All and only primitive Pythagorean triples are generated by the UAD tree of Berggren et al., the Fibonacci boxes FB tree of Price, and a further UMT tree presented here.
Necessary and sufficient conditions are established for primitive Pythagorean triple preserving matrices and it is shown that UAD, FB and UMT are the only trees with a fixed set of matrices and a single root. Some coordinate and area results are obtained for the UAD tree.
Further trees with varying children are possible, such as filtering the Stern-Brocot or Calkin-Wilf trees of rationals.
Read draft 4,
"Draft" means complete and believed correct as far as it goes, but likely to be rearranged, expanded, contracted, etc.
The LaTeX source uses PGF for pictures. It generates "postscript specials" so if you process to DVI you'll need a DVI viewer supporting those.
Document copyright 2013 Kevin Ryde. Permission is granted for anyone to make a copy to read. The PostScript and PDF renditions contain 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.
This page Copyright 2013, 2014 Kevin Ryde.
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