We study cubic polynomial maps from $\C$ to $\C$ with a critical orbit of period $p$. For each $p>0$ the space of conjugacy classes of such maps forms a smooth Riemann surface with a smooth compactification $\overline S_p$. For each $q>0$ I will describe a dynamically defined tessellation of $\overline S_p$. Each face of this tessellation corresponds to one particular behavior for periodic orbits of period $q$. (Joint work with John Milnor.)