This story has two main characters. The first is pi.

Pi represents the ratio of a circle’s circumference to its diameter. Put another way, a circle’s edge is always about 3.14 times longer than its width.

But the geometry doesn’t matter for our story. We only care about pi’s digits.

Our second character is more of a concept: the idea of a walk, but where the walker lets randomness decide their path.

We’ll eventually arrive at this image.

To get there, we’ll take a meandering journey across 150 years, connecting people and mathematical discoveries. Here we go!

We begin with one man and his obsession with pi, all the way back in…

It was the spring of 1853. You might remember it as the time when Gail Borden applied to patent his process for condensed milk in the US (his name is on the tin).

The kids were cuckoo for condensed milk.

But I digress—Gail Borden is not our pi-obsessed man.

At the exact same time over in England, in the small coal-mining town of Houghton-le-Spring, a schoolmaster sent a book to the printer. Back then photos were rare, but I found this impressive ox from the town.

The schoolmaster’s name was William Shanks. He was 41. The book was about the most mouth-watering mathematical constant—the sweetened condensed milk of math—pi, calculated and written out to 607 decimal places. Further than anyone had written it before.

In the mornings before work Shanks would compute, and in the afternoons he would check his morning’s work. Each day the same thing. He’d write out the meticulous calculations, all by hand.

He spent nearly two and a half years producing a number that resulted in 607 characters.

Like me, you might be wondering what that felt like, to spend a day, or more, resulting in a single new digit. So I sharpened my pencil. I opened up Machin’s formula, the gold standard to calculate pi accurately during Shanks’ day. It looked simple enough:

π = 16 arctan(1/5) - 4 arctan(1/239)

I won’t attempt to explain the math, because I can’t. But thanks to YouTube and my decades-old training in long division, here is what it looks like to accurately calculate pi’s first decimal place by hand.

Aside from having to watch some basic videos to re-learn how to carry numbers, it was pretty straightforward. But each digit gets increasingly more computationally taxing to calculate. As the desired decimal place grows, more terms need to be calculated, to even greater precision.

Shanks knew his effort did nothing for his reputation as an (amateur) mathematician. Possibly something, though, for his reputation as a Computer. Back then that was a job for humans, a person who computes. He was darn good.

Shanks said it would not:

If it wasn’t for the money or the fame, why do it at all? He had no need for that precise a value of pi. No one did. His motivation was simple and relatable: he wanted to make something original and fill his free time. That something:

What do you do at the end of a long day? Shanks manually computed the digits of pi.

After publication, he set aside his daily ritual in favor of more thrilling computations like Euler’s constant and prime tables. For reasons unknown, he picked it back up to hammer out 100 more digits, bringing his full total to 707 in his 1873 publication.

How many pencils (or pens) did Shanks wear through? What was the physical toll? My obsession with how he calculated pi started to mirror his obsession with producing it. Lucky for me, number enthusiast Erwin Engert calculated pi to 100 decimal places, and documented the process. He estimated that Shanks’ calculations to arrive at 707 decimal places would fill over 5,000 pages. More pages than the entire stack of the Harry Potter series.

Practically speaking, the 707 decimal places are superfluous. The most sophisticated engineering challenges of modern day don’t need more than 15. But the digits of pi started taking on a life of their own outside of their geometric confines.

The first mathematician to closely interrogate pi’s digits was Augustus De Morgan. In a posthumous publication, he made a passing observation about the lack of 7s represented in Shanks’ original 607 computation.

He wasn’t so much troubled by it as curious. After all, random sequences have uneven counts too. But it raised a question: Was a low count of 7s a property of pi, or just ordinary fluctuation in a random pattern?

Counting the digits was one way to look for a pattern. Someone else thought to draw them. The man, the myth, the legend, John Venn. I can hear your fawning from the future.

After his 1880 smash hit, the Venn Diagram, he was (possibly) searching to recreate his earlier success. He was trying to determine whether a sequence can be called random when you know exactly how it was made.

