The Fano plane is an arrangement of 7 points on 7 lines. An automorphism of the Fano plane corresponds to permuting the points, while conserving the relation between them. These automorphisms correspond to triangles in a tiling of the Klein quartic. Moving through all the automorphisms is achieved by following a Hamiltonian path through a Cayley graph of the automorphism group, which can be represented on the Klein quartic. The results are an artistic illustration of an automorphism, and an animated walk through an automorphism group.