## Friday, June 29, 2012

### Origami Geometry

I've been looking into origami-based geometry recently, on and off, and it's quite a fascinating subject. Geometry is based on constructions (it is the use of constructions that led C. S. Peirce to note that geometrical arguments can draw conclusions that strictly speaking go beyond the premises by, so to speak, abstract experiments with what may be but need not be). Our geometry developed by using three basic kinds of construction devices:

straightedge (or ruler)
compass
neusis (i.e., markings on a straightedge)

Neusis was known to the Greeks but usually depreciated by them; it was often regarded by them as insufficiently noble and abstract. And, indeed, the rejection of neusis shows that they were more interested in the abstract than applications, because neusis allows you to prove things that mere straightedge and compass don't. Regardless, Euclidean geometry is based wholly on straightedge and compass, and, indeed, can be regarded as a general theory of straightedge-and-compass construction: Euclid's geometry starts with a set of basic constructions with straightedge and compass that you are allowed, and then builds everything from there.

The reason for these particular instruments of construction is due to the way the ancient Greeks did geometry -- stick in the sand, chalk on slate, or what have you. It would be entirely possible, however, to do geometry with nothing but folded paper. Hence origami geometry, which has been studied at some length in the past two or three decades. It turns out that origami geometry allows for some very powerful mathematics. There are seven basic constructions possible using folds, which become the axioms of origami geometry; these axioms, having been first discovered by Jacques Justin, are generally known as Huzita-Hatori axioms:

(1) Given two points, there is a unique fold passing through both.
(2) Given points p1 and p2, there is a unique fold that leaves p1 on top of p2.
(3) Given lines l1 and l2, there is a fold that leaves l1 on top of l2.
(4) Given a point p1 and a line l1 there is a fold perpendicular to l1 that passes through p1.

You can do a fair amount with these four alone, but the geometry is weaker than Euclidean geometry. To beef it up a bit, you can add:

(5) Given points p1 and p2, there is a fold passing through p2 that leaves p1 on top of l1.

With these five constructions you can do everything that you can do with straightedge and compass. We can take it another step, though:

(6) Given points p1 and p2 and lines l1 and l2, there is a fold that leaves p1 on l1 and p2 on l2.

This is a neusis axiom, and when this construction is allowed, you can prove things that can't be proven by Euclid. The seventh axiom doesn't actually add anything new; but it's often thrown in because it is still one of the seven possible basic constructions:

(7) Given a point p1 and lines l1 and l2, there is a fold perpendicular to l2 that leaves p1 on l1.

Algebraically speaking, you can think of straightedge-only constructions as powerful enough to prove things describable as linear equations. Adding the compass lets you do quadratic equations generally (in effect, the compass lets you find the geometrical equivalent of square roots in a consistent way). Neusis allows you to do cubic equations (in effect, the compass lets you find the geometrical equivalent of cube roots in a consistent way). Quartic equations are based on square roots of square roots, but quintic equations and so forth are beyond any of these construction methods to achieve. However, apparently, by adding to the Huzita-Hatori axioms so that more complicated constructions are allowed, you can start solving quintic equations and even higher, thus stretching geometry to cover a little more. (There's bound to be some stick-in-sand equivalent of these complicated multifolds, but I haven't ever come across anyone who identified what it would be, and I don't have the mathematical ability myself to guess at what would be required beyond straightedge, compass, and neusis to quintisect an angle.)

There is a world of mathematics in a little folded paper.