|Jeremy Hylton : weblog : 2003-10-01|
Wednesday, October 01, 2003
Ethan Berkove gave a fun talk titled "Mathematics and Origami" today. He covered several interesting mathematical problems related to folding. He also brought some impressively detailed pieces, including an eagle and a leopard.
The first problem was folding a paper into thirds, which is equivalent to trisecting an angle or duplicating the cube. The latter two are classic problems from ancient Greece; it is impossible to solve with a straight edge and compass. It is possible via folding.
The key idea is to fold two points to two lines. This provides a method to trisect an angle. Tom Hull has some good notes on this kind of geometric construction. Berkove mentioned Hull as one of the leading researchers in the field.
The next problem was flat folding. Given a pattern of creases on a flat sheet of paper, can the paper be folded flat? There are two kinds of creases, called mountain and valley creases, depending on whether the paper runs up or down from the crease. The Kawasaki condition says that the sum of alternating angles around a single vertex is always pi in a flat fold.
Not all crease patterns will fold flat. The example he showed involved a triangle. The three vertexes involved satisfying the Kawasaki condition, but aren't compatible with each other. In general, deciding whether a set of creates will fold flat is NP-hard. Again, Hull has a good page on flat folds.
The third problem had to do with folds and cuts. He started with this simple problem: Can you fold a square piece of paper so that you can cut a smaller square out of the middle of it with a single, straight cut? Yes. Fold on the diagonals and cut the end off the resulting triangle.
Berkove mentioned on solution to the fold-and-cut problem, due to Erik Demaine, Martin Demaine, and Anna Lubiw. Any planar graph can be cut out by making some set of folds and a straight cut.
The last problem had to do with coloring. He showed several nice pieces that he had made. One was a soccer ball frame made with red, white, and blue paper. I didn't take good notes here, but he related it to the Four Color Theorem. See Hull's color notes, too.
I wish I had asked a question about the design techniques for the very intricate origami pieces he showed. He mentioned that a chief challenge was extracting corners; the eagle had six corners (feet, wings, beak, and one other). The challenge is to "pull corners out of a flat piece of paper."