All and only primitive Pythagorean triples are generated by three trees of Firstov, among which are the UAD tree of Berggren et al. and the Fibonacci boxes FB tree of Price and Firstov.

Alternative proofs are offered here for the conditions on primitive Pythagorean triple preserving matrices and that there are only three trees with a fixed set of matrices and single root.

Some coordinate and area results are obtained for the UAD tree. Further trees with varying children are possible, such as filtering the Calkin-Wilf tree of rationals.

Read draft 8,

- PostScript
`triples.ps.gz`

(about 458k) - PDF
`triples.pdf`

(about 588k, 43 pages) - LaTeX source
`triples-8.tar.gz`

(about 198k, and sig)

Also by plain HTTP or by FTP or by RSYNC if desired. The sig file is a Gnu PG ascii armoured signature generated from my key.

"Draft" means believe correct as far as it goes, but might be rearranged, expanded, contracted, etc at some time. Sections and equations will renumber. The LaTeX source uses PGF for pictures.

Document copyright 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde. Permission is granted for anyone to make a copy for the purpose of reading it. Permission is granted for anyone to make a full complete verbatim copy, nothing added, nothing removed, nothing overlaid, for any purpose. The PDF rendition contains fonts which are Copyright American Mathematical Society and licensed under the Open Font License.

See Math::PlanePath::PythagoreanTree for Perl code implementing the trees. The corresponding PythagoreanTree section of the PlanePath image gallery has some pictures.

See Trees of Rationals for a similar treatment of positive rationals.

This page Copyright 2013, 2014, 2015, 2016, 2017, 2019, 2020, 2022 Kevin Ryde.