S

Summation Formula

An interactive living book and software repository dedicated to reconstructing Archimedes' circle measurement ("Circvli Dimensio") using original geometric proportions. Leveraging SageMath for symbolic validation and Python/Perl for high-precision numerical computation via decimal/big-number libraries, this project bypasses modern algebraic transformations to prove the flawless accuracy of antike geometric frameworks.

It now expands this foundation into a unified algebraic framework capable of handling higher-order transcendental structures up to \pi^4 and 1/\pi^4, including additive transcendental invariants like \log(2). By introducing a specialized mathematical Gain Function, the engine dynamically bridges ancient geometry with hyper-convergent modern infinite series—such as those of Ramanujan, the Chudnovsky brothers, and the 2024 Saha-Sinha string theory iterations. Through precise lambda (\lambda) parameter tracking to bypass critical systemic singularities and prevent degeneration into the slower Madhava series, the optimized, dual-action polygon generators achieve absolute numeric stability, delivering hundreds of flawless decimal places without relying on circular trigonometric functions.

And much more!