Abstract: In recent years, we are witnessing a rise of AI and machine learning methods for scientific discovery and hypothesis creation. Despite the strides in other fields of science, a persistent challenge lies in the creation of formulas for mathematical constants.In the landscape of formula creation, there is no straightforward ‘’distance metric'' between two samples that can guide progress. Formulas are either true or false, with no continuous adjustments that can enhance their correctness.The absence of a systematic method left the realm of formula discovery elusive for automated methods. In this work, we propose a systematic methodology for categorization, characterization, and pattern identification of such formulas. We demonstrate this methodology on Polynomial Continued Fraction formulas, which are ubiquitous in their intrinsic connections to mathematical constants, and generalize many mathematical functions and structures.We discover organizing metrics for the space of polynomial continued fractions. We test our methodology on a set of 1,768,900 such formulas, identifying many known formulas for mathematical constants, and discover previously unknown formulas for $\pi$, $\ln(2)$, Gauss, and Lemniscate constants. The uncovered patterns enable a direct generalization of individual formulas to infinite families, unveiling rich mathematical structures. This success paves the way towards a generative model that creates continued fractions fulfilling requested mathematical properties, potentially accelerating by orders of magnitude the rate of discovery of useful formulas.