Four Visions of π

At school, π is introduced as something almost embarrassingly simple: the ratio of a circle's circumference to its diameter. A familiar constant. A geometric fact. A decimal that begins 3.14159 and continues forever.

But then one stumbles upon formulas like the ones shown above — involving infinite sums, products, continued fractions, and mysterious constants like 1103 and 396 — and the natural response is astonishment. How did anyone arrive at these? Who looks at a circle and somehow finds factorials and nested fractions?

The answer is that they usually did not begin by chasing π directly. Each of these formulas emerged from a different mathematical landscape. One came from thinking about angles and arcs. Another from patterns in families of integrals. Another from recursive algebra. The last from the deep arithmetic of elliptic and modular functions. π appeared because it lives at the crossroads of all of them.

That is what makes the history of π so beautiful. The number stays the same, but the ways of seeing it keep changing.

Madhava: Building an Angle Out of Infinite Corrections 14th or 15th century, Kerala, India

The most intuitive of these stories begins with Madhava of Sangamagrama, the great mathematician of the Kerala school.

His route to π starts with a special angle:

So if one could express $\arctan(x)$ as an infinite series, then substituting $x = 1/\sqrt{3}$ would give $\pi/6$, and hence π.

That is exactly what happens. One begins with the simple algebraic identity

which is just a geometric series in disguise. Integrating term by term gives

Now substitute $x = 1/\sqrt{3}$, multiply by 6, and Madhava's formula appears.

Seen this way, the formula no longer looks like an arbitrary miracle. It feels almost inevitable. First comes a simple series. Then integration. Then a special angle from elementary geometry. The pieces lock together naturally.

The deeper intuition is even more beautiful. An angle is being built from an infinite sequence of corrections. Start with a crude approximation. Then correct it. Then correct the correction. Then keep going. The curve is unfolded into an endless alternating process.

What makes Madhava especially impressive is that he did not merely find a series for π. He found a good one. If one uses $x = 1$, one gets the famous later Leibniz formula $\pi/4 = 1 - 1/3 + 1/5 - \cdots$, which converges painfully slowly. Madhava's choice $x = 1/\sqrt{3}$ is far wiser, because the powers shrink much faster. This was not only theoretical insight. It was computational intelligence.

Wallis: Finding π in a Pattern of Areas 1656, England

John Wallis arrived by a very different road. He was not primarily trying to produce a dazzling π formula. He was studying patterns in areas and integrals.

In the modern formulation, we define $I_n = \int_0^{\pi/2} \sin^n x\, dx$ and then show, by integration by parts, that these integrals satisfy a recurrence: $I_n = \frac{n-1}{n}\, I_{n-2}$.

This recurrence splits naturally into even and odd cases. The even chain carries a factor of $\pi/2$, while the odd chain does not. Comparing the two and pushing the pattern to its limit yields Wallis's famous product:

But the historical intuition is perhaps even more interesting. Wallis studied families of definite integrals that could be computed for whole-number exponents. He arranged the results in tables, noticed a pattern, and then did something bold: he extended the pattern to the half-integer case. Why half-integers? Because the quarter-circle area involves $\sqrt{1 - x^2}$, and that square root corresponds to exponent $1/2$.

So Wallis was effectively asking a daring question: if a family of formulas behaves beautifully for ordinary integers, might the same pattern continue to the case that describes the circle?

That act of interpolation was the real leap. He did not yet possess our modern language of limits, beta functions, or gamma functions. What he had was pattern recognition, symbolic confidence, and a willingness to trust that the structure extended beyond the cases already computed.

Madhava reached π by adding corrections. Wallis reached it by multiplying balance factors. Each factor in the Wallis product is only a tiny adjustment, but together they build up to $\pi/2$. It is a different kind of infinity, and it reveals a different side of π.

Brouncker and the Continued Fraction: Turning Pattern into Recursion 17th century onward

The continued fraction often shown beside these formulas is

Popular images sometimes attach Gauss's name directly to it, but historically the specific formula belongs more naturally to the Brouncker–Wallis line. Brouncker transformed Wallis-type product structure into a continued fraction, while Gauss, much later, developed the broader theory that explains why such things arise so naturally.

Why should π appear in a continued fraction at all? A continued fraction is what happens when refinement becomes recursive. Instead of saying, "Here is a number, then a correction, then another correction," one says, "Here is a number, and its correction has the same nested structure as the whole expression." Each stage opens into another stage of the same form.

This particular formula is best seen as Wallis's world turned inside out. What appeared there as a product of ratios becomes here a recursive denominator structure. The odd numbers $1, 3, 5, 7, \dots$ and the squares $1^2, 2^2, 3^2, \dots$ are not ornamental. They are the visible trace of a deeper self-similar pattern.

Gauss's later contribution was to show that this phenomenon was not a one-off curiosity. In the larger world of hypergeometric functions, continued fractions arise with striking regularity. Ratios of related functions often collapse into beautiful nested forms. Since trigonometric and inverse trigonometric functions belong to that larger family, π enters naturally as one special value among many.

Ramanujan: The Hidden Tunnel 1914, India

Then comes Ramanujan.

His formula for $1/\pi$,

does not merely converge. It races. Each new term contributes an astonishing number of correct digits. The first time one sees it, it can feel almost supernatural.

And in a sense, this is because Ramanujan was no longer operating in the visible world of circles, arcs, or elementary integrals. He was operating in a far deeper landscape: elliptic integrals, modular equations, special algebraic values, and the hidden symmetries of hypergeometric series.

Modern mathematics explains this formula through the theory of elliptic and modular functions. Certain hypergeometric series are intimately related to elliptic integrals. Elliptic integrals, in turn, are tied to π. Modular transformations connect one representation to another, often turning a slow approximation into a dramatically faster one. At special algebraic points — sometimes called singular moduli — the resulting formulas become extraordinarily efficient.

That is why the constants in Ramanujan's formula are not random curiosities. Numbers like 9801, 396, 1103, and 26390 are arithmetic fingerprints of a very special modular setting. Later mathematicians, including the Borweins in the twentieth century, showed how Ramanujan's formulas fit rigorously into this larger theory.

But even after the theory is understood, some mystery remains. For Madhava and Wallis, one can imagine the chain of reasoning step by step. For Ramanujan, we can describe the mathematical territory, but it is much harder to reconstruct the inner flash by which he saw such identities. He seems to have had an uncanny intuition for where modular and hypergeometric worlds would align.

If Madhava walks to π, and Wallis balances his way there, Ramanujan seems to take a hidden tunnel.

Four Visions of the Same Constant

These four formulas are not four variations of the same trick. They are four different visions of the same constant.

That is why formulas for π can look so wildly different from each other. π is not only the number of the circle. It is also a meeting point of analysis, algebra, recursion, and arithmetic structure.

In that sense, the history of π is almost a miniature history of mathematics itself. A simple constant first seen in geometry keeps reappearing as human thought grows more abstract. The number does not change. The surrounding intuition deepens.

And perhaps that is why these formulas continue to fascinate. They are not just efficient ways to compute digits. They are windows into the many hidden worlds in which π quietly lives.