Behold! The tesselation of the upper half-plane by the extended modular group $\bar{\Gamma} := \{z \mapsto \frac{a z + b}{c z +d} | a,b,c,d \in \mathbb{Z}, ad-bc=1 \} \cup \{z \mapsto \frac{a \bar{z} + b}{c \bar{z} +d} | a,b,c,d \in \mathbb{Z}, ad-bc=1 \}$. This picture was created by using a C++ program to output SVG code. I'll publish it as soon as it's done.