  We give an overview of how to construct continued fractions on the Heisenberg
group $\mathbb{H}$, the projective and planar Siegel models of the group, and
how to perform computations on the group using matrices. We discuss and work
with some recent results of Vandehey on the classification of which elements of
the Heisenberg group have eventually periodic continued fraction
expansions.Then we examine and work with major theorems in ergodic theory to
explore results concerning growth of denominators and digit frequency, relating
the results to hyperbolic geometry.
