2025/04/08 by N. J. Wildberger, Dean Rubine · 2 voices · 2 citations
Mathematics · Computer Science · Physics and Astronomy · #Mathematics and Applications #Polynomial and algebraic computation #Advanced Mathematical Theories and Applications
paper · pdf · doi:10.1080/00029890.2025.2460966
openalex publication_date 2025/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/15
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.