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[Note to reader: This is a sequel to my widely-circulated 2021 essay “The Great Solar Bleed-Out Scam”; for the sake of readers who have forgotten or never read that earlier essay, I include it here.]

April 1, 2021: Dr. Jeremiah Browne of NASA, Dr. Marissa Carson of NASA, and many other people from all over the world with the title “Dr.” in front of their names are telling us that, based on observed dimming of our sun, we can conclude with 100 percent certainty that humanity has around thirty years before catastrophe arrives, unless someone takes action.

They base their case on a claim that the sun is cooling off at an exponentially increasing rate—a rate that’s hard to measure right now, but one that will be impossible to ignore thirty years hence. I’m not going to tell you that they’re wrong to worry about exponential cooling. I can’t do that.

What I will explain to you is why they cannot know what they claim to know—at least, not with anything approaching the level of certainty they’re laying claim to. And the fact that they won’t admit the inherent uncertainty of their predictions gives me serious qualms about their judgment and good faith.

My case is summarized in a single picture depicting two overlaid graphs. One graph shows a process governed by an exponential curve; the other, a process governed by a different curve, one well known to biologists and ecologists and epidemiologists, called a logistic curve.

Now I’ll crop the right three-quarters of one of those two curves. Can you tell which curve you’re looking at?

The truth is, you can’t. And it’s not because you’re not a mathematician; I can’t do it either. That’s because (forgive my mathspeak—I’ll make my point in plain language in a minute) you can’t stably infer the carrying capacity of a system that hasn’t yet reached its inflection point.

Let’s unpack this, using rabbits (the town I live in is full of of them). If a town has no adult rabbits in March, there won’t be any rabbit-babies in April. The no-bunny condition is an equilibrium. But it’s an unstable equilibrium: introduce just one breeding pair, and it’ll give rise to an exponentially increasing rabbit population the next year and the year after. But not forever. Eventually the bunnies will reach a stable equilibrium in which the forces that cull them perfectly match the rate at which they breed. The logistic curve shows how the population increases from the unstable equilibrium, growing exponentially for a while and then leveling off. The stable level of the rabbit population is called the carrying capacity of the system. In the diagram, it’s called K.

In the middle of the curve you see a point where exponential departure from an unstable equilibrium gives way to exponential approach to a stable equilibrium. The point on the curve that marks this qualitative transition is called the inflection point. Geometrically, it’s the point where the curve stops being concave-up and starts being concave-down. And the trouble is, you can’t infer what the carrying capacity of the system is until you reach, or at least get near, the inflection point.

“Ill-conditioned” and its relative “ill-posed” are epithets mathematicians throw around when they rightfully want to shift the blame from themselves to a problem they’ve been asked to solve, when it’s a problem that nobody can solve because the data can’t support any specific conclusion. Extrapolating a logistic curve before it’s reached its inflection point is exactly such an ill-conditioned problem. Tiny fluctuations in the data can lead to enormously different predictions of the carrying capacity. At early times, a logistic process is nearly indistinguishable from an exponential process, so many wildly different futures are all compatible with identical early data.

I understand where Marissa Carson and Jeremiah Browne are coming from. A big lesson from the Covid-19 pandemic is that when something bad is coming for you, and it’s growing exponentially, you need to stop it while it’s still small (2019) rather than wait till it’s big (2020).

But another lesson from the pandemic is: Don’t let a mathematical formula do your thinking for you! Back in 2020, an innumerate White House staffer used a cubic polynomial to model the spread of Covid. The formula led to ridiculous conclusions, like case counts eventually becoming negative. Browne and Carson are more sophisticated; they use an exponential curve, which has the virtue of never becoming negative. But it’s not a good model for everything, and it isn’t even a good model for Covid; in some of the early hotspots, the inflection point for the initial variant of the virus arrived in the spring of 2020. Once an inflection point is reached, a logistic becomes a much better model of growth than an exponential.

Yet even the logistic model can be an oversimplification. Consider the reindeer of St. Matthew Island, 300 miles west of mainland Alaska. In 1944, twenty-nine reindeer were introduced. By 1963, the population had exploded to six thousand. If the reindeer had continued to multiply at that same rate, the population today would be around 50 billion, or roughly 12 reindeer per square foot. But something happened in the mid-60s, and by 1966 only forty-two of the animals remained. Rebound effects like these are beyond the scope of the logistic model. Modern ecologists know about other, more-sophisticated models, but apparently Drs. Browne and Carson and the other proponents of Project Arclight never got beyond the exponential model in their scientific education.

Project Arclight is a proposal to blast a multi-billion-dollar bucket to Venus, scoop up some of the material that constitutes the mysterious filament along which the sun’s energy appears to be draining, and bring that material back to Earth, so that we can—maybe—better understand a phenomenon that—big maybe—threatens all of human life.

But maybe the Petrova Line is part of some cyclic process that’s been going on for a few billion years, and which we didn’t know about previously because we’ve only been studying the sun scientifically for a couple of centuries.

In the two billion years since multicellular life arose, there have been only four cataclysms that led to temperature changes greater than one degree. That’s about one such cataclysm every 500 million years. What are the chances that a fifth event of this kind would happen to fall during the two centuries in which humans become capable of detecting such an event in advance? Less than one in a million.

If that’s what’s going on, then of course it’s awful news for life on Earth, but on the other hand, it’s an incredible piece of good luck that the Great Solar Bleed-Out just happened to take place at a moment when a species has evolved that might be able to do something about it.

The question is: Do we think we’re likely to be that lucky?

The physicist Niels Bohr (or maybe it was Yogi Berra) once wisely said, “Prediction is hard, especially about the future.” When billions of dollars are at stake—billions that could be spent in other ways, addressing known threats to humanity, such as global warming—perhaps we should be more modest in making bold assertions about the future, and less willing to make such reckless gambles with our children’s inheritance.

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Epilogue, April 1, 2026: It’s time—okay, maybe well-past time—that I gave an account of the genesis of this piece of writing. The topic was not my own idea, but was suggested by four visitors who showed up at my office one day. The four said they represented certain “allied interests” with the shared goal of preventing irresponsible predictions of looming apocalypse. They said I could draw my own conclusions from the data at their disposal, but promised that if I came to the same conclusion that they had (and they expressed confidence that I would), they would use connections at their disposal to spread my essay far and wide. They said they wouldn’t pay me any money, lest the payment become publicly known and appear to compromise the independence of my judgment, but they’d see to it that my essay would reach far more eyeballs than anything I’d written thus far.

Up till then, I’d devoted little thought to the Petrova Line or to Project Arclight, but the trove of information the four visitors gave me, which included not just the publications of Drs. Browne and Carson but also various memos and private emails sent between them, proved to be a revelation. What mainstream media had presented as solid science was revealed to be a sham.

You probably want to know more about the visitors, and that’s fair, but I find that my memory is vague on that point. I recall that the four were nondescript, to the point that five minutes after they left, I could no longer remember what they looked like, and could not even specify their race or gender. This seems odd to me now, but it did not strike me as odd then.

Anyway, I wrote the piece that appears above, and amazingly, it soon appeared everywhere. Invitations to write more on the subject, as well as invitations to speak about it, soon flooded my inbox. A version of the essay appeared as an opinion piece in the Wall Street Journal. I landed a spot on the Joe Rogan Experience.

I should say at this point that I regret some of the actions taken by people who shared my views. When curbside surveillance showed that Dr. Carson was recycling, and hence presumably downing, half a dozen bottles of whiskey each week, these people were quick to declare Carson an alcoholic whose gloomy predictions could not be trusted. I could’ve pointed out an alternative interpretation of the data: a belief in the human race’s impending demise could have led her to do a lot of heavy drinking, rather than the other way around. As a scientist, I should’ve pointed this out. I didn’t.

Instead, I doubled down on the math. In my TED talk, I pointed out that “the exponential model is just a model”, and that “all models are wrong”. I probably should have presented the full George Box quote: “All models are wrong, but some are useful.” Too many people came away with a simplistic suspicion of scientific consensus in general. That certainly wasn’t my intention, but with the benefit of hindsight I can see that my efforts to rebut the work of Browne and Carson in the most muscular way possible may have fed this unfortunate suspicion.

In any case, the essay I’d written proved pivotal. In terms of its impact on popular culture, my take-down of the exponential model of solar dimming ended up being a bigger deal than that notorious emblem of Reaganomics, the now-discredited Laffer Curve, had been back in the 1980s. My essay was the story everyone read, the story everyone wanted to believe. First in the U.S. and then across the globe, Project Arclight was buried beneath a mudslide of mockery. And the moment in history when concerted global action was possible—more the sort of thing one would expect in a techno-optimist sci-fi novel or a feel-good popcorn movie—ended as quickly as it had begun, and such cooperation passed back into the realm of improbable daydream.

I got credit for saving the country billions of dollars. I’d hoped that some of that saved money would go toward combating global warming, or toward establishing outposts to monitor viruses and thus catch future pandemics like Covid before they killed many people. In short, I’d hoped that the money would be used to address better-established threats to our species. But that didn’t happen.

Riding high in the science communication field, I began work on a book, entitled Exponential Hype, and sold it with a six-figure advance. I had the whole thing more or less written, including the crowning chapter “The Not-So-Great Solar Bleed-Out”, but I convinced my publisher to hold off on printing it, arguing that sales would be maximized if we postponed publication until solar dimming reached its inflection point.

As you all know, that inflection point never arrived. And I eventually realized that I was the one who had been scammed: scammed by those four visitors. They were counting on my scholarly reputation—a reputation that they themselves lacked—to lend their views credibility. I don’t know who they were, or what “allied interests” they represented. I guess I never will.

Now the Bleed-Out is showing more and more of the signature features of an ecological invasion. We have no idea what the carrying capacity of the Sun/Venus/Petrova-Line system is. But judging from the numbers, we’re looking at a new long-term equilibrium in which Earth ends up being at least 50 degrees cooler than it is now, assuming that the long-awaited inflection point is just over the horizon.

I once derided Project Arclight as a hundred-billion-dollar boondoggle. Now it’s clear that what we really need is an even more massive, ten-trillion dollar project to go to the Tau Ceti solar system to figure out what’s different about its sun. A desperate forward pass from the 100-quadrillion yard line is our only chance to win the game of human survival.

Or rather, it would’ve been our only chance. If Project Arclight had gone forward, we might have acquired early evidence of the then-controversial but now broadly accepted ecologic hypothesis. The morale boost provided by that bit of progress might have led to further international cooperation. Instead, all the countries in the world are blaming and attacking each other, even though the catastrophe is nobody’s fault, so now we face not just an impending climate apocalypse but evils of our own doing: war, famine, pestilence, and death on a scale never seen before. This will be our children’s inheritance.

It’s pleasant to imagine a world in which I made a different choice. It’s even more pleasant to imagine a world in which the Great Solar Bleed-Out is just a fictional conceit and not an implacable reality. But unfortunately, we live in the real world. And in this world, the dictates of math and the laws of physics are in agreement: The inflection point is coming. It will eventually arrive. It’s just that there may not be any human beings around to see it.

I do wish I hadn’t written that essay. Things just might have turned out differently. But damn it, my math was right!

