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By DesiredUsername (Mon Jul 31, 2006 at 03:31:08 AM EST) (all tags)
HOWTO: Construct an equilateral triangle.


I've known for a long time that origami operations can prove all geometric theorems that compass + straight-edge operations can prove and a lot of things that the C+SE can't prove. What I haven't known is how to do that.

So I finally got a list of orgamitric axioms and tried to prove the first Proposition in Euclid's Elements (a copy of which I got for Bmas).

In the process, I found that the "complete" list of origametry axioms still needs to be supplemented by Euclid's definitions and "common notions" and possibly even some of his axioms. For instance, Euclid takes as a given that all right angles are equal--if the origami axioms don't do that, how are you going to prove it? Or maybe a sufficiently clever proof wouldn't need that? My guess is that the word "axiom" is being used rather loosely to just apply to the "what operations are possible", not "what things do we take as true".

Anyway, if you have a (natively?) SVG-enabled browser (i.e. FireAnt 1.5 or later), check it out here.

Honestly, I had a hard time not cleaning this up more. I think I will eventually be compelled to go back and prove the steps I left unproved, figure out exactly what "common notions" I need, etc and demote construction of an equilateral triangle to some later Proposition. Alternatively, maybe once I am more familiar with how the origametric axioms work, I can find a more direct construction. Also, I think it might be clearer to show the construction in more traditionally origami form, rather than the traditionally geometric form.

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props by tps12 (2.00 / 0) #1 Mon Jul 31, 2006 at 03:52:49 AM EST
IAWTSTCIMTOFRTTTGF.

Clearer and look cooler, even by DesiredUsername (2.00 / 0) #2 Mon Jul 31, 2006 at 04:02:51 AM EST
But do you know how long it takes to generate all that SVG by hand? (SVG + Javascript was really the whole point of this exercise.) I guess I'll have to recreate origami fold language to do it for me unless that guy a) makes it work on Lunix b) outputs SVG and c) makes it available for download.

(Oh wait, there's also origami shape language, but there's still no download and even less image generation.)

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Universal propositions by Dr H0ffm4n (2.00 / 0) #3 Tue Aug 15, 2006 at 11:12:38 PM EST
How would you even go about proving a universal proposition such as 'all right angles are equal' using real folding paper? Surely all you can achieve is demonstration of individual cases?

I don't know by DesiredUsername (2.00 / 0) #4 Wed Aug 16, 2006 at 02:06:00 AM EST
And yet the list doesn't include it in the axioms, while Euclid does.

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I'd guess it's not needed by Dr H0ffm4n (2.00 / 0) #5 Wed Aug 16, 2006 at 06:44:59 AM EST
When used for solutions to 2nd and 3rd order equations.

Or

The abstraction of the origametrical axioms is more powerful than paper folding alone.

Or

Angles aren't even mentioned in the axioms, except if you count the concept of perpendicular. It could be that the notion of congruent right angles is implicit in that concept.

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