Finite Interpretations of a Hyper-Catalan Series Solution to Polynomial Equations and Visualizations

The solution to the general univariate polynomial equation has been sought for centuries. It is well known there is no general solution in radicals for degrees five and above. The hyper-Catalan numbers $C[m_2,m_3,m_4,\dots ]$ count the ways to subdivide a planar polygon into exactly $m_2$ triangles, $m_3$ quadrilaterals, $m_4$ pentagons, etc. Wildberger and Rubine (2025) show the generating series $\mathbf{S}$ of the hyper-Catalan numbers is a formal series zero of the general geometric polynomial (meaning, general except for a constant of $1$ and a linear coefficient of $-1$). Using a variant of the series solution to the geometric polynomial that has the number of vertices, edges, and faces explicitly shown, We prove their infinite series result may be viewed as a finite identity at each level, where a level is a truncation of $\mathbf{S}$ to a given maximum number of vertices, edges, or faces (bounded by degree). We illustrate this result, as well as the general correspondence between operations on sets of subdivided polygons and the algebra of polynomials, with figures and animations generated using Python.