Lemniscate Constant: Geometry, Transcendence, and Gauss

Join us as we explore the Bernoulli lemniscate and its iconic constant—the ratio of its perimeter to its diameter. We trace its transcendental nature, uncover Gauss’s arithmetic-geometric mean link, and map the web of connections to elliptic integrals, beta and gamma functions, and the Riemann zeta function. From elegant integral visuals to series analogies with Viète’s and the Wallace product for pi, plus astonishing computations pushing digits into the trillions, this episode reveals how the lemniscate constant sits at the crossroads of geometry, analysis, and number theory.

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