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Comment by not2b | Hacker News Reader

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.

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

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.

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.

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).

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

345 and 435 would require two binary trees.

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.

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.

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.

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