Orbit of complex unit $i$ under moebius tranformation in $SL_2(\mathbb{Z})$

Question

Say I have the group $SL_2(\mathbb{Z})$ (over the Ring of integers) and let it act (via group action on the upper half complex plane $H = \{z \in \mathbb{C} | \Im(z) > 0 \}$ through moebius transformation $$ \begin{bmatrix} a &b \\c&d \end{bmatrix}.i = \frac{ai +b}{ci+d}$$
I want to know the orbit of the complex unit $i$. Is there an elegant description of these orbits other than the obvious one $$ orb(SL_2(\mathbb{Z})) = \frac{ac+bd +i}{c^2+d^2}?$$

Answer 1

In the modular diagram, which depicts the standard $SL_2(\mathbb{Z})$-invariant tiling of $H$, consider each triangular tile $T$ having two finite vertices and one ideal vertex. The midpoint of the side of $T$ connecting the two finite vertices is in the orbit of $i$ (for example, if $T$ is the shaded triangle in that diagram then you get $i$ itself). Furthermore, this set of edge midpoints is exactly the orbit of $i$.