This article delves into the intriguing world of the number $\pi$, uncovering its definition, why its exact value can never be precisely known, how it connects to waveforms, and why it might hold the story of the entire cosmos.

Among all mathematical constants, $\pi$ stands out as one of the most essential. It represents the ratio between a circle’s circumference and its diameter. Regardless of a circle’s size or where it is in the Universe, this ratio remains constant: the circumference equals $\pi$ times the diameter. Likewise, the area of the circle equals $\pi$ times the square of its radius.

Capturing the essence of $\pi$

Determining the exact value of $\pi$ turns out to be far more complex than one might assume. That’s because $\pi$ is classified as an irrational number, meaning it cannot be expressed as a simple fraction. Its decimal representation goes on forever without settling into a repeating pattern.

From ancient Babylon to classical Greece and from early Chinese mathematicians to medieval Indian and Arabic scholars—even the Old Testament (see 1 Kings 7:23)—many attempted to understand and define $\pi$. One major challenge was the difficulty of measuring curved lines precisely. A breakthrough came in the 3rd century BCE, when Archimedes devised a method using regular polygons to trap the circle between them, approximating its circumference. His technique is depicted below:

Here, you see circles enclosed by two polygons: pentagons (left), hexagons (center), and octagons (right). Since calculating the perimeter of a polygon is straightforward (just sum up its sides), these shapes provide estimates for the circle’s perimeter. As the number of sides increases, the approximation becomes more accurate. Once you estimate the circumference, dividing it by the diameter yields an estimate for $\pi$.

Archimedes thereby obtained the following bounds for $\pi$:

Converted to decimals (and slightly rounded), these bounds narrow $\pi$ down to just a few digits beyond the decimal point.

It’s evident that by increasing the number of polygon sides, one could approach the true value of $\pi$ as closely as desired. What Archimedes was essentially using was the idea of a limit: a sequence approaching a certain value—in this case, the circle’s circumference. This concept predates calculus by about a millennium.

Later, with the development of calculus in the 17th and 18th centuries, mathematicians began uncovering new, seemingly unrelated ways to compute $\pi$. These methods used infinite series and products. One particularly elegant approach involves an alternating series of fractions: starting with 1, subtracting 1/3, adding 1/5, subtracting 1/7, and so on. As the sequence continues, the sum converges on $\pi/4$:

This identity is astonishing: a simple alternating sum of odd-numbered reciprocals converging on a number so profoundly tied to geometry. The formula is attributed to both Wilhelm von Leibniz and James Gregory, although Madhava of Sangamagrama, an Indian mathematician, may have discovered it centuries earlier.

Several other surprising expressions for $\pi$ exist (some appear at the end of this article). Despite their beauty, many of them are not efficient for calculating $\pi$ to high precision—especially without a computer. For example, computing 100 terms in the Madhava-Gregory-Leibniz series and multiplying by 4 yields a value accurate to only two decimal places, a disappointing payoff for the effort involved.

Until the mid-1900s, only around 600 digits of $\pi$ had been correctly calculated. But the arrival of modern computing changed everything. Today, numerous algorithms are employed to extend the known digits of $\pi$ further than ever before. As of July 2024, the record stands at an astonishing 202,112,290,000,000 decimal digits—over 202 trillion.

The cosmos encoded in digits

Here are the first 50 digits after the decimal point in $\pi$:

$\pi = 3.14159\ 26535\ 89793\ 23846\ 26433\ 83279\ 50288\ 41971\ 69399\ 37510.$

These digits never repeat, and they appear random, though they are not generated randomly. Throughout history, many have tried to uncover hidden patterns or meanings in $\pi$’s digits—numerologists, mystics, and mathematicians alike—but without success.

Mathematicians suspect that $\pi$ is a normal number. That means each digit from 0 to 9 appears with equal frequency (one-tenth of the time), every possible two-digit combination appears with equal frequency (one in a hundred), every three-digit sequence one in a thousand, and so on. Imagine rolling a fair 10-sided die infinitely many times; you’d expect a similar distribution. A number whose digits exhibit this statistical uniformity is called normal.

Though unproven, if $\pi$ truly is normal, it implies that every finite sequence of digits exists somewhere within its decimal expansion. Through numerical encoding, that means every book ever written, every sentence ever spoken, every human name—past, present, and future—can be found somewhere within $\pi$. Hidden among its digits might be the complete works of Shakespeare, the secrets of the Universe, or even a blueprint for humanity’s future. Little wonder $\pi$ continues to captivate so many minds.

The perfect wave

In reality, most applications don’t require knowing more than a few dozen digits of $\pi$. For example, calculating the circumference of a circle the size of the observable Universe only needs $\pi$ to 39 digits—enough to ensure an error smaller than a hydrogen atom. Engineers and scientists often use far cruder approximations, as no measuring device or computer can process an infinite number of digits.

