## Origami and math

I’ve given a number of talks/run a number of events recently about origami and mathematics.  This is not directly related to my research at all, but it’s a pleasant hobby that produces some pretty results, both in terms of mathematics and origami.

The connection I like the most is related to origami as an axiomatic system.  We follow the Greek notion that we should ask the question, “Using certain simple geometrical tools, what sorts of constructions can we make?” but replace the prefered tools of compass and straightedge by origami.  In other words, imagine not that the infinite plane of Euclid is a piece of paper that we draw on but instead that it is a piece of paper that we fold.  In this setting, certain constructions are very easy: for example, given two points, we can construct the (unique) perpendicular bisector of the segment joining them simply by folding one point onto the other.  Similarly, it’s easy to construct the (unique) fold that passes through two given points. There are several other such axioms; the most involved (called “O6”) asserts that given two points and two lines, we can construct a fold that carries the first point onto the first line and simultaneously carries the second point onto the second fold.  (Try it yourself!)

Famously, the Greeks weren’t able to give a complete answer to their version of our question: in particular, they were neither able to come up with a construction to trisect an arbitrary angle (that is, to construct a line which divides the angle into two parts, one of which has twice the measure of the other) nor to prove that such a construction is impossible.  It wasn’t until the tools of Galois Theory were developed roughly 2000 years later that the question was settled.  In particular, Wantzel proved in 1837 that it’s not possible to trisect an arbitrary angle using compass and straightedge (a fact that has not prevented numerous mathematical cranks from subjecting math professors to bizarre missives on this and related subjects).  However, it turns out that it’s possible (and actually quite simple) to trisect an arbitrary angle using origami.

The short explanation of why this is possible is the fold given by axiom O6 arises as the solution of a cubic equation, so is more powerful (or at least differently powerful) than the compass and straightedge, which can only solve quadratic equations.  (This also explains a subtle shift in my language above –there may be more than one fold carrying two given points onto two given lines.)  Note that there are still plenty of limits on what you can do with origami — squaring the circle is still (provably) out of reach, for example.  Tom Hull’s generally excellent website has information on this and more.

My talks typically consist of me going on for twenty minutes or so, followed by a mass lesson on the construction of (one variant of) the Sonobe unit, a simple origami unit that can be used to create beautiful polyhedra with some interesting mathematical properties.  I hope to have slides from one version of my talk up soon; until then, here are materials from the event I ran as part of the Cambridge Science Festival (all PDF): a handout on the Sonobe unit, a slideshow about origami and math, and a handout on origami with business cards.  Both handouts have links to other interesting websites; your favorite search engine can probably bring you much more.  Some of the origami I’ve made using the Sonobe unit is available here.

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