A family of representations for the modular group

Marked boxes are configurations of points and lines in the real projective plane that comprise the initial data for Pappus’ Theorem. Using Pappus’ Theorem, Richard Schwartz defines a group of box operations G that acts on the set of marked boxes. Based on Schwartz’ article “Pappus’s Theorem and the Modular Group”, we prove in detail that G is isomorphic to the modular group M, and that there is a faithful representation of M into the group of projective \(\mathcal{G}\). Additionally, we investigate the fractal structure of Pappus Curves, which are topological circles associated with G-orbits of convex marked boxes, and numerically estimate their box dimension.