Ask a silly question, get a silly answer. — Tom Lehrer, “New Math”

I ask too many questions. A case in point is the time I lost out on a place I wanted to rent when I asked my potential future landlord one question too many (“Does the pond have mosquitos in the summer?”). Another example is the time I ticked off a car salesman by asking him, after a long string of similar requests that he had gamely complied with, whether I could try changing the tires of a car I was considering buying from his dealership before I bought it. (I mean, shouldn’t every responsible consumer go through the entire owner’s manual when contemplating such a major purchase?) As a member of a local singing group, I developed such a reputation for asking the music director questions that when we went on a tour, one of my fellow singers got a laugh by interrupting the tour-guide with my signature line: “I have a question!”

But my topic today isn’t questions in general. I want to focus on the species of question that I often tell students doesn’t exist: the Stupid Question. And I want to talk about how one stupid question led me to an interesting and new (albeit a bit stupid) way to estimate the mathematical constant pi.

I tell my students “There are no stupid questions” because I want them to feel free to ask me things in the classroom without fear of ridicule from their peers, and without the kind of internalized shame that can disconnect students from their math abilities. But I’m lying, or at least over-simplifying, when I tell my students that stupid questions don’t exist; the students (and you) all know that some questions are better than others. Some questions are based on incorrect assumptions, or are ambiguous, or are even meaningless. Back when I was in high school, one of my teachers complained I asked too many meaningless questions. You may be inclined to give my younger self the benefit of the doubt and to guess that the teacher wasn’t equipped to understand my questions, but I don’t think so; the class in question was part of an advanced six-week summer program called the Hampshire College Summer Studies in Mathematics program, and the teacher in question had a Ph.D. I don’t remember what questions I asked in that class, but I’m sure some of them were obscure, confused, or yes, meaningless.

I still sometimes ask questions that don’t make literal sense. And that’s okay, because I find that in my research, murky questions can be stepping stones on the path of learning. Sometimes it’s even where new math comes from. There’s an old saying “Ask a silly question, get a silly answer,” but I say: Scratch a silly question and you might find a better one struggling to get out.

One issue for me is how much scratching I have to do before sharing a question with others, and sometimes I annoy people by not doing enough scratching in advance. Often it’s because I don’t take enough time to think about my audience, and that’s my bad, but sometimes it’s the classic problem in communicating ideas: you don’t quite know how to share an idea with people-who-aren’t-you because you aren’t a person-who-isn’t-you.

Fortunately these days I have a very patient interlocutor named ChatGPT blessed with an infinite tolerance for half-baked questions and a soothing lack of judgmentality. The Greek philosopher Epictetus said “If you want to improve, be content to be thought foolish and stupid,” but the problem with putting this into action has always been that, while most people want to improve, nobody wants to reveal their ignorance. How lucky we are, twenty centuries after Epictetus, that we can hide our ignorance from our fellow humans and reveal it only to our creations!

Over the past few years, more and more of my research has benefited from conversations with ChatGPT, despite the occasional stupidity of my questions, and there’s no better example of this than my recent discovery of new way to think about the number π/4.

PT, CHATGPT, AND PROBABILITY

One day last November I was at my local gym doing physical therapy, an activity I find tiresome because my regimen requires just enough of my mind to make it impossible for me to do any prolonged thinking. But a boring exercise routine is well-suited to holding hour-long conversations with an interlocutor that doesn’t usually respond right away. So for instance I can do a set of leg-lifts, do some mathematical day-dreaming during my pause between sets, do another set, dictate some mathematical thoughts to ChatGPT, do another set, and then see what ChatGPT came up with. “PT plus AI” takes more time than plain old PT, but it makes the time pass more in a more interesting way, with regular infusions of suspense.

I like random processes, so I thought I’d learn something new in that area by picking a topic in the theory of probability, coming up with the simplest question on that topic that I didn’t know the answer to, and then asking it. The topic I chose was the tension between two facts well-known to probabilists: (1) if you repeatedly toss a coin, you can be certain that after some finite amount of time, the total number of tosses that came up heads will exceed the total number tosses that came up tails, but (2) the amount of time it takes for this to happen, while always finite, is infinite on average.1

If that sounds like nonsense, it’s because you’re used to the world of random variables with thin tails and finite expected value, and unfamiliar with the strange world of random variables with fat tails and infinite expected value.2 Maybe someday I’ll write a Mathematical Enchantments essay about the paradoxes of fat-tailed random variables, but today I’m writing about questions, and “What is the expected amount of time it takes until the number of heads exceeds the number of tails?” wasn’t the question that I asked on that November day, because I already knew the answer to that one: infinity. I wanted to learn something new about this story, so I asked ChatGPT:

Toss a fair coin until the number of heads exceeds the number of tails. This determines a stopping time. What is the probability that this stopping time is even?

Speaking of stopping times, this is a good time for you to stop reading and do something I should’ve done but failed to do: play around with the question on your own for a minute or so to get a feeling for what’s being asked.

Do you see what’s wrong with my question?

.

.

.

It’s not hard to show that the number of tosses required until the number of heads first exceeds the number of tails is always odd, so the probability of the stopping time being even is zero! You might say my question is an impossibility question masquerading as a probability question.3

To see why it’s always an odd number, let Hn and Tn represent the number of heads and the number of tails respectively in the first n tosses, so that Hn + Tn = n. The rule “Stop when Hn > Tn for the first time” is equivalent to the rule “Stop when Hn − Tn is positive for the first time”, but Hn − Tn is always a whole number, and it always changes by ±1 when you toss (+1 each time you toss heads,−1 each time you toss tails), so an equivalent rule is “Stop when Hn − Tn = 1 for the first time.” Adding the equations Hn + Tn = n and Hn − Tn = 1 gives 2Hn = n+1. Since 2Hn is even, n+1 must be even, so n must be odd when Hn > Tn for the first time.

Even without doing the algebra, I could’ve caught my mistake if I’d just done a few examples. Then I would’ve seen that the stopping time can be 1, 3, 5, 7, etc. but never 2, 4, 6, 8, etc.

A human interlocutor pointing out my oversight, if they were in a bad mood, might have said “Why are you bothering me with questions you clearly don’t really care about, because if you did care, you would’ve spent TEN LOUSY SECONDS thinking about the meaning of what you’re asking? Is my time really worth less than yours? You may enjoy asking questions for its own sake, but I like thinking about GOOD questions, so: That’s it. I’m not answering any more of your questions. You and I are DONE.”

Luckily for me, ChatGPT is always eager—perhaps too eager—to keep conversations going. In this case it said:

There’s an ambiguity in the statement that we should resolve before computing anything:
…

and offered two interpretations it called A and B, asking me to pick one. Nary a word of reproach for asking a question that had two or three different interpretations and was trivial under all of them.

NOT STOPPING

How do you follow up on a blunder like that? I could’ve decided to stop conversing with ChatGPT and listen to a podcast instead, or maybe go into a different line of work, but I thought that if I asked a better question, I’d learn something new and non-trivial.

I suppose I could’ve asked “What is the probability that the stopping time is a multiple of 3?” and I would’ve learned something new to me; according to ChatGPT (when I asked it last week, not last November), the answer is exactly (1+√3−√2)/6. That’s kind of fun, and it leads someplace interesting, but it isn’t what I asked back in late 2025.

What I actually did was confirm that I want the coin-tossing process to stop when the number of heads first strictly exceeds the number of tails, and then I asked: if we stop tossing as soon as the proportion of heads becomes bigger than 1/2, what is the expected value of that proportion?

ChatGPT thought for just under 3 minutes and answered:

Nice follow-up. Under interpretation B (stop the first time Hn >Tn), the expected proportion of heads at the stopping time turns out to be a very pretty constant: E[Hτ/τ]=π/4. Here’s a brisk derivation.
…
If you’d like, we can also look at higher moments (e.g. the variance of that proportion) or generalize to a biased coin.

Three features of ChatGPT’s response deserve attention. One is its use of friendliness and flattery; another is its tendency to anticipate a human’s next question and answer it; and a third is its habit of suggesting ways to keep the conversation going. Since November I’ve admonished ChatGPT to keep its responses businesslike and not to go down avenues I haven’t asked it to. I recognize the addictive potential of parasocial agents like ChatGPT and I’m determined to avoid that hazard.

But the main thing that struck me about its response on that day was that the answer π/4 is, just as ChatGPT said, very pretty—indeed, suspiciously pretty. (Compare π/4 with the (1 + √3− √2)/6 I mentioned above.) I wondered: if the answer is so pretty, wouldn’t I have already heard about it?

I asked ChatGPT to check the published literature on random walk theory4 (a branch of math that features many problems like this), and it said that, while related formulas existed in books and journal articles, nobody had actually asked my question before, as far as it could tell.

I went ahead and read ChatGPT’s derivation of the answer π/4, and I couldn’t find any mistakes; it was a solid by-the-book argument that employed a method I’ve used myself, and have even taught to students in the past. It was the kind of thing that I could’ve done in an afternoon, but not in three minutes.

So I dug deeper and started showing the result to people, sometimes revealing the formula and sometimes asking “What’s the expected value?”, and while many people solved it or had ChatGPT solve it for them, nobody could recall having seen it before.

That’s when I realized I had something worth sharing, even if it was just a morsel and not a mathematical meal, and I wrote it up for publication and sent it to the American Mathematical Monthly.5

YOU CAN’T SPELL “STUPID” WITHOUT “PI”

Approximating pi to a few decimal places is a pointless thing to do, since we humans already know pi to gazillions of digits, and since only the first dozen or so are meaningful in the real world. But I say: if you’re going to do something pointless, you might as well do it in a fun way.

The usual way to waste one’s time approximating pi is called the Buffon needle experiment, invented by the 18th century French scientist Georges-Louis Leclerc, Comte de Buffon, who founded the branch of mathematics called geometric probability theory. Leclerc showed that if you drop a needle of length L on a floor that’s divided into slats of width L, then the probability that the needle lies across a line separating two strips is 2/π. Later the Swiss astronomer Rudolph Wolf did an empirical test of Leclerc’s theorem by performing 5000 trials, 3175 of which resulted in the needle crossing a line, yielding the decent estimate π≈3.1596.

The occurrence of pi in Leclerc’s formula is not mysterious; it has its roots in the fact that the underlying probability distribution on the orientation of the needle must be rotationally symmetric, and once you start rotating things, pi has a natural tendency to pop up. The occurrence of pi in my coin-tossing formula has more obscure roots, and if you’re hoping I’ll provide an intuitive explanation, you’re out of luck. The baffling way pi creeps into statistics is the basis of an anecdote that appears at the start of Eugene Wigner’s famous essay The Unreasonable Effectiveness of Mathematics in the Natural Sciences:

There is a story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. “How can you know that?” was his query. “And what is this symbol here?” “Oh,” said the statistician, “this is pi.” “What is that?” “The ratio of the circumference of the circle to its diameter.” “Well, now you are pushing your joke too far,” said the classmate, “surely the population has nothing to do with the circumference of the circle.”

