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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