Pi was the perfect specimen. It gave Venn what his argument needed:

To illustrate this last point, Venn drew a path that used pi’s digits to choose the direction of the pen strokes. He assigned the digits 0–7 to the directions of an eight-point compass and walked through pi digit by digit. 8s and 9s had to sit this one out.

Venn’s verdict:

I recreated Venn’s path with code so you could see it animated. However, mine turned out differently than his. I checked and re-checked my numbers, and even drew it out by hand to confirm, but Venn’s path departed dramatically from mine at the 24th decimal place.

I scoured the internet to see if anyone could confirm my suspicions. I only found one mention—a lone comment in a brief blog post about Venn’s pi path.

Mark hit the mark. Here is Venn’s path, with the actual digits from Shanks’ 707 decimal places of pi.

While the result is quite a bit different, they both look random to me.

We will briefly wander away from pi and continue Venn’s thread. In 1905, Karl Pearson threw down a challenge he dubbed “the problem of the random walk.”

He asked fellow readers in the journal Nature, if a man walks one yard at a time, turning in random directions, and repeats this n times, how far from his starting point is he likely to end up?

Sound familiar?

Lord Rayleigh responded with a “well actually”—he solved this problem 25 years prior with sound waves. It was about the square root of n, or after 100 steps, only about 10 yards away.

Pearson replied with an excuse that he was into other stuff and was unaware of Rayleigh’s prior work. Perhaps there were other excuses too, since he famously summed up this correspondence with:

Pearson gave a name to Venn’s wandering pen. “Random walk” stuck, and is still the term used in mathematics today.

Years passed. Nothing much happened of note. But then in 1921—in Europe again—a mathematician was having his own problems with walking.

Hungarian mathematician George Pólya enjoyed his leisurely walks in the woods, finding them a fine place to think about his work. One day he ran into one of his students out on a stroll with his fiancée. Nothing unusual. But then they crossed paths again. And again. Things were getting awkward and Pólya was embarrassed.

It was an accident. But how likely was it?

Instead of confronting the problem head-on, Pólya retreated to an abstract and hypothetical world, as one does. He reduced the problem to a two-dimensional grid and used a simplified four-point compass to find an answer to his definitely-not-snooping incident.

He posed a question similar to Pearson’s, though used the more elegant term “promenaders” in place of “walkers.” Simply put, given an infinite walk and infinite grid, will the probability of two promenaders meeting be a certainty?

He found that this concept of two walkers meeting can actually be decomposed and reframed to ask the question, will a walker return to its origin? Yes. A simple random walk on a two-dimensional grid returns to its starting point with probability one. Given unlimited time, it returns infinitely often.

He published a proof in the German journal Mathematische Annalen, which I don’t understand, but which looks thorough and compelling.

He also found that in a three-dimensional space, the probability of a return drops to 34%. This gave rise to the saying by Shizuo Kakutani :

You may now be wondering if our pi thread got lost. Same. So let’s bring back our main character. If pi’s digits are random, does a pi promenader in Pólya’s world return to its starting point? Drumroll, please…

Sadly, no, it does not. But 707 digits are certainly not a lot compared to infinity. Maybe we just need more.

Back to pi, for real. Shanks’ calculation had become authoritative through reuse and time. No one dared match his Herculean effort to go further.

In 1937, the Palais de la Découverte, a science museum, opened in Paris. In it was a room entirely dedicated to pi. Shanks’ laborious digits were celebrated as a giant ceiling installation around the room.

Pi’s digits had gone from an idle-time filler, to a published reference, to a museum display. His work would be immortalized.

Until 1946.

British mathematician D. F. Ferguson duplicated Shanks’ by-hand efforts. He used a similar-but-different Machin-esque formula and computed 620 digits of pi.

Ferguson compared his result to Shanks’ digits. Everything was peachy until the 528th digit. Somewhere along the way, Shanks’ unflappable exactness faltered. Ferguson discovered that everything Shanks had computed after that point, from the 528th to 707th digit, was entirely wrong.

Oh, poor Shanks.