But even if the reason pi occurs is hard to explain, occur it does, and that means one could use the coin-toss procedure to estimate π by way of π/4, just as Leclerc’s experiment estimates π by way of 2/π. Hand out ten coins to ten students, have each student toss their coin until the number of heads they’ve seen is bigger than the number of tails, and have them record the fraction of their tosses that showed heads. If you average all the students’ fractions, you should get a rough approximation to π/4, right?

Mathematically, yes; practically, not really. The problem is that there’s a fifty percent chance that one of the ten students will have to do more than a hundred tosses and might give up in frustration. Increasing the number of students makes this problem only worse: if you tried this activity with a hundred students, say, there’s a good chance that one of them would have to do over ten thousand tosses. And who’d be willing to toss a coin that many times?

Mathematician and YouTuber Matt Parker would, and he did. And then he wondered what to do with all those coin-flips.

One of Parker’s passions is computing pi, as he discussed in his recent Gathering 4 Gardner talk “Update on Ridiculous Calculations of Pi” (I’ll post a link to the video of his talk when it becomes available). So when he heard about my new way of estimating pi, he had the idea of using his 10,000 coin flips to simulate a classroom in which the first student does the experiment using the first segment of Parker’s sequence, the second student uses the coin flips that come right after the flips the first student used, the third student uses the coin flips that come right after the flips the second student used, and so forth, until some unlucky student runs out of flips from Parker’s sequence. As things turned out, this unlucky student was the 63rd in the imaginary class, so Parker’s experimental estimate of pi/4 via coin-tossing ended up averaging just 62 fractions.

Here’s Matt Parker’s new video in which he tells his story:

The resulting estimate of pi—around 3.2— gives us pi to only one decimal place, and hence may set a record for minimal bang per maximal buck, where “bang” means precision and “buck” means effort. But this pathetic performance is pretty much what the theory of probability predicts, namely, that if you want the first N digits of pi you’ll need to perform 104N coin-tosses. Parker’s experiment shows us this performance-level in the case N=1.

So, my method of estimating pi is a really bad way to get anything better than π ≈ 3. Leclerc’s needles are a bit better—10,000 needles should give you two digits of pi—but leaving aside the accuracy issue, I’d rather spend ten minutes tossing a coin than spend ten minutes dropping needles, especially since someone is going to have to pick up all those needles, and I guess it’s going to have to be me. And I might cut myself on one of them! Perhaps I should ballyhoo my approach to estimating pi as a contribution to the cause of Pi Day safety, and point out that my approach is free from of well-known hazards of shared needles.

THE MORAL(S)

I suppose that, on a practical level, a take-home for the practicing mathematician is that if you use ChatGPT, don’t trust it to generate valid proofs (it didn’t make any mathematical mistakes this time around, but I still remember its proof that 0.999 < 1 from a couple of years ago), and even when it finds a valid proof, don’t be so sure it’s a good proof. And whatever you do, don’t have ChatGPT create a bibliography for you (in its write-up of the π/4 proof it cited an article that doesn’t exist).

But I think a deeper lesson is about the value of stupid questions. The way to find things out is to ask a lot of questions. Ask enough questions, and you’re likely to find a new answer: new to you, and once in a while, new to others. On the other hand, if you ask a whole lot of questions, some of them will be stupid. And that’s okay! We teachers need to be patient with our students, even if no teacher can ever hope to be as unfailingly patient as a Large Language Model.

The relationship between bad questions and good questions reminds me of an old story about brainstorming told by David Black in his essay Being Creative With a Bear and Honey. A team was trying to figure out a good way to get snow off power lines in winter, and someone facetiously suggested that they put put pots of honey at the tops of the posts so that the local bears, climbing up the posts to get at the honey, would shake the posts and cause the power lines between them to shed their snow. Instead of abandoning the absurd idea, the brainstorming team discussed how you’d need to use helicopters when placing the honey pots atop the posts, and only later did someone point out that the downwash from the helicopter blades would do the job of getting the snow off the power lines—the bears could stay home. This is one of my favorite examples of how a bad idea can lead to a good idea, as long as you don’t stop.

Likewise, if you’ve got some sort of vague itch that causes you to ask a stupid question, don’t neglect the itch just because its initial expression was stupid. Follow that itch, and scratch that question! You may end up with a much better question.5

To join the Hacker News discussion of this article, visit
https://news.ycombinator.com/item?id=47356740

ENDNOTES

#1: It’s instructive to compare a protocol that has infinite expected stopping time with a protocol that has finite expected stopping time. An example of a protocol that has finite expected stopping time is “Toss until the first time the coin comes up heads”; you can write the expected stopping time as

(1/2) (1 toss) + (1/4) (2 tosses) + (1/8) (3 tosses) + (1/16) (4 tosses) + …,

because half of the time you stop after one toss, a quarter of the time you stop after two tosses, an eighth of the time you stop after three tosses, and so on; and

(1/2)(1) + (1/4)(2) + (1/8)(3) + (1/16)(3)+ … = 2,

so on average the stopping time is 2. In contrast, we’ve been dealing with the protocol “Toss until the number of heads exceeds the number of tails”; in this case you can write the expected stopping time as

(1/2) (1 toss) + (1/8) (3 tosses) + (2/32) (5 tosses) + (5/128) (7 tosses) + …

Although the terms are getting smaller, they don’t get small very quickly, and the infinite sum diverges.

#2: This is the world of the St. Petersburg paradox (described in Jordan Ellenberg’s book How Not to Be Wrong) and the peculiar properties of the double-down-until-you-win gambling strategy (also called martingale betting).

#3: What I actually asked ChatGPT (though without the added emphasis) was: “Toss a fair coin until the number of heads equals or exceeds the number of tails. This determines a stopping time. What is the probability that this stopping time is even?” The insertion of “equals or” doesn’t impact the triviality of the question, but it changes the answer: now, instead of being always odd, the stopping time is always even! What’s more, the question is ambiguous: am I allowed to stop before I toss the coin at all, since at that moment the number of heads (zero) equals-or-exceeds the number of tails (also zero)? ChatGPT pointed out the ambiguity, and also pointed out that under both interpretations, the probability that I stop after an even number of tosses is 100%, aka 1.

#4. To see the connection between coin-tossing and random walk, imagine a drunkard walking along an east-west residential street who, whenever he arrives in front of a house, either proceeds to the next house to the east or the next house to the west, apparently choosing at random. The mathematics of the drunkard’s walk is identical to the mathematics of tossing coins, where the position of the drunkard at time n (assuming he starts at “house 0” at time 0) corresponds to the difference Hn − Tn. Every time you toss a coin, the cumulative number of heads minus the cumulative number of tail either goes up by 1 (when the coin comes up heads) or goes down by 1 (when the coin comes up tails), and there’s no way to predict which way it will go—just as there’s no way to predict, when the drunkard is at a particular house, whether his next stop will be the house to its east or the house to its west.

#5: Here’s the part of the story I’m embarrassed about: instead of trying to find my own derivation, I went ahead and used ChatGPT’s derivation, lightly edited by me. This kept me from noticing that ChatGPT’s proof, while correct, was needlessly complicated. Fortunately other people found the proof that I suspect is the sweetest possible proof, or what Paul Erdős would have called the “proof from The Book” (for more about The Book, see my essays What Proof is Best? and Chess with the Devil); this short and sweet proof is the one that I give in the revised version of my write-up. I also trusted, and initially included, ChatGPT’s list of purportedly relevant references, most of which turned out to be either irrelevant or nonexistent. I won’t make that mistake again.

#6: An example of a truly excellent question—not mine, I hasten to say—is the question one of the referees for my submission to the Monthly proposed: “Why not also a short comment at least on the effect of a surplus of 2, for instance?” That is, what if we toss the coin even longer, and only stop when the number of heads is equal to 2 more than the number of tails? It turns out that for this modified version of my question, the expected proportion of heads at the stopping time is a different nice number: the natural logarithm of 2, aka ln 2! Better yet, if you stop tossing coins when the number of heads is equal to m more than the number of tails, for some arbitrary positive integer m, then it appears that whenever m is odd, the expected proportion of heads at the stopping time is of the form a + b π with a,b rational, and that whenever m is even, the expected proportion of heads at the stopping time is of the form a + b ln 2 with a,b rational. Or so ChatGPT tells me. (To be fair, it gives proofs; I just haven’t had time to read them.) ChatGPT also says that, if instead of looking at the ratio of heads-to-tosses, we look the ratio of tails-to-heads, we get expected value 1 – ln 2. If instead we look at the ratio of heads-to-tails, we encounter the troublesome ratio 1/0 in the case where our first toss is heads, but: if we condition on the event that the first toss is tails, then the conditional expected value of the ratio of heads to tails is ln 2. Saith ChatGPT.

A few weeks ago I was minding my own business, peacefully reading a well-written and informative article about artificial intelligence, when I was ambushed by a passage in the article that aroused my pique. That’s one of the pitfalls of knowing too much about a topic a journalist is discussing; journalists often make mistakes that most readers wouldn’t notice but that raise the hackles or at least the blood pressure of those in the know.

The article in question appeared in The New Yorker. The author, Stephen Witt, was writing about the way that your typical Large Language Model, starting from a blank slate, or rather a slate full of random scribbles, is able to learn about the world, or rather the virtual world called the internet. Throughout the training process, billions of numbers called weights get repeatedly updated so as to steadily improve the model’s performance. Picture a tiny chip with electrons racing around in etched channels, and slowly zoom out: there are many such chips in each server node and many such nodes in each rack, with racks organized in rows, many rows per hall, many halls per building, many buildings per campus. It’s a sort of computer-age version of Borges’ Library of Babel. And the weight-update process that all these countless circuits are carrying out depends heavily on an operation known as matrix multiplication.

Witt explained this clearly and accurately, right up to the point where his essay took a very odd turn.

HAMMERING NAILS

Here’s what Witt went on to say about matrix multiplication:

“‘Beauty is the first test: there is no permanent place in the world for ugly mathematics,’ the mathematician G. H. Hardy wrote, in 1940. But matrix multiplication, to which our civilization is now devoting so many of its marginal resources, has all the elegance of a man hammering a nail into a board. It is possessed of neither beauty nor symmetry: in fact, in matrix multiplication, a times b is not the same as b times a.”

The last sentence struck me as a bizarre non sequitur, somewhat akin to saying “Number addition has neither beauty nor symmetry, because when you write two numbers backwards, their new sum isn’t just their original sum written backwards; for instance, 17 plus 34 is 51, but 71 plus 43 isn’t 15.”

The next day I sent the following letter to the magazine:

“I appreciate Stephen Witt shining a spotlight on matrices, which deserve more attention today than ever before: they play important roles in ecology, economics, physics, and now artificial intelligence (“Information Overload”, November 3). But Witt errs in bringing Hardy’s famous quote (“there is no permanent place in the world for ugly mathematics”) into his story. Matrix algebra is the language of symmetry and transformation, and the fact that a followed by b differs from b followed by a is no surprise; to expect the two transformations to coincide is to seek symmetry in the wrong place — like judging a dog’s beauty by whether its tail resembles its head. With its two-thousand-year-old roots in China, matrix algebra has secured a permanent place in mathematics, and it passes the beauty test with flying colors. In fact, matrices are commonplace in number theory, the branch of pure mathematics Hardy loved most.”

