Legendre’s Singular Modulus

In 1811, Legendre published an identity between integrals involving the constant k:= sin ⁡π12 that inspired Abel to create his brilliant theory of complex multiplication. Then k reappeared as eccentricity ellipse whose arclength Ramanujan computed explicitly in terms gamma functions with rational arguments. Finally, appeared a consequence three-body choreography along Bernoulli’s lemniscate. We develop these results detail well mentioning random walks on cubic lattice and renormalization period simple pendulum. All this shows wonderful unity underlying seemingly different branches mathematics physics.