Life of π

In the legion of famous numbers, π (Pi) stands as a superstar. This irrational, never-ending decimal has captivated humanity for nearly 4,000 years. π is the ratio of a circle’s circumference to its diameter, a simple concept with unexpected traces. Over centuries, π evolved from rough ancient estimates to a transcendent constant entwined with our modern understanding of mathematics, physics, and even culture. Join this story of π: a journey through time, from pyramid builders and Greek geniuses into the very fabric of our universe.

Birth of π

π (Pi) wasn’t invented, it was discovered. When early humans first noticed that all circles, regardless of size, share the same ratio between circumference and diameter, they stumbled upon something fundamental about our universe. π emerges naturally from the geometry of a circle, perhaps, the most perfect shape in nature.

The Babylonians around 1900–1680 BC used π ≈ 3.125, as evidenced by a clay tablet. Meanwhile, the Ancient Egyptians recorded a formula implying π ≈ 3.1605 in the Rhind Papyrus (c. 1650 BC). These values were approximations used for practical construction, maybe even pyramids but that has been debated.

A huge leap forward came with Archimedes of Syracuse (287–212 BC). Often called the greatest mathematician of antiquity, Archimedes devised an ingenious method to pin down π more accurately. He inscribed and circumscribed regular polygons around a circle and calculated their perimeters. He squeezed the circle’s circumference between lower and upper bounds and proved π is between 3 10/71 and 3 1/7 (approximately 3.1408 < π < 3.1429). This was a remarkable result, and the first known algorithmic approach to π!

Across the world, others pursued π with equal zeal. In 5th-century China, Zu Chongzhi used a similar polygon method to reach an astonishing approximation: 355/113, accurate to 6 decimal places (3.1415926...). Zu Chongzhi’s method was lost to time, but his approximation remained legendary – it stood as the world’s most accurate π value for nearly a millennium. The achievement implies he effectively used a polygon with over 24,000 sides in his calculation. The Chinese were initially far ahead of the West in finding digits of π, primarily because they had a symbol for zero (we may dive deep into that one some other time).

Then after around 700 years, Madhava, an Indian Mathematician would develop an infinite series for π. Soon after in the Arab world, mathematicians like Al-Khwarizmi and Al-Kashi refined π further (al-Kashi, in 15th-century Persia, computed π to 16 decimal places using a polygon 2^28 sides). Each culture’s attempt to measure circles contributed a πece to this unfolding puzzle.

For most of history, π lacked a single name – it was just “that circle ratio”. Mathematicians would write out words or phrases to discuss it. This changed in the 18th century. In 1706, William Jones, a self-taught British mathematician, first used the Greek letter π to denote the circle constant. The choice of π was a nod to the Greek word periphery (περιφέρεια) and perimeter, as these relate to a circle’s circumference. But it was Leonhard Euler, the prolific Swiss mathematician, who truly popularized π. Euler adopted the symbol in 1737, and his enormous influence cemented π as the standard notation. Thus, from the 1700s onward, π became part of the language of mathematics.

π Everywhere

What makes π truly remarkable is how it appears in places seemingly unrelated to circles:

In the Natural World

• The spiral patterns in sunflower seeds, pinecones, and nautilus shells follow the Fibonacci sequence, which relates to π through the golden ratio.
• The probability that two randomly chosen numbers are relatively prime (sharing no common factors) is $6/π^2$ which is so unexpected!

In Fundamental Physics

• Heisenberg's uncertainty principle, which defines the limits of what we can know about quantum particles, contains π in its formulation.
• Einstein's field equations of general relativity, describing how matter warps spacetime to create gravity, incorporates π.
• The wave function in quantum mechanics, describing the probability of finding a particle in a specific location, uses π in its mathematical expression.

In Statistical Phenomena

• The normal distribution (bell curve), which describes countless natural phenomena from human heights to measurement errors, has π embedded in its formula.
• π is half the period of sine and cosine waves when we measure angles in radians, mathematicians prefer radians for this very reason: π creates elegance. When we say 180°, that’s just π in disguise. This means π is at the foundation of oscillatory motion.
• Even something as seemingly simple as a pendulum contains π, connecting time measurement to this constant.