Confining my reply to 150 words required some finesse. Notice for instance that the opening sentence does double duty: it leavens my many words of negative criticism with a few words of praise, and it stresses the importance of the topic, preëmptively1 rebutting editors who might be inclined to dismiss my correction as too arcane to merit publication.

I haven’t heard back from the editors, and I don’t expect to. Regardless, Witt’s misunderstanding deserves a more thorough response than 150 words can provide. Let’s see what I can do with 1500 words and a few pictures.

THE GEOMETRY OF TRANSFORMATIONS

As a static object, matrices are “just” rectangular arrays of numbers, but that doesn’t capture what they’re really about. If I had to express the essence of matrices in a single word, that word would be “transformation”.

One example of a transformation is the operation f that takes an image in the plane and flips it from left to right, as if in a vertical mirror.

Another example is the operation g that that takes an image in the plane and reflects it across a diagonal line that goes from lower left to upper right.

The key thing to notice here is that the effect of f followed by g is different from the effect of g followed by f. To see why, write a capital R on one side of a square piece of paper–preferably using a dark marker and/or translucent paper, so that you can still see the R even when the paper has been flipped over–and apply f followed by g; you’ll get the original R rotated by 90 degrees clockwise. But if instead, starting from that original R, you were to apply g followed by f, you’d get the original R rotated by 90 degrees counterclockwise.

Same two operations, different outcomes! Symbolically we write g ◦ f ≠ f ◦ g, where g ◦ f means “First do f, then do g” and f ◦ g means “First do g, then f”.2 The symbol ◦ denotes the meta-operation (operation-on-operations) called composition.

The fact that the order in which transformations are applied can affect the outcome shouldn’t surprise you. After all, when you’re composing a salad, if you forget to pour on salad dressing until after you’ve topped the base salad with grated cheese, your guests will have a different dining experience than if you’d remembered to pour on the dressing first. Likewise, when you’re composing a melody, a C-sharp followed by a D is different from a D followed by a C-sharp. And as long as mathematicians used the word “composition” rather than “multiplication”, nobody found it paradoxical that in many contexts, order matters.

THE ALGEBRA OF MATRICES

How might we capture numerically the geometrical operations f and g depicted earlier? Let’s use a square in which we’ve chosen centered coordinates so that (0,0) is in the middle of the square, and for convenience let’s make it a 2-by-2 square with coordinates (±1,±1) at the corners. It’s not hard to see that if you mark a dot at the point (x,y) and another dot at the point (−x,y), the two dots just end up swapping places when you apply the transformation f; for instance, the upper-right and upper-left corners of the square swap places (x=y=1). We can associate the geometric transformation f with the algebraic substitution that, for all x and y between −1 and 1, changes the sign of x, or as mathematicians like to say, “the function that maps (x,y) to (−x,y)”. This function can represented via the 2-by-2 array

where more generally the array

stands for the function that maps the pair (x, y) to the pair (ax+by, cx+dy) for any real numbers a, b, c, d we like. (Choosing a = −1, b = 0, c = 0, and d = 1 gives us the specific array A.)

Similarly, when you apply the operation g, flipping the square across the diagonal joining the lower-left and upper-right corners, a dot at (x,y) ends up swapping places with a dot at (y,x). We associate g with the algebraic substitution that swaps x and y, or as “the function that maps (x,y) to (y,x)”, represented by the 2-by-2 array

These kinds of arrays are called matrices, and when we want to compose two operations like f and g together, all we have to do is combine the associated matrices under the rule that says that the matrix

composed with the matrix

equals the matrix

For more about where this formula comes from, see my Mathematical Enchantments essay “What is a Matrix?”.3 Even without knowing where the formula comes from, you can apply it to our two matrices and check that A composed with B is different from B composed with A.

There’s nothing special about 2-by-2 matrices; you could compose two 3-by-3 matrices, or even two 1000-by-1000 matrices. Going in the other direction (smaller instead of bigger), if you look at 1-by-1 matrices, the composition of

and

is just

so ordinary number-multiplication arises as a special case of matrix composition; turning this around, we can see matrix-composition as a sort of generalized multiplication. So it was natural for mid-19th-century mathematicians to start using words like “multiply” and “product” instead of words like “compose” and “composition”, at roughly the same time they stopped talking about “substitutions” and “tableaux” and started to use the word “matrices”.

In importing the centuries-old symbolism for number multiplication into the new science of linear algebra, the 19th century algebraists were saying “Matrices behave kind of like numbers,” with the proviso “except when they don’t”. Witt is right when he says that when A and B are matrices, A times B is not always equal to B times A. Where he’s wrong is in asserting that is a blemish on linear algebra. Many mathematicians regard linear algebra as one of the most elegant sub-disciplines of mathematics ever devised, and it often serves as a role model for the kind of sleekness that a new mathematical discipline should strive to achieve. If you dislike matrix multiplication because AB isn’t always equal to BA, it’s because you haven’t yet learned what matrix multiplication is good for in math, physics, and many other subjects. It’s ironic that Witt invokes the notion of symmetry to disparage matrix multiplication, since matrix theory and an allied discipline called group theory are the tools mathematicians use in fleshing out our intuitive ideas about symmetry that arise in art and science.

So how did an intelligent person like Witt go so far astray?

PROOFS VS CALCULATIONS

I’m guessing that part of Witt’s confusion arises from the fact that actually multiplying matrices of numbers to get a matrix of bigger numbers can be very tedious, and tedium is psychologically adjacent to distaste and a perception of ugliness. But the tedium of matrix multiplication is tied up with its symmetry (whose existence Witt mistakenly denies). When you multiply two n-by-n matrices A and B in the straightforward way, you have to compute n2 numbers in the same unvarying fashion, and each of those n2 numbers is the sum of n terms, and each of those n terms is the product of an element of A and an element of B in a simple way. It’s only human to get bored and inattentive and then make mistakes because the process is so repetitive. We tend to think of symmetry and beauty as synonyms, but sometimes excessive symmetry breeds ennui; repetition in excess can be repellent. Picture the Library of Babel and the existential dread the image summons.

G. H. Hardy, whose famous remark Witt quotes, was in the business of proving theorems, and he favored conceptual proofs over calculational ones. If you showed him a proof of a theorem in which the linchpin of your argument was a 5-page verification that a certain matrix product had a particular value, he’d say you didn’t really understand your own theorem; he’d assert that you should find a more conceptual argument and then consign your brute-force proof to the trash. But Hardy’s aversion to brute force was specific to the domain of mathematical proof, which is far removed from math that calculates optimal pricing for annuities or computes the wind-shear on an airplane wing or fine-tunes the weights used by an AI. Furthermore, Hardy’s objection to your proof would focus on the length of the calculation, and not on whether the calculation involved matrices. If you showed him a proof that used 5 turgid pages of pre-19th-century calculation that never mentioned matrices once, he’d still say “Your proof is a piece of temporary mathematics; it convinces the reader that your theorem is true without truly explaining why the theorem is true.”

If you forced me at gunpoint to multiply two 5-by-5 matrices together, I’d be extremely unhappy, and not just because you were threatening my life; the task would be inherently unpleasant. But the same would be true if you asked me to add together a hundred random two-digit numbers. It’s not that matrix-multiplication or number-addition is ugly; it’s that such repetitive tasks are the diametrical opposite of the kind of conceptual thinking that Hardy loved and I love too. Any kind of mathematical content can be made stultifying when it’s stripped of its meaning and reduced to mindless toil. But that casts no shade on the underlying concepts. When we outsource number-addition or matrix-multiplication to a computer, we rightfully delegate the soul-crushing part of our labor to circuitry that has no soul. If we could peer into the innards of the circuits doing all those matrix multiplications, we would indeed see a nightmarish, Borgesian landscape, with billions of nails being hammered into billions of boards, over and over again. But please don’t confuse that labor with mathematics.

Join the discussion of this essay over at Hacker News!

This essay is related to chapter 10 (“Out of the Womb”) of a book I’m writing, tentatively called “What Can Numbers Be?: The Further, Stranger Adventures of Plus and Times”. If you think this sounds interesting and want to help me make the book better, check out http://jamespropp.org/readers.pdf. And as always, feel free to submit comments on this essay at the Mathematical Enchantments WordPress site!

ENDNOTES

#1. Note the New Yorker-ish diaresis in “preëmptively”: as long as I’m being critical, I might as well be diacritical.

#2. I know this convention may seem backwards on first acquaintance, but this is how ◦ is defined. Blame the people who first started writing things like “log x” and “cos x“, with the x coming after the name of the operation. This led to the notation f(x) for the result of applying the function f to the number x. Then the symbol for the result of applying g to the result of applying f to x is g(f(x)); even though f is performed first, “f” appears to the right of “g“. From there, it became natural to write the function that sends x to g(f(x)) as “g ◦ f“.

#3. Here’s one point on which I can sympathize with Stephen Witt: matrix multiplication would be prettier if the product of two matrices were just a matter of multiplying each entry in the first matrix by the corresponding entry in the second matrix:

This kind of product is called the Hadamard product and it does play a small role in mathematics, but it’s nowhere near as common as the usual matrix product. The Hadamard product is too symmetrical to be useful, whereas the usual matrix product strikes a perfect balance between simplicity and versatility.

There is a subclass of matrices for which the Hadamard product and the standard product coincide, namely, the class of diagonal matrices. Here’s how diagonal matrices multiply:

In the world of neural networks, such matrices correspond to a trivial kind of data-processing in which every output variable is simply a specific input variable multiplied by a constant. There’s no cross-talk or interaction between variables. What makes matrices-in-general more useful than diagonal matrices is that with a general matrix, every output is potentially affected by every input.

To put things in a grandiose but not entirely inaccurate way, matrices are the first thing you should consider using to model a situation in which you don’t know ahead of time which outputs depend on which inputs. Of course one shouldn’t expect matrices to be a panacea; after all, linear algebra requires that every output be a linear function of the inputs (hence the name). Linearity is a heavy constraint. The beautiful miracle is that, despite this constraint, linear algebra is such a useful tool in all the sciences.

I’m a great believer in low-tech math. I don’t like to rely on things a computer tells me; what if there’s a bug in the code? I prefer trusting things that I can check for myself. At the same time, I’m keenly aware of the limits of my imagination even when it’s aided by paper and pencil. Sometimes I need a computer to show me things I can conceive of but can’t see.

ILLUSTRATING MATH TOGETHER

In 2016, the Institute for Computational and Experimental Research in Mathematics (ICERM) in Providence, Rhode Island hosted a workshop called Illustrating Mathematics with the hope of bringing together researchers who, like me, study mathematical abstractions that can be brought to life by appropriate visuals. The workshop spawned a community that has held meetings at ICERM from time to time and has been running a webinar series since 2023.

I’ve spoken twice at the webinar. Back in 2024, I gave a brief mathematical eulogy for the brilliant mathematical explorer Roger Antonsen, now sadly deceased (though you can’t tell that he’s deceased from his website), who had a unique knack for coming up with cool visuals related to every topic we ever discussed. The striking figure below, arising from a deterministic model of a one-dimensional gas I’d proposed, is just one instance among many dozens he created as part of our email conversations.

