I'm a high school math teacher who came to the profession from an economics background who is slowly working through a math major and beyond (complex analysis is next...)
I'm probably not allowed to post links yet, but the 4th image on the wikipedia page for Pythagorean Triples blows my mind.
http://en.wikipedia.org/wiki/Pythagorea ... sRev08.svg
I "discovered" the Pythagoras and Plato primitive generators on my own (they appear to be the upper and lower bounds for triples,) but am at a loss as to finding a pattern for the rest of them. There seem to be many paths they take, but I wondered if those are connected.
I've tried reading the wiki on it, and it's beyond me
Any insights?
Thanks.
Pythagorean Triples
Moderators: gmalivuk, Moderators General, Prelates
Re: Pythagorean Triples
The graph seems to revolve around Euclid's formula for primitive pythagorean triples: a triple a^2 + b^2 = c^2 is displayed centered at the coordinate (a,b). For any valid m and n you plug in, it spits out a unique pythagorean triples, and all of them can be specified like that. Proving that all triangles generated by the formula are pythagorean should be easy: just check that a^2 + b^2 is equal to c^2 if you substitute a, b, and c with the ones used in the formula.
The proof of necessity(that all primitive pythagorean triples are given by the formula) given by Wikipedia doesn't seem very hard for me, and I'm not more educated than you. It's admittedly a mouthful, but it doesn't use any complicated concepts or weird tricks. Is there something in particular you don't understand?
The curves seem to be an attempt to put the pythagorean triples on a grid according to their generators m and n rather than according to their sides: if you graphed the triangles ordered by m and n, instead of their sides, they would be diagonal lines.
The proof of necessity(that all primitive pythagorean triples are given by the formula) given by Wikipedia doesn't seem very hard for me, and I'm not more educated than you. It's admittedly a mouthful, but it doesn't use any complicated concepts or weird tricks. Is there something in particular you don't understand?
The curves seem to be an attempt to put the pythagorean triples on a grid according to their generators m and n rather than according to their sides: if you graphed the triangles ordered by m and n, instead of their sides, they would be diagonal lines.

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 Joined: Wed Jan 30, 2013 6:02 pm UTC
Re: Pythagorean Triples
Upon a rereading, and your encouragement, I totally get it now. I must have been tired when I first went looking. Your explanation of the curves connecting similar m and n values is what was eluding me. Thanks. Pretty neat.
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