π Three Ways
π without a circle in sight. Three formulas, three different mathematical worlds — sums, products, nested radicals — all yielding the same number. Each was discovered independently and convinced its discoverer that something deep was going on with the integers themselves.
Basel sum (1734)
Σ 1/n² = π²/6
Euler — solving Basel's problem
Wallis product (1656)
π/2 = ∏ (2n / (2n−1)) · (2n / (2n+1))
Wallis — Arithmetica Infinitorum
Vieta's formula (1593)
2/π = (√2/2) · (√(2+√2)/2) · …
Vieta — earliest infinite product for π
three windows on the same number
Basel (Euler 1734): sum of reciprocal squares. Euler did this by treating sin x / x as an infinite product of its zeros at ±π, ±2π, ±3π, … — comparing coefficients with its Taylor series gives Σ 1/n² = π²/6. Convergence is slow (1/n²) — 100 terms give about 2 digits of π.
Wallis (1656): an infinite product over even/odd ratios. The famous "double infinite product" — used as the first proof that π is transcendental-flavored. Convergence is even slower than Basel.
Vieta (1593): the oldest known infinite formula for π. Each new term takes square roots of the previous one. Converges exponentially — 20 terms give 12 digits of π. Vieta proved this by computing the area of a 2ⁿ-gon inscribed in a unit circle and taking n → ∞. Half-angle formulas turned that geometric idea into the nested-radical formula above.
Three windows on the same number from three different centuries. The fact that they all agree is itself evidence that π is more than a circle's ratio: it's a constant that turns up wherever oscillation, periodicity, or orthogonality appears in mathematics.