  In 1981 W.L. Edge discovered and studied a pencil $\mathcal{C}$ of highly
symmetric genus $6$ projective curves with remarkable properties. Edge's work
was based on an 1895 paper of A. Wiman. Both papers were written in the
satisfying style of 19th century algebraic geometry. In this paper and its
sequel [FL], we consider $\mathcal{C}$ from a more modern, conceptual
perspective, whereby explicit equations are reincarnated as geometric objects.
