Twist folds explained 
[Mar. 14th, 200810:01 am]
Callicrates

This is a very, very rough explanation of what's going on in the image I posted in my last entry.
Each of the images in the series is an example of what's called a twist fold. This is a concept in origami that goes back about 30 years to when Shuzo Fujimoto, a Japanese professor who published his investigations back in the mid70s. A twist fold is a polygon (usually equilateral and equiangular) with pleats radiating out from each of its faces. When the pleats are all folded over at once, the central polygon twists in one direction or another and the entire thing folds flat.
For the sake of argument, I'm going to talk only about the regular polygons from here on out. Twist folds with irregular polygons are still under investigation.
A single pleat in a twist fold emerges from one side of the polygon. In the figure below, we're looking at side AB of the central octagon. The pleat consists of a mountain fold emerging from point A (the heavy, solid lines labeled 03) and a valley fold, parallel to the mountain fold, emerging from point B (the thin lines labeled a, b, c). The critical feature here is the angle between the pleat and side AB. In theory, it can take on any value between 0 and 90 minus a bit. In practice, there's a much more elegant set of possibilities.
We're going to define the angle with a chord that subtends the polygon. We're interested in the number of vertices that fall above that chord. Take a look at lines 0, 1, 2 and 3. Those are so labeled because of the number of vertices they isolate. Line 1, for example, cuts off vertex B from the rest of the polygon. Line 2 cuts off vertex B and one other. Line 3 cuts off B and 2 others. Line 0 is a degenerate chord: there are no vertices that fall strictly above it.
When we actually fold these, we find that the more vertices are above a pleat, the smaller the "aperture" in the center of the finished project. This is what we see happening from left to right in the image below. It turns out that every regular Ngon admits floor(N/2) of these possibilities. For the octagon here that's 3. For the 16gon I used in the progression it's 7. The reason it's only N/2 is that once the mountain fold (lines 03) becomes collinear with the diameter of the polygon, the aperture in the center closes completely and the angle cannot be increased any farther.
I know this is rough. If it's unclear, ask questions and I'll do my best to answer.


