Origeometry

February 13, 2007 6:45 AM

What if Euclid had been Japanese? There are traditionally stated and proved theorems about origami. And MetaFilter has previously explored modular origami (as well as the boring old artistic kind), which has a geometric foundation. However, origami itself is a powerful mathematical framework that allows one to, for instance, solve the famously insoluable problem of trisecting an angle. More generally: Traditional geometry solves quadratic equations, origami solves cubic ones. (Many more mathematical items about and using origami can be found in the excellent mathematics teachers' book: Project Origami: Activities for Exploring Mathematics, most of which are unfortunately not findable online).
posted by DU (9 comments total)

The item I most desperately wanted to include was a recasting of Euclid's Elements into the origami axioms, but such a thing does not seem to exist. I actually did Proposition I myself at one point, but of course I can't link that here. I think such a project would be awesome beyond belief.

Also, since I discovered the extra calculational power that low-tech origami provides, it has been a dream of mine to write a story where ancient Chinese/Japanese mathematicians and engineers figure out how to, say, launch a rocket to the moon using not the slide-rules of the 60s but simple squares of paper to do all the calculations. Figuring out what cubic equation to use, let alone actually solving it using origami and writing the story, is beyond my powers however.
posted by DU at 6:52 AM on February 13, 2007 [2 favorites]

Also also: 'insoluble" and much more about origami cubics (and other gems) in a PDF I just found by Robert Lang.
posted by DU at 6:58 AM on February 13, 2007

"Beware! My new Crane style is Turing-complete!"
posted by kid ichorous at 9:35 AM on February 13, 2007 [3 favorites]

What, no link to Robert Lang's own site?

(See also current New Yorker. )
posted by IndigoJones at 11:37 AM on February 13, 2007

"Beware! My new Crane style is Turing-complete!"


HA!
posted by The Power Nap at 1:18 PM on February 13, 2007

awesome post
posted by rks404 at 8:11 PM on February 13, 2007

Too sleepy to look, is this just glide-reflections?
posted by fleacircus at 1:59 AM on February 14, 2007

It's not origeometry - it's just oregano! ;^) Not glide-reflections but neusis construction.

The axioms of Euclidean geometry are describing the limitations of doing constructions with a straight edge and compass. There were always solutions to trisecting an angle, etc. but the millenial challenge was doing it within Euclidean geometry. This origami stuff, where you're sliding the edges of the paper around until points on the edges line up, is just a neusis construction like the two-and-a-half-thousand-year-old Archimedes Trisection. It's good for cross-disciplinary teaching but it doesn't solve insoluble problems or get you any closer to launching a rocket than Archimedes or Hippias was.
posted by XMLicious at 5:13 PM on February 14, 2007

The part about launching rockets was just for an alternative history-type story. For the purposes of fiction, I think the word "insoluble" (using the methods at the time) is acceptable even if not literally true.
posted by DU at 5:53 AM on February 15, 2007