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A binary tree of all Pythagorean triples
- sevensor 2y agoThis is wonderful! I’d somehow missed the classical enumeration of Pythagorean triples. I learned them as magic numbers. That structure alone is worth the price of admission.
- not2b 2y agoYou changed the article's title to an incorrect one. The tree of primitive Pythagorean triples is ternary, not binary. Each node has three children.
- generationP 2y agoBoth conventions are valid. You call it binary when you view it as a rooted tree, or ternary if you view it just as a graph.
- nyrikki 2y agoBut it _all_ triples? > I sketch how the stereographic projection of the Stern–Brocot tree forms an ordered binary tree of Pythagorean triples, which can be used to compute best approximations of turn angles of points on the circle and finally trigonometric functions The permutation and stack problem in the page seem to indicate this is a potential method for approximations, but insufficient for _all_ That said I am reading this on mobile and may have missed something.
- not2b 2y agoThe ternary tree contains all primitive triples (where the GCD of the terms is 1), where a<b<c. So it contains (3,4,5) but not (6,8,10) or (4,3,5).
- nyrikki 2y agoYes, but the binary projection does not according to the link. 345 and 435 would require two binary trees.
- AnotherGoodName 2y agoI think skipping transposed values is fine though. You could just mirror the output at 45degrees for that if you wanted it. It does hit all distinct triples including the multiples of triples so it’s more inclusive of everything than the ternary tree.
- hakmem 2y agoYou can see both triples are contained in one binary tree using the big diagram in section 3. The triple [3 4 5] has the "path" RR. The triple [4 3 5] the path R.
- feoren 2y agoKeep reading. The Barning-Hall tree is ternary, but this article is mostly devoted to the Stern-Brocot tree, which is binary.
- ColinWright 2y agoFrom the article: "I sketch how the stereographic projection of the Stern–Brocot tree forms an ordered binary tree of Pythagorean triples ..." ... and ... "Before that, I briefly recapitulate the classical enumeration of Pythagorean triples and the ternary Barning–Hall tree." So this article is about the binary tree representation.
- matt3210 2y agoVery nice! nit: website isn’t mobile friendly
- lollobomb 2y agoWow, this is extremely cool! Only problem, the JS slows my Firefox almost to freezing, is it normal?
- hakmem 2y agoYes, this is normal. I am sorry, I am working on a more efficient implementation. The JavaScript of this page does a lot of number crunching. It is actually doing arithmetic on the Stern-Brocot tree. It is all written in ClojureScript and not really optimized yet. I mention in the paper that I do not even use TCO. Anyway, thank you - and all the people here - for the kind words! I am so happy that my article was so well received today.
- akomtu 2y agoA simple trick to solve nearly all freezing problems: move the computations to a background thread, aka Worker in JS terms.
- aabhay 2y agoAnother trick — use comlink (https://github.com/GoogleChromeLabs/comlink https://github.com/GoogleChromeLabs/comlink) to make the worker thread into essentially an async API of your local methods.
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- dpunjabi24 2y agoBeautiful. Thanks for sharing.
- keithalewis 2y agoThere are Pythagorean triples (a, b, c) for which there do not exist integers m, n with a = m^2 - n^2, b = 2mn, c = m^2 + n^2.
- FredrikMeyer 2y agoLearned something new today. For other interested: https://en.wikipedia.org/wiki/Pythagorean_triple#Generating_a_triple https://en.wikipedia.org/wiki/Pythagorean_triple#Generating_...
- tromp 2y agoBut all such triples are non-primitive. I.e. they are of the form (ka, kb, kc) with k>1. So all Pythagorean triples can be generated as a scaled primitive triple (a,b,c) that is generated by Euclid's formula from m,n.