The Story of Pi

From ancient geometry to modern mathematics — 4,000 years of chasing infinity

~250 BC

Archimedes Polygon Method

Archimedes · Greece
π≈n⋅sin⁡ ⁣(πn)\pi \approx n \cdot \sin\!\left(\frac{\pi}{n}\right)
linear
~1400

Madhava-Leibniz Series

Madhava of Sangamagrama · India
π=12∑k=0∞(−1)k3k(2k+1)\pi = \sqrt{12}\sum_{k=0}^{\infty}\frac{(-1)^k}{3^k(2k+1)}
linear
1656

Wallis Product

John Wallis · England
π2=∏k=1∞4k24k2−1\frac{\pi}{2}=\prod_{k=1}^{\infty}\frac{4k^2}{4k^2-1}
linear
1676

Leibniz Series

Gottfried Wilhelm Leibniz · Germany
π4=∑k=0∞(−1)k2k+1\frac{\pi}{4}=\sum_{k=0}^{\infty}\frac{(-1)^k}{2k+1}
linear
1735

Basel Problem

Leonhard Euler · Switzerland
π26=∑k=1∞1k2\frac{\pi^2}{6}=\sum_{k=1}^{\infty}\frac{1}{k^2}
linear
1813

Gauss-Legendre Algorithm

Carl Friedrich Gauss · Germany
π=4an+121−∑j=1n2j+1cj2\pi = \frac{4a_{n+1}^2}{1-\sum_{j=1}^{n}2^{j+1}c_j^2}
linear
1914

Ramanujan's Series

Srinivasa Ramanujan · India
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}}
exponential
1989

Chudnovsky Algorithm

Chudnovsky Brothers · Ukraine/USA
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}}
exponential
1995

Bailey-Borwein-Plouffe

Bailey, Borwein & Plouffe · Canada
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)
exponential
For centuries, each formula appeared as its own island of mathematical insight. Then, in 2024, researchers revealed they were all connected through a single unifying framework.

The Conservative Matrix Field

For 2,000 years, mathematicians discovered pi formulas that seemed unrelated. In 2024, researchers found a unifying framework — the Conservative Matrix Field — that connects virtually all of them.

51%

Series

Summation formulas that add up infinitely many terms. This is the largest family, including Leibniz's alternating series, Ramanujan's rapidly converging series, and the Chudnovsky algorithm. The CMF framework shows they all arise from the same matrix recurrence with different parameter choices.

43%

Continued Fractions

Nested fraction expansions that converge to pi. The Gauss-Legendre algorithm and classical continued fraction representations belong to this family. Under the CMF lens, each continued fraction corresponds to a specific matrix factorization path.

6%

Products

Infinite products like the Wallis product, where pi emerges from multiplying an endless chain of rational factors. Though the smallest category, these products are unified with the other families through the same matrix field structure.

Conservative Matrix Field

The 2024 CMF paper showed that nearly all known pi formulas can be unified through a single matrix framework. Formulas cluster into three families.

Series51%Continued Fractions43%Products6%CMF

Key Insight

The CMF framework suggests there may be an infinite number of pi formulas waiting to be discovered. Every valid parameter set in the matrix field yields a new formula for pi — and infinitely many parameter sets exist.

Fun Facts

Fastest convergence Bailey-Borwein-Plouffe ~14 digits per term
Slowest convergence Wallis Product Needs thousands of terms for a few digits
Oldest formula Archimedes Polygon Method ~250 BC · Greece
Total span 2245 years From ~250 BC to 1995