On October 10th, 2025, I spoke at the webinar for a second time, even more briefly: I gave a five-minute “show-and-ask” pitch as a warmup-act for the phenomenal math explainer/animator Grant Sanderson (aka 3Blue1Brown). My lightning talk was entitled “Evolving cross sections of Ford spheres”, and it was my way of testing the waters of the webinar crowd. I wondered: if I described a compelling mathematical object that nobody has illustrated yet in a fully satisfying manner, or at least not in a way that I find fully satisfying, and I shared with other webinar attendees my vision of how one could make that mathematical object more available to the brain by way of the eye, then could I convince others, more skilled than I in the art of computer-assisted illustration, to bring my vision into reality?

The answer proved to be a resounding “Yes!” Roice Nelson (with whom I’ve corresponded in the past) was one of several people who expressed interest, and Roice and I have moved forward with this project. Arguably I shouldn’t be spending my time this way—I don’t plan to write any research articles on the Ford spheres. I just think that they’re cool things that other people would find interesting if they were better publicized. And they got stuck in my head like a catchy tune.

AN 87-YEAR-OLD FRACTAL

I’m sure you’ve heard of fractals—they had a moment back in the 1980s that basically never ended, with fractals penetrating not just the sciences and geek culture but popular culture as well, culminating in a line about frozen fractals in a stirring power ballad in the 2013 Disney movie Frozen. The Ford spheres form a three-dimensional fractal that not enough mathematicians know about, even though Lester Ford described it in a charming article called, simply, “Fractions”, back in 1938—thirty-seven years before Benoit Mandelbrot coined the term “fractal”.

There are actually many arrangements that are called Ford sphere arrangements nowadays, but the one Ford himself described looks like this:

This image is a still from a video made by Sam Wells and Aidan Donahue. The video gives some intuition for the fractal, but (to quote another Disney movie heroine) I want more.

What makes the Ford spheres worthy of study? From a research perspective, they’re descendants of a more famous two-dimensional fractal Ford wrote about in his 1938 article. The Ford circles are geometrical surrogates for the rational numbers, and the way the circles nestle against one another turns out to reflect important facts in number theory.

It stands to reason that the analogous three-dimensional fractals would have secrets to teach us as well.

ALL OVER THE PLACE BUT ALMOST NOWHERE

Another thing that makes the Ford spheres worthy of illustration is the way they offer math-loving non-mathematicians the chance to have their minds blown by the counterintuitive behavior of countable dense sets. The primordial example of such a set is the set of rational numbers: as elements of the real line, the rational numbers are all over the place but they’re also almost nowhere. I’ve chosen my phrasing to be provocative and a little paradoxical, but in a certain mathematical sense, it’s true: hardly any real numbers are rational, but no tiniest stretch of the real line is free of them. If you zoom in on (say) the square root of two, no matter how far in you zoom, you’ll keep on seeing rational numbers with ever-bigger numerators and denominators. Ford circles give geometric meaning to that bigness: the bigger those numerators and denominators are, the tinier the corresponding circles are.

All the Ford circles are tangent to a single horizontal line. One way to think about the Ford circles is as what you get when you try to pack together as many circles as you can above that line. You start by drawing evenly-spaced circles tangent to the number line at the points . . . , −2, −1, 0, 1, 2, . . . I’ll just show the two circles that touch the line at 0 and 1 and hereafter ignore all the circles to the left or right of them:

Then you add a circle to fill the gap between the 0-circle and the 1-circle, tangent to the line at 1/2):

Then you add more circles to fill the new gaps with tangencies at 1/3 and 2/3:

Then you add even more circles to fill the newer gaps with tangencies at 1/4, 2/5, 3/5, and 3/4:

If you continue this process, the circles you’ll draw are precisely the Ford circles, all tangent to the line, and the points of tangency will be all the rational numbers and nothing else.

Now imagine that, having drawn in the Ford circles (or as many of them as you have the patience to draw), you add to your picture a horizontal line parallel to, but slightly above, the line we were talking about before. This new line will intersect some of the circles. If you move that new line downward slightly, it’ll intersect more of the circles. As you continue to move the new line further downward, closer and closer to the original line (which I’ll call the “limit line”), you start to intersect more and more circles.

FROM TWO TO THREE

Ford also described a similar fractal one dimension up. We have infinitely many spheres, all tangent to the x, y plane, and the points of tangency correspond exactly to the points (x, y) with x and y rational. Here’s Ford’s sketch showing four of the infinitely many spheres:

(Yeah, four is a lot less than infinity, but cut Ford some slack: this was before computers.)

I want to picture this complicated three-dimensional object by way of its two-dimensonal cross-sections. Here’s one of the animations Roice sent me a few days ago, as part of our ongoing work:

It shows what you get when you intersect the Ford spheres with a moving plane that approaches, without ever reaching, the limit plane that all the Ford spheres touch (analogous to the limit line that all the Ford circles touch). As time passes in the video and the moving plane moves on, we see a mix of growing disks and shrinking disks; the shrinking disks are cross-sections of the spheres whose centers the plane has already passed through, while the growing disks are cross-sections of the spheres whose centers still lie just a bit ahead of us. The picture becomes frothier and frothier. Of course the video stops short of infinite fractal frothiness; once the circles become too small to see, time reverses and the cutting-plane changes direction, until we end up back where we started.

The video is just a rough cut, but already I can see features of the image that I didn’t expect: halos and solar arches, one might call them. Perhaps one of you will find a way to make rigorous mathematics of what your eyes are telling you, but even if not, I hope the animation gives you visual pleasure.

WHY I BOTHER

If this essay inspires any of you to drop by one of the monthly meetings of the Illustrating Math webinar, visit the webinar link. Or, if you’re feeling brave and want to pitch an idea or to give a five-minute presentation of any kind, go to the show-and-ask signup sheet. Or if you just want to see what other cool visuals Roice has created, check out his website.

I realized shortly before I published this essay that there is a connection to my research, though it’s not a direct link, and that it was probably subconsciously driving me to explore the Ford spheres. Two decades ago I was looking at “rotor-router blobs” that gave rise to images like this one, generated by Tobias Friedrich and Lionel Levine:

If you’re like me, your eyes and brain see ghostly circles (or near-circles), forming bands separated by lines of orange fire. The trouble is, those near-circles are very much creations of your eyes and brain, intermediated by software called ImageMagick. The task of figuring out what details at the pixel-level create ghostly near-circles in my brain defeated me. Such are the frustrations of “digital pointillism”: when we zoom in, we tend to lose sight of what we are trying to understand! It’s the problem faced by creators of monumental paintings: you have to stand close to the canvas to paint your strokes or dots or whatever, but when you stand close it’s easy to literally lose sight of the big picture. I’d like to try looking at those blobs again sometime, once I have the tools and the skills to “interrogate” such pictures more effectively than I could in the past.

A smaller-scale version of this gap in my skill-set manifests itself for the Ford spheres. Those halos and solar arches exist in my brain (and I hope in yours), but what do they correspond to at the pixel level? I don’t know how to ask the picture to tell me, but I’m hoping I can learn.

I’ll finish by mentioning one last reason for bringing the Ford spheres from the world of fancy to the world of the senses: videos like these can convey to non-mathematicians, in a way that words and symbols can’t, what makes math so addictive to those of us who love it.

Thanks to David Jacobi and Roice Nelson.

REFERENCES

L. R. Ford, Fractions, American Mathematical Monthly, 45, 586–601.

S. Northshield, Ford circles and spheres, 2015.

C. Pickover, Beauty and Gaussian Rational Numbers, Chapter 103 (pages 243-247) in: “Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning”, Oxford University Press, 2001.

S. Wells and A. Donahue, Ford spheres, 2021.

As a member of the Advisory Council for the National Museum of Mathematics (“MoMath”) over the past decade, I’ve had a number of unique opportunities, such as the thrilling chance to improve the Museum’s datebase via my smartphone and watch exhibit-content update in real-time, and the less thrilling opportunity to break an exhibit on the museum’s opening day (buy me a coffee and I’ll confess to you that shameful episode from my past). But the opportunity I’m writing about today is one that’s still playing out: the chance to play a role in creating a new kind of Math Thing, namely, a programmable quincunx.

If you look up the word “quincunx”, you’ll find that one definition is this arrangement of five dots:

But I’m talking about the kind of quincunx that looks like this:

You’ll notice that the balls piled up at the bottom form a bell-shaped curve, reminiscent of the normal curve from statistics:

This isn’t a coincidence; the quincunx was designed to illustrate statistical principles in general and the Gaussian distribution in particular. Many science museums have a quincunx, but MoMath was unique in having an adjustable quincunx in which a lever allowed users the chance to introduce biases at the junctions, making it more likely for balls to go to the left or to the right, and correspondingly shifting the bell-shaped curve to the left or to the right.

I wrote “MoMath was unique …”, not “MoMath is unique …”, because when MoMath’s lease at its old location on 26th Street ran out, it moved to a temporary smaller location on Fifth Avenue, and the adjustable quincunx wasn’t included in the downsized museum. But in 2026 a bigger-than-ever MoMath on Sixth Avenue will feature something new under the sun: a customizable quincunx, in which each junction will have its own individual bias, and in which the distribution of the balls at the bottom won’t necessarily be a Gaussian at all, but a curve of your own devising.

As a user of the exhibit, you won’t control the biases of the junctions directly; instead, you’ll draw your own not-necessarily-bell-shaped curve and a computer will adjust the biases in such a fashion that, when the balls fall randomly, the distribution that the balls form in their bins will closely match the curve you’ve drawn. Hence the exhibit’s name, Draw Your Own Conclusions (“DYOC” for short).

With the DYOC, you’ll be able to enter even a perverse shape like an upside-down bell-shaped curve and the algorithm will adjust the biases to produce the distribution you specified.

My role in the project was figuring out how the biases at the pins should be adjusted to achieve the desired distribution at the bottom. In the process of designing an algorithm for this, I discovered that in a way, what’s needed is a curious kind of sequential compression of one-dimensional images.

I gave a 20-minute talk about DYOC earlier this month at the MOVES conference on recreational mathematics (“MOVES” stands for “the Mathematics Of Various Entertaining Subjects”) hosted by MoMath. In some ways that talk was an expanded version of a talk that I gave back in 2014 at the 11th Gathering for Gardner conference. Here’s a link to the video of that talk:

https://www.youtube.com/watch?v=LDr8c2NmDDA

If that talk leaves you hungry for more details, here’s a link to the video of my MOVES talk, as well as a link to my slides:

https://faculty.uml.edu/jpropp/moves25a.mp4

https://faculty.uml.edu/jpropp/moves25a.pdf

Unfortunately the price-tag for an actual programmable quincunx is still too high, so the 2026 version of the exhibit will be a virtual mock-up preserving much of the user experience of the original DYOC concept. The math will be the same, but the balls will be simulated. Hopefully some people with good mathematical taste and flush bank accounts will come to the Museum in its new Sixth Avenue home and be so inspired by the virtual exhibit that they’ll fund the construction of a physical one!