π also appears in Euler's identity ($e^{iπ} + 1 = 0$), often considered the most beautiful equation in mathematics (I would eventually like to cover all the numbers in this equation). It shows that rotating a point π radians (180°) in the complex plane gives –1, and somehow, exponentiating $e$ by an imaginary multiple of π captures that rotation. It’s baffling, beautiful, and incredibly real, connecting five of the most fundamental constants in Mathematics. If you don't find that beautiful or thought-provoking, I salute you for reading this far into the article.

This is just the tip of the iceberg, I think we don’t fully realize how this number has encircled different aspects of our life and reflects a profound truth about our universe and its mathematical structure of reality. The reason is that our universe is filled with cyclic and wave-like phenomena everywhere. Wherever there's rotation, oscillation, or periodicity, pi naturally emerges.

When physicists search for a "Theory of Everything," they're looking for the underlying principles that explain why constants like pi have the values they do. Some theorists even speculate that in alternate universes, fundamental constants might differ—but pi would remain unchanged, as it emerges from the nature of geometry itself i.e. circumference / diameter.

π Fandom

When I stated that π is a superstar, it is no joke. March 14 (3/14) is officially recognized as Pi Day, passed by U.S. congress to promote interest in the field of STEM and celebrate one of the most recognizable celebrities of Maths. π has firmly rooted its place in high science and pop culture.

With the advent of electronic computers, the number of known π digits exploded. In 1949, an early computer (the ENIAC) ground out 2,037 digits of π in 70 hours – more than all prior human history had uncovered. By the 1970s, algorithms by John Machin’s spiritual heirs (like the Gauss–Legendre algorithm) and digital computers pushed it to hundreds of thousands, then millions of digits. As of 2024, π has been calculated to an utterly mind-boggling 202,112,290,000,000 digit i.e. over 202 Trillion Digits by StorageReview Lab.

We can use π to the 38th digit to be precise up to the diameter of the hydrogen atom, so it begs the question about the infatuation of the human-race to find even more digits of π. Our fascination with pi reflects something deeply human — our desire to find order in chaos and unquenchable curiosity. Finding the next digit of pi also seems to be kind of a stress test for computers which shows the technological prowess of that field.

Iconically, in a 1967 Star Trek episode, Mr. Spock defeats an evil computer by asking it to calculate π to the last digit, exploiting the fact that π’s decimals never terminate. There is also the movie Pi (1998) by Darren Aronofsky which is a psychological horror thriller about a mathematician obsessed with finding patterns in π’s digits. I will update this article once I have seen it! There are numerous other pop culture references about this fascinating number.

Legacy of π

Pi’s journey is far from over. It’s an infinite story – literally, since exploring its digits or patterns is a task with no end. But the legacy of π is secure: it has transformed human thought by linking the unexpected.

Four thousand years ago, π was a number to approximate for builders; today it’s a symbol recognized worldwide, worn on T-shirts, toasted on Pi Day, and taught in every school.

What can we learn from π? For one, perseverance. Generations of mathematicians pushed the boundary of knowledge bit by bit (or now byte by byte) to uncover π’s secrets. This perseverance led to unimagined discoveries – the development of calculus and infinite series came in part from trying to understand π better.

Pi also teaches open-mindedness: who would think that a number about circles would appear in probability or number theory? By keeping an open mind, mathematicians found π cropping up in integrals with no circles in sight. It’s a hint that mathematics has a beautiful unity underlying its diversity.

No matter how advanced our civilization becomes, the circle ratio remains the same, connecting us to ancient surveyors and future spacefarers alike. We all look up at the moon (a circle) and share π in common in describing it. Next time you slice a pizza (an edible circle), give a little nod to π. This constant has been a constant companion to human progress: from the first wheel to the latest quantum physics. The story of π is, in a sense, a circle itself – unending, looping through history, connecting and coming “full circle” in surprising ways. And as we continue to write digits of π or find new formulas, we are all part of that never-ending story.

References (π)

• Exploratorium – “A Brief History of π (π)”
• University Of Washington - “π and the Great Pyramid”
• Jet Propulsion Laboratory - “How Many Decimals of Pi Do We Really Need?”
• StorageReview - “Lab Breaks Pi Calculation World Record with Over 202 Trillion Digits”
• Baylor University - “Baylor Math Chair Explains Magic, Mystery of π (Pi)”
• Perimeter Institute - “The History of Pi (no, not us)”
• The Varsity - “A brief history of pi”