When I first read that I felt for the guy. But Shanks never knew of his mistake. Maybe that was better.

As I dug into Shanks’ error, two sad facts emerged.

The first was when the error appeared. In his book’s preface from 1853, dated February 28th, Shanks wrote:

He had initially stopped calculating and was planning to publish 527 decimal places. An April 30th addendum stated how he extended the values to 607 decimals. The slip-up came quite literally in the first digit in that two-month sprint to add 80 more.

The second fact was his misplaced intuition. Dr. Rutherford, the previous record holder, separately calculated and confirmed pi up to 440 decimal places. Shanks wrote:

There were indeed no errors in the next 86 places, but there was one in the 87th.

Many mathematical historians have been drawn to this notorious miscalculation. How did a man who spent years calculating by hand suddenly slip up? A brief micro-mathematical forensic investigation was in order.

The short version: Shanks forgot to write a zero. Oops! Skip ahead.

The long version: In 2014, a science writer at American Scientist named Brian Hayes did exactly that.

Instead of searching for the correct answer, Hayes searched back for the wrong one, reverse-engineering what happened.

Remember Machin’s formula? No, probably not. A refresher:

π = 16 arctan(1/5) - 4 arctan(1/239)

It contains two arctan series. Think of each as a long list of fractions that get smaller and smaller, and you add them all up. Every fraction in the list is a “term.”

The two sides are lopsided, though. The 1/239 terms shrink and die out fast. The 1/5 terms shrink slowly. To reach his desired accuracy, Shanks had to calculate 506 individual fractions for just that side of the equation, each written out to 709 decimal places.

Pedantic note: Shanks needed to calculate two extra “guard digits” at the end to prevent rounding errors. So for 707 decimal places that would be 709. You’ll notice that shift below.

Shanks had also published his intermediate scratch work, including the final sums for both arctan series. Hayes measured exactly how far Shanks’ final value strayed from the truth. Then he walked back through all 506 terms to spot the culprit.

Suspect one: term 248

To calculate this specific term, (1/5)^497/497, Shanks was doing standard long division by hand. At the 531st decimal place he skipped a zero.

Every digit after it slid one column to the left. (This was actually first spotted by our other Shanks superfan, Erwin Engert.)

If you are still paying attention, you may have noticed that the typo happened at the 531st decimal place, but Shanks’ final value of pi broke earlier, at the 528th. Because Machin’s formula multiplied the erroneous series by 16, it magnified the error and caused a chain reaction of carried digits that caused the mistake to appear three places earlier.

Suspect two: term 72

Ferguson—the man who was also passionate enough to calculate pi by hand—actually flagged this anomaly back in 1946. Hayes found that if you dropped five digits at that point and slid everything over, you could reproduce Shanks’ value.

With both of these errors accounted for, Hayes reproduced Shanks’ value up to 601 digits on the 1/5 side of the equation.

Suspects three and four: still at large

At digit 601, the 1/5 side fractured again, and the trail went cold for detective Hayes.

A similarly unexplained fracturing occurred on the 1/239 side starting at the 592nd decimal place.

Because both sides are ultimately combined into one final number, pi was corrupted from the first error all the way to the end.

Perhaps the sadder and more relatable part of this cold case is that Shanks knew he had made clerical errors.

In the preface to his 1873 publication, Shanks issued an apology for a few mismatched digits, explaining he had lost track of his own scratch work revisions. Confident he had found the only typo, he wrote:

So there were a handful of errors, all seemingly in those months before publication in April 1853. None of them were computing mistakes, but rather clerical errors. What surprises me more is the sheer amount of numbers he had to calculate, track, and copy by hand, and that there weren’t more—a testament to both his endurance and focus.

Before we speed into the future, let’s quickly revisit the past.

The OG skeptic, De Morgan, counted 44 sevens in the original 607 decimal places. The true version has 50 sevens. Part of the shortfall De Morgan noticed was not a property of pi at all; it was the error.

Statistically speaking, the quantity of 7s is nothing out of the ordinary.