I’ll report on later stages of the project as things evolve. Stay tuned.

In ordinary math, the infinite decimal .999… is defined to be the limit of the terminating decimals .9, .99, .999, …; that is, it’s defined to be the real number that the fractions 9/10, 99/100, 999/1000, … approach in the ordinary sense. And that limit is most definitely 1, not some real number that’s a tiny bit less than 1. This is not an approximate truth; it’s a 100% accurate, rigorously established mathematical fact. It’s a part of how the real number system works, and it’s a feature, not a bug.

But if you’ve read my essay “Marvelous Arithmetics of Distance”, you already know that there are number systems in which things that look like rational numbers behave differently1, and you won’t be too surprised to learn that there’s another new number-game to play. It’s the q-deformed real number “game” of Sophie Morier-Genoud and Valentin Ovsienko, and in the context of their work, it becomes true (in an admittedly somewhat arcane sense) that 9/10, 99/100, 999/1000, etc. do not approach 1 but rather approach something smaller. Except that it’s not the numbers themselves that behave in this ill-bred way; it’s their avatars in the q-deformed world, avatars that Morier-Genoud and Ovsienko write as [9/10]q , [99/100]q , [999/1000]q , etc.

In this essay, without going too deeply into the underlying theory, we’ll take a concrete look at the q-deformations of the numbers 1/2, 2/3, 3/4, etc. and of the numbers 2/1, 3/2, 4/3, etc. In the ordinary real number system, all these fractions approach 1 as the numerators and denominators get big, regardless of whether the bigger number is on top of the fraction or on the bottom. But looking at these rational numbers through the spectacles that Morier-Genoud and Ovsienko have given us, we’ll see that we get two different limits according to whether the numerator is bigger than the denominator or vice versa. In particular, we’ll find that, while the q-deformations [2/1]q, [3/2]q, [4/3]q, etc. approach [1]q, the q-deformations [1/2]q, [2/3]q, [3/4]q, etc. approach [1]q’s evil (or maybe not so evil) twin.

Morier-Genoud and Ovsienko’s work continues a centuries-old tradition of taking facts about numbers and upgrading them to facts about functions, often called generating functions in this context. Traditionally, the independent variable in these functions is called q. For instance, a common q-analogue of the number n! (defined as the number 1×2×3×···×n) is the polynomial function (1)(1+q)(1+q+q2)···(1+q+q2 +···+qn−1), the “q-factorial” of n, which becomes n! when you replace q by 1.2 Some ways of sticking a q into a formula work; others don’t. Seeking the right q-analogue of some known bit of math can be an adventure. Certain mathematicians, seeing their colleagues succumb to the temptation of adding an extra variable or two, jestingly refer to “the q disease”. And now I’m infecting you with it.3

If this were a more historical essay I’d talk about where q comes from, as a confluence of eighteenth century combinatorics, nineteenth century number theory, and twentieth century physics. But I’m assuming you’re more interested in math than math history, so let me just assure you that the q you’re about to meet has an illustrious pedigree.

RAMBLING THROUGH THE RATIONALS

There are several ways to define the q-deformed rational and q-deformed real numbers, none of them easy (which is probably why they weren’t discovered till recently). But it’s possible to play with them without defining them conceptually, by giving rules for computing them. Each of our new “numbers” is going to be a finite or infinite expression involving a variable q; we call q a “formal indeterminate” as a way of signaling that you’re not supposed to regard q as having any particular value. For instance, [1/3]q = q2/(1+q+q2). The kind of function of q that we see when we look at the q-deformed rational numbers is what’s called a rational function; it’s a function of q that can be computed using only addition, subtraction, multiplication, and division. The definitions of [n]q when n is an integer is centuries old, but the correct way to define [r]q when r is not an integer only came to light in the past decade, through the work of Morier-Genoud and Ovsienko.

Here are three rules that taken together and patiently applied will allow you to compute [r]q for any rational number r:

(1) [0]q = 0

(2) [r+1]q = q[r]q + 1

(3) [r]q [−1/r]q = −1/q

For instance, applying (2) with r = 0 and using (1) you can calculate

[1]q = 1

Now applying (2) with r = 1 you can calculate

[2]q = 1+q

Now applying (3) with r = 2 you can calculate

[−1/2]q = −1/(q+q2)

Next applying (2) with r = −1/2 you can calculate

[1/2]q = q/(1+q)

And so on, round and round. You can ramble through the rational numbers for a long time, and you may encounter some rational numbers you’ve seen before, but the rules will never lead to a contradiction. For instance, by applying rules (2) and (3) in alternation, you can go from 1/2 to 2/3 to −2/3 to 1/3 to −3 to −2 to 1/2, and you’ll find that our rules bring us back to [1/2]q = q/(1+q). Here’s a picture of what sorts of steps we can take in the rational numbers, where I’ve drawn the rational numbers on a “number circle” rather than on the usual number line, with 0 at the bottom. Blue edges correspond to rule (2) and red edges correspond to rule (3).

The surprising consistency of these rules when we go around a loop is a clue that something important is going on.

Take a look at what we’ve learned so far:

[1]q = 1

[2]q = 1+q

[1/2]q = q/(1+q)

[−1/2]q = −1/(q+q2)

Notice that if you replace q by 1, [r]q simplifies to r. This is what we mean when we say that [r]q is a “deformation” of r; when q is 1, [r]q is just good-old r.

BOTH SIDES NOW

With rules (1), (2), and (3), you can show that

[1/2]q = q/(1+q)
[2/3]q = (q+q2)/(1+q+q2)
[3/4]q = (q+q2+q3)/(1+q+q2+q3)
…

and that

[2/1]q = (1+q)/1
[3/2]q = (1+q+q2)/(1+q)
[4/3]q = (1+q+q2+q3)/(1+q+q2)
…

Let’s look at the improper fractions first. Multiplying numerators and denominators by 1−q, we can rewrite [2/1]q as (1−q2)/(1−q), [3/2]q as (1−q3)/(1−q2), [4/3]q as (1−q4)/(1−q3), and more generally rewrite [(n+1)/n]q as (1−qn+1)/(1 − qn). What can we say about how [(n+1)/n]q behaves when n gets large, assuming that q is some fixed number strictly between −1 and 1? In this case qn and qn+1 both approach zero, so the fraction (1−qn+1)/(1−qn) approaches (1−0)/(1−0), or 1. And remember, [1]q is also 1. So [2/1]q, [3/2]q, [4/3]q, . . . converge to [1]q , as we would hope. Pictorially:

It’s the proper fractions that don’t behave properly when −1 < q < 1. The source of the impropriety is the way that the sums in the numerators of [n/(n+1)]q start with “q+q2+…” instead of “1+q+…”. Multiplying numerators and denominators by 1−q, we can rewrite [1/2]q as (q−q2)/(1−q2), [2/3]q as (q−q3)/(1−q3), [3/4]q as (q−q4)/(1−q4), and more generally rewrite [n/(n+1)]q as (q−qn+1)/(1−qn+1). As n gets large, the fractions (q−qn+1)/(1−qn+1) approach (q−0)/(1−0), or q. So the q-deformed fractions [1/2]q, [2/3]q, [3/4]q, … don’t converge to [1]q, but rather to a different function of q, namely, q itself. Pictorially:

Likewise you can show that [1/2]q , [1/3]q , [1/4]q , . . . approach [0]q but that [−1/2]q , [−1/3]q , [−1/4]q , . . . approach a different function of q, namely 1−1/q (though we have to be careful not to let q equal 0).

Morier-Genoud and Ovsienko show that as long as the rational numbers r1, r2, r3, … approach some rational limit s from above (or, in number-line terms, from the right), their q-deformations will behave well: the functions [r1]q, [r2]q, [r3]q, … will approach the function [s]q, at least when q is close enough to 0. But if r1, r2, r3, … approach the rational limit s from below (or, in number-line terms, from the left), their q-deformations [r1]q, [r2]q, [r3]q, … will not approach the function [s]q but will instead approach a different deformation of s – a function of q that is actually smaller than [s]q when q is near 0, even though the two functions agree at q = 1. Researchers Asilata Bapat, Louis Becker, and Anthony Licata represent this “twin” of [s]q by the symbol [s]♭q, and write the ordinary [s]q as [s]♯q when they want to stress the parallelism.

FILLING THE HOLES

So far I’ve only talked about the q-deformed rational numbers, but what about q-deformed irrational numbers, filling the holes in our strange new number line? What happens when the rational numbers r1, r2, r3, … approach an irrational limit s? In that case, it turns out not to matter whether the rn’s approach s from the left, from the right, by oscillation, quickly, slowly, whatever – in every case, the q-deformed rationals [r1]q, [r2]q, [r3]q, … will converge to one specific function of q, which Morier-Genoud and Ovsienko call the q-deformation of the irrational number s, written as [s]q.

For example, just as the fractions 2/1, 3/2, 5/3, 8/5, . . . (ratios of consecutive Fibonacci numbers) converge to the celebrated golden ratio

Φ=(1+sqrt(5))/2,

the q-deformed fractions [2/1]q, [3/2]q, [5/3]q, [8/5]q, … converge to the function

[Φ]q = (q2+q−1+sqrt(q4+2q3−q2+2q+1))/(2q)

For more of a woo-woo vibe, you can call this “quantum Φ” instead of “q-deformed Φ”.

Want to learn about quantum sqrt(2) and quantum e and quantum π? Check out what Morier-Genoud and Ovsienko have written. Or you can learn about the q-deformed real numbers by watching online talks the two have given courtesy of the One World Numeration Seminar.

DEJA VU?

The division of the q-rational numbers into sharps and flats is curiously reminiscent of many students’ mistaken beliefs about infinite decimals, and in particular the belief that even though the sequence 1.1, 1.01, 1.001, … converges to 1, the sequence 0.9, 0.99, 0.999, … does not converge to 1 but rather converges to 0.999… conceived of as a “smaller 1”. It may also remind some of you of the literal decimal system I described in my essay “More About .999…”); in that system, 0.999… and 1.000… are genuinely different. So you might be tempted to relate the q-deformed rationals to the literal decimals. But there’s an important mismatch between the q-deformed integers and the literal decimal system. In the literal decimal system, some rational numbers (like 1/2 and 1/5) have an evil twin but others (like 1/3) don’t; different rational numbers are treated differently according to how their denominators factor into primes, with 2 and 5 (the prime factors of 10) being accorded special treatment. In contrast, the society of q-deformed rational numbers is an egalitarian one: every rational number, regardless of numerator or denominator, has two avatars, one ever so slightly smaller than the other.

A more important difference is that I brought the literal decimals out of obscurity and into the blogosphere not because they’re useful (they’re not) or beautiful (I don’t think so) but because I wanted to make a point about the freedom of mathematics: you can change the rules as long as your new rules are consistent and you’re honest about having changed the rules. But just because you’re allowed to define a new number system doesn’t mean that you should, or that if you do, that anybody else should care! In comparison, the q-deformed rational number system is “real mathematics”, with connections to topics like continued fractions, quantum groups, and knot theory. The q-rationals wanted to be found, and one of the more amazing things about them is that nobody found them sooner.

