## Is there a topograph for Pythagorean triples?

I have reading Allen Hatcher's notes on quadratic forms. Naturally, draw a pictures encoding all the values of a quadratic forms in a topographs. These are build by iterating the parallelogram identity:

$$2 Q(\vec{v})+2Q(\vec{w}) = Q(\vec{v}+\vec{w}) + Q(\vec{v}-\vec{w})$$

These can be found in Ch 1 of The Sensual (Quadratic) Form by John H Conway.

They are many interesting related to Farey fractions, circle packings, Voronoi tesselations and Kleinian groups.

I am interested points $x,y,z \in \mathbb{Z}^3$ in the quadratic form $Q(x,y,z) = x^2 + y^2 - z^2$ vanishes. It's known such triples exhibit a ternary tree structure. One can multiply vector $(x,y,z)$ by any of

$$\left[\begin{array}{ccc}1 &-2 & 2 \\ 2& -1& 2\\ 2& -2 & 3 \end{array} \right] \text{ or } \left[\begin{array}{ccc}1 &2 & 2 \\ 2& 1& 2\\ 2& 2 & 3 \end{array} \right] \text{ or } \left[\begin{array}{ccc}-1 &2 & 2 \\ -2& 1& 2\\ -2& 2 & 3 \end{array} \right]$$ and get another Pythagorean triple. The result is an $\Gamma(2)$ action on the Pythagorean triples.

If I had to guess, the topograph would be somehow dual to the hyperbolic tessellation associated to the congruence group. The vertices of the "topograph" would be (similar to) the Farey fractions and the relation would involve 6 numbers instead of 4. I wonder what it could be.

Having a topograph for solutions of quadratic forms rather than values is not without precent. It's been done for Appolonian circle packings and Markoff triples. The topograph itself, has extension to ground fields other than $\mathbb{Q}$.

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 The Hatcher notes are very nice. I have used the topograph method to answer a few questions on MSE, when the represented targets were a bit too large for the Lagrange method (adjacent indefinite forms, essentially continued fractions) to work. I had not realized anyone else had paid attention to the idea. – Will Jagy Oct 29 at 3:23 Perhaps this question is moot? The topograph is gives a binary tree structure to the values of a quadratic form. For Pythagorean triples we exhibit a ternary tree with a $\Gamma(2)$ action. It remains to overlay the Pythagorean triples in the faces of the $\Gamma(2)$ Poincare disk tiling. – John Mangual Oct 29 at 4:20