Pi Playground

Explore 2,000 years of formulas for the world's most famous constant — from Archimedes' polygons to the Chudnovsky algorithm.

In 2024, a single framework called the Conservative Matrix Field was shown to unify virtually all of them.

Archimedes Polygon Method

Archimedes · ~250 BC
π≈n⋅sin⁡ ⁣(πn)\pi \approx n \cdot \sin\!\left(\frac{\pi}{n}\right)

Approximates pi by computing the perimeter of a regular polygon inscribed in a unit circle. Starting with a hexagon (6 sides), the number of sides is doubled each step, and the perimeter approaches pi.

linear

Madhava-Leibniz Series

Madhava of Sangamagrama · ~1400
π=12∑k=0∞(−1)k3k(2k+1)\pi = \sqrt{12}\sum_{k=0}^{\infty}\frac{(-1)^k}{3^k(2k+1)}

The earliest known series for pi, discovered by Madhava of Sangamagrama in Kerala, India. It multiplies √12 by an alternating series involving powers of 1/3, converging faster than the basic Leibniz series.

linear

Wallis Product

John Wallis · 1656
π2=∏k=1∞4k24k2−1\frac{\pi}{2}=\prod_{k=1}^{\infty}\frac{4k^2}{4k^2-1}

An infinite product for pi discovered by John Wallis. Each factor is a ratio of consecutive even and odd squares, and the product slowly converges to π/2.

linear

Leibniz Series

Gottfried Wilhelm Leibniz · 1676
π4=∑k=0∞(−1)k2k+1\frac{\pi}{4}=\sum_{k=0}^{\infty}\frac{(-1)^k}{2k+1}

The simplest series for pi: an alternating sum of odd reciprocals. It converges very slowly — thousands of terms are needed for even a few digits of accuracy.

linear

Basel Problem

Leonhard Euler · 1735
π26=∑k=1∞1k2\frac{\pi^2}{6}=\sum_{k=1}^{\infty}\frac{1}{k^2}

Euler solved the famous Basel problem, proving that the sum of reciprocal squares equals π²/6. Taking the square root recovers pi, though convergence is slow.

linear

Gauss-Legendre Algorithm

Carl Friedrich Gauss · 1813
π=4an+121−∑j=1n2j+1cj2\pi = \frac{4a_{n+1}^2}{1-\sum_{j=1}^{n}2^{j+1}c_j^2}

The Gauss-Legendre algorithm uses the arithmetic-geometric mean to compute pi. It doubles the number of correct digits with each iteration, reaching full float64 precision in about 25 steps.

linear

Ramanujan's Series

Srinivasa Ramanujan · 1914
1π=229801∑k=0∞(4k)!(1103+26390k)(k!)4⋅3964k\frac{1}{\pi}=\frac{2\sqrt{2}}{9801}\sum_{k=0}^{\infty}\frac{(4k)!(1103+26390k)}{(k!)^4\cdot 396^{4k}}

Ramanujan's extraordinary series for 1/π converges at about 8 digits per term. It was used in 1985 to compute over 17 million digits of pi.

exponential

Chudnovsky Algorithm

Chudnovsky Brothers · 1989
1π=12∑k=0∞(−1)k(6k)!(13591409+545140134k)(3k)!(k!)3⋅6403203k+3/2\frac{1}{\pi}=12\sum_{k=0}^{\infty}\frac{(-1)^k(6k)!(13591409+545140134k)}{(3k)!(k!)^3\cdot 640320^{3k+3/2}}

The fastest known series for pi using standard arithmetic. Each term produces about 14 new correct decimal digits. It has been used to compute trillions of digits.

exponential

Bailey-Borwein-Plouffe

Bailey, Borwein & Plouffe · 1995
π=∑k=0∞116k(48k+1−28k+4−18k+5−18k+6)\pi=\sum_{k=0}^{\infty}\frac{1}{16^k}\left(\frac{4}{8k+1}-\frac{2}{8k+4}-\frac{1}{8k+5}-\frac{1}{8k+6}\right)

The BBP formula can extract any hexadecimal digit of pi without computing preceding digits. It converges about 1.2 decimal digits per term.

exponential