On Pi Day, Puzzling Over the Most Famous Transcendental Number

Depending on how you look at it, Pi Day is a festival of nerds, a math class gimmick, a chance to pause and reflect on the infinite, or just a good excuse to eat a lot of sugar. For these and many other reasons  people around the world have taken March 14th to ponder geometry’s most popular constant.

Pi (π), a number used to measure the circumference of circles, is approximately 3.1415. I say approximately because pi is a transcendental number, meaning that its decimal digits actually go on forever, unpredictably. As far as mathematicians can tell, those numbers never settle into any kind of pattern.

This numerical lawlessness is quite the thorn in the side of human pattern-loving brains. It’s like peeling back the skin of the universe and finding pure randomness.

After the existential surprise dies down, you’re left with confusion. A transcendental number is more unnatural than unicorns or dragons, which are, after all just combinations of things in the world (big lizards with bat wing; horses with horns).

Transcendental numbers are unimaginable. It would literally take you beyond the end of time to envision infinite grains of sand. They also define some of the basic observable properties of the universe. It’s this combination–the ubiquity, the centrality, the ungraspability—that makes them so alluring.

Four millennia of fascination

Pi is not the only transcendental number, nor is it the only one that’s an important constant. The number e, which is used to calculate logarithms, is also transcendental, as is Chaitin’s constant, which is used to determine probabilities. But pi has always had a special appeal. It is at once profoundly simple—the ratio of the circumference of a circle to its diameter—and infinitely complex.

The New Testament mentions a man who makes a metal bowl for a temple that’s 10 cubits in diameter and 30 cubits around the circumference, putting pi at 3. A Babylonian clay tablet from somewhere between 1900 and 1600 BCE sets the constant at 3.1250. An Egyptian papyrus from 1650 BCE goes with 3.1605. In 250 BCE, the Greek mathematician Archimedes discovered a geometrical approach to calculating pi, proving that pi is somewhere between 3.1408 and 3.1429. Today, supercomputers can calculate over 13.3 trillion digits of pi.

It’s not just mathematicians who seem to be entranced by pi. Each year on March 14th, many people mark the time at 1:59 and 26 seconds—in other words, the time 3.14 1:59:26. Others will recite digits of pi from memory. There are people who have memorized tens of thousands of digits, reciting them like mantras.

In medieval times, the philosopher Moses Maimonides wrote that pi’s reality has not been found, only approximated. “The ratio between a diameter of a circle and its distance around is not known,” writes Maimonides, who goes on to recommend 22/7 as an approximation.

More recently, Pi, the main character in 2001 novel Life of Pi, calls pi an “elusive, irrational number with which scientists try to understand the universe.”

“Like ‘pi’, life is not finite,” author Yann Martel explained in an interview. “And so I didn’t make the title ‘The Life of Pi’: I deliberately left out the definite article. That would have denoted a single life.”

Similarly, the 1998 thriller Pi revolves around Max, a mathematician who tries desperately to find an equation for the universe, a pattern in the seeming randomness.

One author even points out that, if you convert the Hebrew letters in one of the names of God to their numerical equivalents and write them around the circumference of a circle, you find yourself looking at 3.1415.

“It’s part of popular culture in a way almost nothing in mathematics is,” explained Jonathan Borwein, a mathematician and expert on pi, and a laureate professor at the University of Newcastle in Australia.

Borwein likens searching for pi to climbing Mt. Everest. “Why does anyone bother climbing Everest anymore?” he asked me. “Because everybody knows about it.” People pay attention to pi partly because it’s so entrenched in culture. Pi is famous, and fame attracts more fame. Still, he adds, “It’s the only math object that goes back to the history of clay tablets that’s still studied seriously, 4,000 years later.”

Pi draws attention not just on its own, but inside equations. For instance, it’s is an essential part of Euler’s identity, e+1=0, an equation which commonly tops “greatest equations ever” polls. Euler’s identity combines the two most important transcendental numbers with i, which equals the square-root of -1.