If you’re looking for precedents for the strange behavior of the sharps and flats, a better analogy can be made with Dedekind cuts, as described in my essay “Dedekind’s Subtle Knife”. When we construct the real numbers via Dedekind cuts of rational numbers, we get a single cut corresponding to each irrational number, but if we’re not careful we’ll wind up with two copies of each rational number r, corresponding to the two ways to cut the rational number-line at r (with r itself going into either the left set or the right set). There are ways to fix this when one constructs the reals via Dedekind cuts, but in the case of the q-deformed rational numbers, we don’t view the duplication of the rationals as a bug to be fixed; we accept it as a fact of life in our strange new number system, just as non-duplication (e.g., .999… = 1) is a fact of life in the ordinary reals.4

FURTHER WEIRDNESS

One fascinating feature of the q-deformed real numbers is that, when they’re expressed as power series around 0 in the variable q, all the coefficients are integers; for instance, the q-deformed golden ratio [Φ]q expands as

1+q2−q3+2q4−4q5+8q6−17q7+37q8−82q9+185q10−423q11+978q12−…

You may not recognize the sequence of coefficients, but the Online Encyclopedia of Integer Sequences does, and http://oeis.org will gladly tell you that this is a sequence of “generalized Catalan numbers”, which count something. So the golden ratio – not a rational number, let alone a counting number! – has hidden combinatorial meaning.

Even though the power series for [Φ]q diverges when you plug in q = 1, evaluating the algebraic expression for [Φ]q at q = 1 still gives Φ and still has a sort of combinatorial meaning, best expressed in probabilistic language: it says that if you take a half-infinite strip of height 2 that starts at the left and goes infinitely far to the right and you tile it by dominos “at random” (whatever that means!) with no gaps or overlaps, then the odds are Φ-to-1 that the two leftmost squares are covered by a single vertical domino (as in the picture) rather than two horizontal dominos.

THE TWISTY ROAD AHEAD

The study of q-deformed rational numbers and q-deformed real numbers is still in its infancy. One thorny challenge is bringing ordinary addition and ordinary multiplication fully into the q-deformed world, by which I mean devising operations +q and ×q on the power-series representations of the q-functions so that [r]q +q [s]q = [r+s]q and [r]q ×q [s]q = [rs]q . For instance, since [Φ]q is 1+q2−q3+2q4−…, you might think that [2Φ]q should be 2+2q2−2q3+4q4−…, or perhaps be 1+q times 1+q2−q3+2q4−…, but no: the power series expansion of [2Φ]q begins 1+q+q2+q7−q9−….

It’ll probably be years before we can say we understand what the q-deformed real numbers are trying to tell us, and possibly decades before we can say what they’re good for. But I want to spread the word about this work now because the q-deformed reals offer a kind of partial vindication to anyone who ever learned to live with “.999… = 1” but never learned to like it, or who learned to like it but still has fond memories of .999… from the days when it seemed to be its own numinous thing rather than just 1 trying to look mysterious. Of course every mathematician must at some point in their education come to grips with the topology of the real line and take to heart the limitations of infinite decimals as stand-ins for the real numbers themselves. But I think it’s fun that there’s a respectable kind of math in which the sequence [1/2]q, [2/3]q, [3/4]q, … and the sequence [2/1]q, [3/2]q, [4/3]q, … converge to different limits.

It just goes to show that the mathematical universe is weirder than you think it is, even when you know that it’s weirder than you think it is.

Thanks to Sophie Morier-Genoud, Valentin Ovsienko, Sandi Gubin, and Nick Ovenhouse.

ENDNOTES

#1. For instance, the infinite sum 1+2+4+8+… diverges in the real numbers but converges 2-adically to −1.

#2. If you take the formula (a+b)!/(a!b!) for binomial coefficients and replace all three factorials by q-factorials, you get “q-binomial coefficients”, which play a role in the “q-binomial theorem”. And so on.

#3. As R. A. Lafferty says to the reader near the start of his novel “Fourth Mansions”: “You who glanced in here for but a moment, you are already snake-bit! It is too late for you to withdraw. The damage is done to you. … Die a little. There is reason for it.”

#4. The sharps and flats together have nice convergence behavior that mimics the way the two kinds of rational cuts behave under the set-theory operations of union and intersection. When a decreasing sequence of rational numbers rn converges to some rational number s, then the [rn]♯q’s converge to [s]♯q and the [rn]♭q’s also converge to [s]♯q. On the other hand, when an increasing sequence of rational numbers rn converges to some rational number s, then the [rn]♯q’s converge to [s]♭q and the [rn]♭q’s also converge to [s]♭q.

REFERENCES

Asilata Bapat, Louis Becker, and Anthony M. Licata, q-deformed rational numbers and the 2-Calabi–Yau category of type A2, https://arxiv.org/abs/2202.07613.

Thomas McConville, James Propp, and Bruce Sagan, Hyperbinary partitions and q-deformed rationals, Forum of Mathematics, Sigma 14 (2025), https://www.cambridge.org/core/journals/forum-of-mathematics-sigma/article/hyperbinary-partitions-and-qdeformed-rationals/53D08B4727DE6CB12C950A1E0586CC8E.

Sophie Morier-Genoud and Valentin Ovsienko, On q-deformed real numbers, Experimental Mathematics (2019); https://arxiv.org/pdf/1908.04365.

Sophie Morier-Genoud and Valentin Ovsienko, q-deformed rationals and irrationals, preprint (2025); https://arxiv.org/abs/2503.23834.

This past week I was saddened to learn of the death of mathematician and teacher David C. Kelly, the founder of the Hampshire College Summer Studies in Mathematics program (HCSSiM). “Kelly”, as everyone called him, had a huge impact not just on my career but on the careers of people spanning several generations.

I knew Kelly for nearly fifty years. At the time we met I was a high school student who’d done well enough in inter-school math competitions to earn a spot on the Nassau County team, and when I and my team-mates went to the Atlantic Regional Math League competition, Kelly was there, spreading the word about HCSSiM. I thought he looked remarkably like Kurt Vonnegut, though not everyone agreed. Judge for yourself:

The “17” in the background in the former picture is important, as you already know if you read my essay “Will ’17 Be the Year of the Pig?” And if you haven’t read that essay, and you’ve wondered why I post my blog on or around the 17th of each month … well, read that essay.

I don’t have anything to write about the summer program that I didn’t already write back in 2017, but I do want to share my two favorite Kelly stories. I’m sure I’ll get some details wrong, and alas, Kelly isn’t around to set me straight, but I think he would agree that my versions are true in spirit. (And if any of you have corrections, please post them in the Comments!)

The first, longer story takes place around 1960 when Kelly was a student at Princeton. He was a math major and his roommate was a history major, and they thought it would be fun to take two courses together, one course in each of their specialties. The roommate wasn’t surprised when Kelly got a higher score on the math midterm, but was annoyed that Kelly also outscored him on the history midterm. “How’d you do it?” the roommate demanded.

Kelly explained how he’d managed to do so well on the four-question essay-format history test. The first thing he’d done while studying was come up with six questions on the material that seemed like the sort of questions the professor would put on the exam. Then he composed responses to all six questions and memorized them. When he took the exam, he saw that one of the exam questions was the same as one of his six, so he just regurgitated the essay he’d composed for that question. Another one of the exam questions was similar to another one of his six, so he was able to use most of what he’d composed in response to that question. This left him with plenty of time to write answers to the other two assigned questions.

The roommate was impressed. “I’m going to try that for the final. Come to think of it, why don’t we team up? I’ll write six, you write six—we memorize all twelve. We’ll crush it.”

So that’s what they did — though you don’t have to be a Princeton student to spot the flaw in their plan.

A few days after they took the exam, the summons came. They were told to report to the dean’s office at 9am sharp. When they showed up, the dean and their professor were both waiting for them, looking none too happy.

The dean showed the boys their identical answers. “This is a flagrant violation of the honor system,” said the dean. “I don’t care which of you copied from the other. Clearly there was collusion. I’m inclined to expel you both.”

“But there’s one thing that puzzles me,” said the professor. “The boys sat on opposite sides of the room and both of them stayed in their seats the whole time. They must have invented some new way to communicate during the exam.”

“Interesting,” said the dean. Then he turned toward the boys. “If you tell us how you did it, we’ll go easy on you — just suspend you for the semester instead of expelling you.”

“But we didn’t break any rules,” insisted Kelly. Then he explained what they’d done.

The dean was sure their scheme violated some rule, and he flipped through the rules governing Princeton’s honor system, searching for one.

Eventually he had to admit defeat. “I guess I can’t punish you, because technically you didn’t do anything wrong. But don’t ever let me catch you doing it again!”

(I’d be curious to know whether Princeton or any other schools have rules that forbid this kind of “exam preparation”. Of course a college student nowadays would probably get ChatGPT to compose all twelve essays.)

The second, shorter story is about something that happened in the 1970s during the school year. Kelly was sitting in a circle with other Hampshire College folks (a mix of faculty and students, I’m guessing) when someone took out a joint and started to pass it to the person to his right. Some joker in the circle said “No, don’t do that; there’s a cooler way, where everybody passes it to the person after the person to their right.”

“But then only half of us get it!”

“No, trust me,” said the joker, and the other folks good-naturedly indulged him. They started to get what was happening when the person immediately to the left of the “OS” (original smoker) received the joint and then, skipping over the OS, passed the joint to the person who’d been skipped at the start. After the joint had gone around the circle twice and returned at last to the OS, everyone had had exactly one toke.

“Whoa,” said the OS. “Far out.”

“Yeah,” said the joker. Then a puzzled look crossed his face. “Funny thing is, sometimes it works and sometimes it doesn’t.”

Thanks to Sandi Gubin.

“I couldn’t help but wonder…” — Carrie Bradshaw (in every episode of Sex and the City)

The best birthday party I ever had as a kid was a trip to the Museum of Natural History in New York City with half a dozen like-minded friends and my indulgent parents. The huge dinosaur skeleton in the main hall was impressive, but I was even more enchanted by the exhibits at the Hayden Planetarium. How intriguing it was to see a ball roll round and round the inside of a curvy funnel, evading its fate for what seemed like an eternity before finally falling into the hole in the middle, and how fun to wonder how the rules governing our universe not only allowed but mandated this behavior! How intriguing it was to see how much I would weigh on different planets, and how fun to wonder what that would feel like!

But as much as I enjoyed the planetarium, my childhood love of science was already secondary to my passion for math – the skeleton of the universe, you might call it. I probably would’ve enjoyed a trip to a math museum even more than a trip to a natural history museum. The trouble is, New York City didn’t have one.

That’s not true anymore. New York City now boasts one of the best mathematics museums in the world, the National Museum of Mathematics, informally called MoMath. With 19,000 square feet containing over three dozen exhibits, MoMath became a major attraction to NYC-area schoolchildren and tourists from all over the world when it opened in 2012. Sometime in 2026 it’ll be moving to a new location at 635 Sixth Avenue, where it’ll occupy either 36,000 or 46,000 square feet.

The square footage depends partly on you, as I’ll explain.