As expected, $\pi$ arises naturally in equations involving circles and spheres. But it also appears in many unrelated contexts—especially in the mathematics of waves. Whether we’re talking about light, sound, radio, water, or electromagnetic waves, all these oscillations can be elegantly described using mathematics rooted in circular motion.

To see why, imagine a unit circle (with radius 1) centered at the origin of a Cartesian coordinate system. The circle’s total circumference is then $2\pi$. Starting at point (1,0), walk counterclockwise around the circle.

Track the vertical ($y$) position as you move. Initially, $y$ is 0, then it increases to 1 when you reach the top of the circle after traveling a quarter-circle ($\pi/2$ units). As you continue, $y$ returns to 0 at the leftmost point (after traveling $\pi$ units), drops to -1 at the bottom ($3\pi/2$), and finally returns to 0 at the starting point (after $2\pi$ units).

This progression of the $y$-coordinate creates a smooth, periodic wave—what we call a sine wave. You can visualize this process using the Geogebra applet below, where point $C$ moves around the circle, and point $G$ traces the $y$-coordinate as a function of distance traveled.

As the point continues around the circle (from $2\pi$ to $4\pi$, then to $6\pi$, and so on), the pattern repeats indefinitely, forming a continuous, perfect wave. Similarly, the $x$-coordinate creates another wave—this one shifted by $\pi/2$ relative to the sine wave.

The red wave represents the sine function, while the green one is the cosine function.

These familiar functions—sine and cosine—are staples of trigonometry, usually introduced in the context of right-angled triangles. But they originate in circular motion, where angles are measured as fractions of a full turn: 90 degrees for a quarter circle, 180 for half, and so on. While basic trigonometry restricts sine and cosine to right-angled triangles, their definitions naturally extend to all angles using the circular model described above.

Decoding messages

At a glance, these waves might seem too orderly to reflect the complex, irregular vibrations we encounter in the real world. For example, if you use a sound visualizer to record yourself singing a single note, the resulting waveform will appear quite erratic. This is because your voice doesn’t produce a single pure frequency—it generates multiple harmonics layered on top of the fundamental tone. Similarly, the radio signals emitted by GPS satellites don’t reach us as smooth, perfect waves. They are often distorted by random interference from other signals in the environment.

What gives sine and cosine waves their exceptional value is the fact that any signal with wave-like behavior can be deconstructed into a collection of pure sine and cosine components. This powerful idea stems from a discovery made in 1822 by the French mathematician Jean Baptiste Fourier. Contrary to the belief that mathematical brilliance peaks early in life, Fourier’s greatest contribution came relatively late. He began his career studying for the priesthood, took part in the French Revolution, survived imprisonment during the Reign of Terror, and later accompanied Napoleon’s expedition to Egypt as a scientific advisor. It wasn’t until he was 55 years old and working at the Académie des Sciences in Paris that he published his groundbreaking work, Théorie analytique de la chaleur.

Fourier arrived at his insight while investigating the movement of heat through solid objects like metal plates. He realized that it was mathematically simpler to represent heat sources as periodic functions—similar to waves. In essence, Fourier’s fundamental theorem states that nearly any repeating curve can be expressed as a (potentially infinite) combination of sine and cosine terms, such as:

$$y = \cos(x) + 4\sin(0.5x) + 0.5\cos(10x).$$

In this equation, $y$ is graphed vertically against the horizontal axis representing $x$.

This seemingly irregular waveform corresponds to the expression y = cos(x) + 4sin(0.5x) + 0.5cos(10x).

Scottish physicist Lord Kelvin once described Fourier’s book as a “great mathematical poem,” a fitting metaphor. Just as poets craft verses from words, complex waveforms can be built from the fundamental elements of sine and cosine functions.

Of course, knowing that such a decomposition exists is only useful if you can identify the actual sine and cosine components of a given signal. Fortunately, mathematicians developed a powerful method known as Fourier analysis, which enables exactly that. This analytical technique can break down any signal into its fundamental waveforms, making it incredibly versatile. It’s employed in interpreting radio signals from GPS satellites to determine your position. In audio engineering, it allows for the modeling of real instrument sounds for digital synthesis or for refining recorded audio. In medicine, it’s essential for reconstructing internal body images from CT scans, and in digital imaging, it’s used to compress files and remove imperfections. The range of applications is vast and continually growing.

The number $\pi$ has taken on a pivotal role, bridging the gap from pure geometry to advanced calculus and beyond, with extensive applications across scientific and engineering disciplines. It truly stands out as an extraordinary constant.

Alternative Representations of $\pi$

$$\frac{\pi^2}{6} = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \dots.$$

$$\frac{\pi^3}{32} = 1 – \frac{1}{3^3} + \frac{1}{5^3} – \frac{1}{7^3} + \dots.$$

$$\frac{\pi^4}{90} = 1 + \frac{1}{2^4} + \frac{1}{3^4} + \frac{1}{4^4} + \dots.$$