Leonhard Euler was an 18th century Swiss mathematician. No one knows for sure if Euler actually came up with Euler’s identity, but its weird simplicity has grabbed the attention of mathematicians and philosophers for a long time.

The balance and the simplicity of the equation is elegant—and also entirely mysterious. PhysicsWorld.com poll participants called it “uncanny and sublime” and “filled with cosmic beauty.”

“Five different numbers, with different origins, built on very different mental conceptions, invented to address very different issues. And yet all come together in one glorious, intricate equation, each playing with perfect pitch to blend and bind together to form a single whole that is far greater than any of the parts. A perfect mathematical composition,” wrote mathematician Keith Devlin. “Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler’s equation reaches down into the very depths of existence.”

Such a simple equation with such far-reaching significance borders on the sublime. As one Physicsworld.com user asks, “What could be more mystical than an imaginary number interacting with real numbers to produce nothing?”

Where infinity merges with the earth

When you dig into pi, you encounter questions that are as much theological as mathematical: is there a pattern to the universe? Or is it fundamentally random? And how do we reckon with the infinite?

Some people have always insisted that there must be a pattern behind pi. If only humans had the right tool or proof, it would be possible to discern some kind of order within those trillions of unfolding digits.

For the moment, though, the digits of pi—that fundamental ratio defining that most basic of shapes—seem random. Facing this can be deeply disconcerting. Is there some kind of arbirariness in the makeup of the world? Or something human beings can’t fathom?

It’s the notion of infinity that really twists people’s brains. Mathematician Georg Cantor, who is often credited with introducing the concept of mathematical infinity, went mad. According to Borwein, this was partly because of Cantor’s effort to grasp the infinite, and partly because he was ridiculed by so many mathematicians who refused to try.

Supercomputers might let researchers expand our understanding of pi. But even then, the task is endless. When people search for pi, they search for the infinite. They’ll never reach it, as long as they’re finite beings with finite imaginations. But there’s some wonder that comes with trying. Some satisfaction that comes with meditating on pi, reciting its digits like a mantra.

Cliff Pickover, a math and science author and the editor-in-chief of the IBM Journal of Research and Development, says that pi’s endless stream of numbers must contain all possible numerical representations. If you calculated pi long enough, you’d eventually find the works of Shakespeare coded in 1’s and 0’s.

“Somewhere inside the digits of pi is a representation for all of us — the atomic coordinates of all our atoms, our genetic code, a coding of our motions and all our thoughts through time, all our memories…. Given this fact, all of us are alive, and hopefully happy, in pi. Pi makes us live forever. We all lead virtual lives in pi. We are immortal,” Pickover writes.

“We exist in pi, as if in a Matrix,” he continues. “This means that romance is never dead. Somewhere you are running through fields of wheat, holding hands with someone you love, as the sun sets–all in the digits of pi. You are happy. You will live forever.”

Why do people keep trying to know pi, even though they know they won’t find it? What pot of gold is there at the end of that search? What happens when you recite the last digit of pi? What’s it like to hold the infinite in your hand?

“When I was a little kid my mother told me not to stare into the sun,” said Max in Pi. “So once when I was six, I did. At first the brightness was overwhelming, but I had seen that before. I kept looking, forcing myself not to blink, and then the brightness began to dissolve. My pupils shrunk to pinholes and everything came into focus and for a moment I understood.”

With every new number, the end recedes, an ever-lengthening ribbon in a universe that has found a way to hold it all. How can a finite universe house something infinite?

It seems impossible. And yet, pi is there, in every sunrise and car tire, every planet and solar system. Are mundane objects the site of the earth’s encounter with infinity?

As scientists plumb the universe, they find only nothingness and subatomic buzz; particles that pop in and out of being. The world, humans included, seems to be made of this patternless hum. It’s no wonder so many mathematicians quake at the concept of infinite randomness.

We see patterns throughout the universe. But physics suggest that it’s a structure built on random subatomic flux. In this view of the world, there’s only one source for patterns to build from. Only one original cosmic raw ingredient. And it looks something like π.

Also on The Cubit: What if animals believe in God?