Comments by "David Terr" (@dcterr1) on "Fibonacci = Pythagoras: Help save a beautiful discovery from oblivion" video.

  1. Like all your videos, I love this one and I learned lots of useful information as well! In particular, I'm quite intrigued by the ternary tree construction of PPTs. In 2012, I published a paper in The Fibonacci Quarterly entitled "Some Interesting Infinite Families of Primitive Pythagorean Triples", in which I describe four infinite families of PPTs all involving Fibonacci and Lucas numbers, as well as the three you mention in this video, corresponding to the left, middle, and right paths through the tree. Now I suspect the other four families in my paper correspond to other simple paths through this tree, most likely involving a repeating finite sequence of left, middle, and right moves up the tree. This leads to several interesting questions. For starters, what's the correspondence between these repeating patterns and the resulting infinite families of PPTs? Furthermore, as in the cases you illustrate here, do the resulting right triangles each converge on a single one, and what are the proportions [a:b:c] of these limiting right triangles? Now I'd very much like to write another paper, or perhaps a series of papers, in which I try to answer these questions. If you'd like, you can collaborate with me! Let me know if you're interested. If so, we can exchange emails.
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  2. Here's an interesting fact, based on the ternary tree of PPTs. By replacing left, middle, and right movements up the tree by the ternary digits 0, 1, and 2, respectively, we obtain an isomorphism between the set of PPTs and the set of 3-adic rational numbers strictly between 0 and 1, i.e., rational numbers in the interval (0, 1) with denominators equal to powers of 3. Furthermore, by considering the set of infinite paths up the tree, we also obtain an isomorphism from [0, 1] to [0, 1], given by x -> lim(u/v) where x is the real number with ternary expansion described above and u and v are the parameters in Euler's characterization of PPTs. Is this isomorphism equal to the identity? If not, what is it?
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