Hyperbolic geometry and real moduli of five points on the line

We show that each connected component of the moduli space of smooth real binary quintics is isomorphic to an open subset of an arithmetic quotient of the real hyperbolic plane. Moreover, our main result says that the induced metric on this moduli space extends to a complete real hyperbolic orbifold structure on the moduli space of stable real binary quintics. This turns the moduli space of stable real binary quintics into the quotient of the real hyperbolic plane by the non-arithmetic triangle group of angles $\pi/3, \pi/5$ and $\pi/10$.