Fundamental mathematical constants appear in nearly every field of science, from physics to biology. Formulas that connect different constants often bring great insight by hinting at connections between previously disparate fields. Discoveries of such relations, however, have remained scarce events, relying on sporadic strokes of creativity by human mathematicians. Recent developments of algorithms for automated conjecture generation have accelerated the discovery of formulas for specific constants. Yet, the discovery of connections between constants has not been addressed. In this paper, we present the first library dedicated to mathematical constants and their interrelations. This library can serve as a central repository of knowledge for scientists from different areas, and as a collaborative platform for development of new algorithms. The library is based on a new representation that we propose for organizing the formulas of mathematical constants: a hypergraph, with each node representing a constant and each edge representing a formula. Using this representation, we propose and demonstrate a systematic approach for automatically enriching this library using PSLQ, an integer relation algorithm based on QR decomposition and lattice construction. During its development and testing, our strategy led to the discovery of 75 previously unknown connections between constants, including a new formula for the `first continued fraction' constant $C_1$, novel formulas for natural logarithms, and new formulas connecting $\pi$ and $e$. The latter formulas generalize a century-old relation between $\pi$ and $e$ by Ramanujan, which until now was considered a singular formula and is now found to be part of a broader mathematical structure. The code supporting this library is a public, open-source API that can serve researchers in experimental mathematics and other fields of science.

Ongoing efforts that span over decades show a rise of AI methods for accelerating scientific discovery, yet accelerating discovery in mathematics remains a persistent challenge for AI. Specifically, AI methods were not effective in creation of formulas for mathematical constants because each such formula must be correct for infinite digits of precision, with 'near-true' formulas providing no insight toward the correct ones. Consequently, formula discovery lacks a clear distance metric needed to guide automated discovery in this realm. In this work, we propose a systematic methodology for categorization, characterization, and pattern identification of such formulas. The key to our methodology is introducing metrics based on the convergence dynamics of the formulas, rather than on the numerical value of the formula. These metrics enable the first automated clustering of mathematical 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 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's 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 formulas fulfilling specified mathematical properties, accelerating the rate of discovery of useful formulas.