Bird Base as well.
Figure Ti-4-10.
There are several commands accessible from the Edit->Stub submenu:
Because of this property of triangulated trees, stub creation can be a useful technique for making a TreeMaker design more easily foldable without need to measure. A different set of techniques for making patterns more easily foldable will be described in the next section.
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