  In this note we are interested in the rich geometry of the graph of a curve
$\gamma_{a,b}: [0,1] \rightarrow \mathbb{C}$ defined as \begin{equation*}
\gamma_{a,b}(t) = \exp(2\pi i a t) + \exp(2\pi i b t), \end{equation*} in which
$a,b$ are two different positive integers. It turns out that the sum of only
two exponentials gives already rise to intriguing graphs. We determine the
symmetry group and the points of self intersection of any such graph using only
elementary arguments and describe various interesting phenomena that arise in
the study of graphs of sums of more than two exponentials.
