I wonder how can one descibe the generators of the triangle group for the tesselation of Poincare unit disk by triangles with angles $\pi/4, \pi/4 , \pi/4 $ in terms of the action of the modular group on a fundamental triangles.In other words, I mean how can one write the generators of this tesselation in terms of Mobius transformations. I'm completely unfamiliar with the theory of hyperbolic tesselations, and there may be many inaccuracies in my understanding and even with the specific terminology i use.
Side remark:
My real purpose is just to verify an historical hypothesis i have on Gauss's tesselation of the unit disk as described in John Stilwell "Mathematics and it's history". Looking at the relevant pages in Gauss's Nachlass (volume 8, p.102-105), I read that the commentor (Robert Fricke) on this fragment of Gauss says that Gauss's drawing (the (4 4 4) tesselation) is intended to be a geometrical illustration for composition of substitutions other then the fundamental generators of the modular group. The substitutions Gauss used are:
$$\frac{128\theta + 37i}{-45i\theta + 13}$$
$$ \frac{121\theta+36i}{-84i\theta+25}$$
Checking the determinants of these substitutions gave $-1$ for the first one and $+1$ for the second one, so this made me suspect that these are Mobius transformations (the only thing that doesn't settle is that $a,b,c,d$ in the Mobius transformation should be real integers, not imaginary integers). But this conclusion is a result of a very shallow reading of Fricke's comments and i lack the proffesional knowledge needed to verify my reading. In addition, there are two drawings in these pages (one on p.103 and the Gauss's tesselation on p.104), and i'm not sure to which drawing Fricke refers.