The modular group is the quotient of the group of integer matrices

with

by the subgroup

, where

** **denotes the

identity matrix. This group acts on the upper half-plane

by

. It turns out that the modular group is generated by just two elements: inversion, given by the matrix

, and translation, given by

. The fundamental domain is the subset of the upper half-plane

formed by all points

such that

and

. The action of the modular group tiles the upper half-plane exactly with images of the fundamental domain. This tiling requires the use of the entire infinite fundamental domain, so it cannot be demonstrated by

*Mathematica*. However, this tiling can be approximated by considering the images of a finite part of the fundamental domain under a finite number of compositions

, where each

and

is either a translation by an integer or a reflection.