This 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.

There 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.

• tromp

  11/22/2024

  But 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.

• FredrikMeyer

  11/22/2024

  Learned something new today. For other interested: https://en.wikipedia.org/wiki/Pythagorean_triple#Generating_...

Beautiful. Thanks for sharing.

Very nice! nit: website isn’t mobile friendly

Wow, this is extremely cool! Only problem, the JS slows my Firefox almost to freezing, is it normal?

• hakmem

  11/21/2024

  Yes, 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

    11/21/2024

    A simple trick to solve nearly all freezing problems: move the computations to a background thread, aka Worker in JS terms.

    • aabhay

      11/22/2024

      Another trick — use comlink (https://github.com/GoogleChromeLabs/comlink) to make the worker thread into essentially an async API of your local methods.

• 11/21/2024

You changed the article's title to an incorrect one. The tree of primitive Pythagorean triples is ternary, not binary. Each node has three children.

• feoren

  11/21/2024

  Keep reading. The Barning-Hall tree is ternary, but this article is mostly devoted to the Stern-Brocot tree, which is binary.

• generationP

  11/21/2024

  Both 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

    11/21/2024

    But 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

      11/21/2024

      The 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

        11/21/2024

        Yes, but the binary projection does not according to the link.

        345 and 435 would require two binary trees.

        • AnotherGoodName

          11/21/2024

          I 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

          11/21/2024

          You 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.

• ColinWright

  11/21/2024

  From 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.