Next up, John Venn. Let’s fix his walk using the true value of pi through for the same number of digits as his original. A walk made from Shanks’ digits and one made from the corrected digits would initially follow the same instructions and then diverge.

While there is a clear difference between them, nothing jumps out to me as noteworthy. Both the old and new look random-ish.

Finally, just for fun, we return to George Pólya. We’ll do an updated version of his walk on the correct 707 digits.

Unsurprisingly, the walker still hasn’t returned to the origin. But 707 digits isn’t a very long walk.

Let’s keep moving forward. In 1948, the museum’s ceiling was corrected with the true value of pi.

A few years earlier, lowercase computers entered the scene. Non-human, electronic, digital, and programmable.

The job title Computer used to belong to people, like Shanks. During World War II, the University of Pennsylvania employed nearly a hundred of them (mostly women) to calculate ballistic trajectories by hand. But then the Army built a machine to do it faster.

Dude, you’re getting a Dell Electronic Numerical Integrator and Computer (ENIAC for short)!

Six of those Computers (Kay McNulty, Betty Jennings, Betty Snyder, Marlyn Wescoff, Fran Bilas and Ruth Lichterman) had to figure out how to program ENIAC with just diagrams since they didn’t yet have clearance to see it.

By hand it would take around 40 hours to calculate a shell’s trajectory. By machine, according to Jennings:

The six pioneering programmers weren’t invited to the celebration dinner, and none were named in the press coverage.

The same year the museum fixed the digits, Professor John von Neumann, like his mathematical forefathers, wanted to see if ENIAC could crank out more decimal places of pi, for the purpose of:

ENIAC computed 2,037 decimal places of pi. There was mayhem in the streets. A surge of babies were named Pi. Okay, neither of those things happened, but the record was shattered and von Neumann got his sweet, sweet digits.

Now there was a healthy serving of sevens. The updated distribution of pi’s digits looked like this:

The pi arms race had begun. As computers got more and more powerful, calculating pi came along for the ride as a sort of benchmark.

In 1961, Shanks was back. Daniel Shanks , that is. No known relationship to William, but you have to wonder if it wasn’t some sort of namesake destiny. Daniel Shanks and John Wrench calculated pi to a whopping 100,000 decimal places, all in less than nine hours.

They tested the digit frequencies almost immediately. That continued to be part of why anyone wanted more digits. The digits’ distribution remained well balanced.

Today, the record for the most calculated digits of pi is 314,000,000,000,000, set in 2025, though the attempt took a Shanks-esque 110 days. Imagine for a moment you are holding Harry Potter and the Deathly Hallows (759 pages). “Deletrius Aparecium!” you shout with enthusiasm. Every single letter in that book vanishes, then turns into a digit of pi. Now imagine that every single person in the US—babies included—is holding their own copy, with their own unique digits. That is 314 trillion, give or take a few.

The amount of compute has exploded so much that I had to chart this with a y-axis on a log scale just to make it readable. Here is another comparison: If I were to render the chart with a linear scale—that is, a 1:1 scale of digits to pixels—it would take you about 9,950 years of non-stop scrolling on your phone to get from Shanks’ 1853 digits to 2025.

To end things, we return to the image at the beginning: a random walk using pi’s digits. In 2012, researchers were exploring walks built from 100 billion digits of pi.

But they didn’t keep track of the number of times it returned to its origin, so I measured it myself to one billion digits of pi. Drumroll, please…

At the 21,344th decimal place, our walker comes home, and again several digits later. Then it wanders off and doesn’t return for the rest of the first billion digits. It took 399,980,598 steps in total, and ended up 18,891 steps away from the origin! So it could be a while before it returns again…

Shanks died in 1882, unaware of Venn’s path, the museum’s celebration, and the error and its correction. But as our forensic investigator Hayes aptly put it:

I can only imagine Shanks, De Morgan, Venn, Pearson, and Pólya tuned into my livestream of pi digits being calculated in real time to guide a random walk.

Want more pi? Tune into our lo-pi radio station. Streaming live, forever.