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 $\mathbb{Q}$.
