Abstract The set Γ def = {(z + w, zw) : |z| ≤ 1, |w| ≤ 1} ⊂ C has intriguing complex-geometric properties; it has a 3-parameter group of automorphisms, its distinguished boundary is a ruled surface homeomorphic to the Möbius band and it has a special subvariety which is the only complex geodesic that is invariant under all automorphisms. We exploit this geometry to develop an explicit and detai...