The Catalan numbers 𝐶𝑚 count the number of subdivisions of a polygon into m triangles, and it is well known that their generating series is a solution to a particular quadratic equation. Analogously, the hyper-Catalan numbers 𝐶𝐦 count the number of subdivisions of a polygon into a given number of triangles, quadrilaterals, pentagons, etc. (its type 𝐦), and we show that their generating series solves a polynomial equation of a particular geometric form. This solution is straightforwardly extended to solve the general univariate polynomial equation. A layering of this series by numbers of faces yields a remarkable factorization that reveals the Geode, a mysterious array that appears to underlie Catalan numerics.