  The plotting of Riemann surfaces by computational software is discussed. The
link between the branches of a multi-valued function $g(z)$, defined on the
range of $g(z)$, and a Riemann surface, defined on the domain of $g(z)$, is
emphasized. The connection between the two is clarified by defining the
\textit{charisma} of the argument $z$ to the function. This approach is used to
plot several surfaces using Maple.