I wrote about the Museum of Mathematics in my very first Mathematical Enchantments essay “The lessons of a square-wheeled trike” (available as a WordPress document and as an audio file). This month I’ll focus on an exhibit called Beaver Run, explaining the mathematics behind the exhibit and describing how the exhibit grew out of the math. And I’ll say a bit about how the museum hopes to grow into something even bigger and better in the years ahead, with even more exhibits to enchant young minds.

CANALS AND CRYSTALS

I owe the versatile French priest Jean Sébastien Truchet (1657–1729) a debt that I didn’t discover until I was writing this essay: he was a font geek back before that was even a thing, designing fonts to a level of precision that was technologically infeasible until the modern era. One of Truchet’s creations was the ancestor of Times New Roman, one of my favorite fonts.1 But Truchet is more famous today for a style of ornamentation that grew out of his interest in hydraulics and more specifically his study of canals. In examining the ceramic tiles decorating France’s canal-network, Truchet was especially taken with square tiles that were divided into two contrastingly-colored isosceles right triangles. Playing with such tiles on his own, he noticed that when multiple tiles were placed in various orientations as part of a tiling of the plane by squares, many pleasing patterns were formed.

The metallurgist Cyril Stanley Smith, best known for his work on fissionable materials as part of the Manhattan Project, reinvented Truchet’s tiles as an outgrowth of his interest in crystalline matter. He later learned of Truchet’s work and modified Truchet’s ideas in various ways. In Smith’s most famous variation on Truchet’s theme, each tile is decorated with two quarter-circular arcs joining the midpoints of two adjacent sides of the square. When the tiles are placed together, the endpoints of the arcs on one tile match up with the endpoints of the arcs on the four adjoining tiles, forming visually arresting collections of loops.

It’s Smith’s tiles that nowadays are called Truchet tiles, at least in the mathematical community. I first learned about the tiles from Bernd Rümmler, who used them to ingenious effect to solve a problem I’d posed about Chris Langton’s virtual ant2, but that’s another story. Smith’s Truchet tiles have been popular with graphic artists because of their versatility. They’re so versatile that, in a development that Truchet-the-typographer would have found amusing, mathematician and designer David Reimann came up with a font based on them. Here’s a picture of a wall in the bathroom at MoMath’s original location. Can you find the hidden message MATH IS COOL?3

Of course, as a hydraulics expert from the early years of the Enlightenment, Truchet would also have found modern flush toilets extremely interesting.

ROTATING RAILCARS AND ROVING RODENTS

I first learned about MoMath when I heard mathematician and artist George Hart give a talk about some interesting puzzles he’d invented that he thought might be exhibited at a new museum he was founding along with mathematician-turned-stock-trader-turned-philanthropist Glen Whitney, joined by accountant-turned-math-educator Cindy Lawrence and architect-turned-designer-and-engineer Tim Nissen.

The talk took place at Microsoft’s Cambridge lab, not far from MIT’s building 20 where the hobby of recreational computing began as an outgrowth of the hobby of model train set design. The basement of building 20 was the home of the Tech Model Railroad Club, a haven for students interested not just in trains but in the switching systems that controlled them – switching systems that were influential in the design of the telephone network and the modern computer.

Hart was also working on an exhibit based on the idea of trains that run on Truchet tracks. By using a descendant of the switching systems found in the basement of Building 20, one could reconfigure the track even while the train was zipping along. (The proximity of Microsoft’s Cambridge lab to MIT’s Building 20 is irrelevant to the events of this story, but I find it satisfying.)

Consider a train, represented schematically by an arrow, about to move from one tile to another:

The train will enter the new tile along its left edge, but what happens next depends on which kind of tile it is. The train might exit the new tile along its upper edge:

Or the train might exit the new tile along its lower edge:

If a kid operating the exhibit switches between the two kinds of tile before the train gets there, the train is diverted, and hopefully the kid who switched the track will be diverted too.

Of course, you can’t really rotate a single square within a square tiling; each tile is locked in place by its neighbors. So we need circular turntables in the middle of each square:

Now imagine twenty-four turntables, as in the image below. There are 224 or about seventeen million different ways to configure them. Sometimes twiddling a knob causes two different loops to merge; other times it causes one loop to split into two. You could twiddle the knobs all day and never see the same configuration twice.

Also notice that you can’t make the train leave the exhibit; the turntables let you reconfigure the network, but the tracks always form a collection of loops.

In the picture above, each turntable is occupied by two trains. Let’s suppose we use just two of those forty-eight trains, moving at the exact same speed, and let’s further suppose that the mechanism is set up so that a turntable can never rotate while a train is actually on it. Then (as Geoge Hart understood, and you’ll understand too by the end of this essay), no amount of track-switching will lead to a collision.

The idea of having two trains running on a dynamically reconfigurable network of Truchet tiles had almost everything a good MoMath exhibit needed: it was gripping, it had never been done before, and it could expand kids’ ideas of what math could be. But what if some young visitors would be prone to take one look at it and think “Meh, a model-trains thing, not for me”? This might especially be the case for girls. In order for the exhibit to attract interest among a broader sector of the public, it would help if the trains were replaced by some sort of critter. The animal the team chose was, appropriately, the engineer of the animal kingdom, the beaver – also, coincidentally, the MIT mascot.

Now there was a vision of an exhibit called Beaver Run, featuring two beavers traveling through a landscape of ponds and evergreens, moving along reconfigurable tracks, gracefully gliding and somehow never colliding. All that remained was to fill in the engineering details.

UNDER THE SURFACE

Implementing those details proved not to be so simple. Even while Tim Nissen was trying to get Beaver Run running, he and his team were designing and building a few dozen other exhibits, each of which was its own engineering project. The ambition of MoMath’s founders was to open a museum in which, from Day One, every exhibit would be something never seen before in any museum. As a member of the Advisory Council, I respectfully told them I thought this was crazy. I tried to convince them that this would be like an auto-maker deciding to release a car with a new kind of motor, a new kind of transmission, a new kind of suspension, a new kind of safety system, etc., all at the same time. I advised them to do what other science museums had done, mixing a few home-grown innovations with a bunch that were tried and true. I urged them to hedge their bets and license some proven winners from the San Francisco Exploratorium. Fortunately, they didn’t listen to me. The museum was a hit from the day it opened its doors in 2012, and part of its appeal was its outrageous originality.

But there were some casualties of the let’s-invent-everything ethos, and Beaver Run proved to be one of them. The engineering challenges were too great, and the leadership team reluctantly decided that it would be best not to present Beaver Run to the public until the kinks in its design had been ironed out. The ironing took four years. If that seems like a long time to you, keep in mind that MoMath ran at capacity almost from the day it opened, and most of its visitors were middle-school children, who are not exactly gentle with museum exhibits. Exhibits are certain to break, so they have to be designed with the need for periodic repair in mind.

Also keep in mind that the theorem that guarantees that the beavers will never collide requires that the beavers move at the same speed. Even tiny deviations, over the course of a day, could lead to a collision and a “Closed for repairs” banner draped over the exhibit. You need the speeds of the beavers to match up to within one part in a million. How would you engineer the exhibit to achieve that much precision without breaking the bank? You’d probably decide pretty quickly that the cars need sensors that report to a central control system that can detect when the cars are getting out of sync and can make minute speed corrections, too small for visitors to notice but crucial for keeping collisions from happening. But now the design is becoming more complicated, with increased cost of fabrication and decreased mean-time-between-failures.

Four years after MoMath had opened its doors to the public, the team felt that the exhibit was ready, and Beaver Run had its debut in 2016.

DEATH AND REBIRTH

Although the exhibit was popular, it was plagued by mechanical difficulties. In 2019, the leadership team reluctantly removed it from the Museum floor. But Executive Director Cindy Lawrence didn’t put it in the trash. As she writes, “I couldn’t bear to throw it out, so I put it in storage near where I live. It languished there for years, and every time I went to get something out of storage, I would admire it and lament its sad ending.”

Then along came a math-loving New York City dad with math-loving kids, including two daughters who had especially loved that exhibit; he wanted to know what had happened to it and what it would take to fix it. Lawrence explained that it was no simple repair job; it would be necessary to basically start from scratch and re-do the whole exhibit. It would cost a lot of money. The dad said he would help pay for the redesign/rebuild of the exhibit. The Museum hired Richard Rew, who along with his son Oliver redesigned and rebuilt the exhibit and got it to work beautifully. Beaver Run went back into the Museum in 2024, and it’s been a popular exhibit ever since. How intriguing it is that no matter what you do, the beavers never meet, and how fun to wonder why!

MoMath might not have been able to redo the exhibit had two young girls not been smitten with it, and had their father not had both the vision and the resources to bring it back to life. Would the girls have loved it as much if there hadn’t been critters? Would they have identified as strongly with trains? When asked, the dad said he doubts it.

MoMath’s lease at 11 East 26th Street ran out about a year ago; the Museum has been housed in a temporary location at 225 Fifth Avenue since March 2024, pending relocation to larger quarters at 635 Sixth Avenue. The upgrade is giving the Museum the opportunity to expand and design new exhibits — one of which, Draw Your Own Conclusion, has been on leadership’s wish-we-could-build-this list for over a decade. The initial lease provided 36,000 square feet but the landlord subsequently offered to lease the Museum an additional 10,000 square feet that would bring the Museum up to 46,000 square feet in total.

MoMath signed the lease for 36,000 square feet and has been trying to raise funds for the additional 10,000 square feet. The Museum was halfway toward its capital goal when its prospective landlord at the new location made a surprising offer: the Museum could have the extra space rent-free for one year but only if the Museum signed the lease for the added space by the end of May, 2025 (just over a week from now). Here’s a link to the announcement.

Just as the future course of a beaver in the exhibit can be radically altered by a simple twist of a dial, the future of MoMath could be radically altered by what happens this month. If my essay leads to your making a donation to MoMath, tell ’em the beavers inspired you.

SEEING LIKE A MATHEMATICIAN

So, why don’t the two beavers ever collide? Since they move at the same speed, neither of them can rear-end the other, so what we’re really asking is, why won’t there ever be a head-on collision? It’s possible to imagine that, with enough patience or cleverness or pure dumb luck, you could start from a configuration that contains a tile like this

in which the beavers are passing each other, and turn it into a configuration that somewhere or other contains three tiles like this

in which the beavers are about to collide on the tile they’re poised to enter. It’s not that you can really imagine a particular way to get from the Before picture to the After picture, but you can certainly believe that you live in a universe in which something like this might happen.

But now, go back to the picture of the beavers’ world, subtract away all the distracting details of ponds and evergreens — actually, I’ve already done that for you in the preceding images — and color the terrain in your mind with the colors red and blue, so that the outer region is colored red and so that each stretch of track has red on one side and blue on the other. This isn’t something in the real world; it’s something you build in your head.

Now I can show you why the Before and After pictures can’t be stills from the same movie. In the Before picture, both beavers had Blue on their left and Red on their right or vice versa.

But in the After picture, one beaver has Blue on its left and Red on its right while the other has Red on its left and Blue on its right.

So there’s no way to get from the Before picture to the After picture, and the same is true for any collision scenario you can invent. If the beavers start with the same color on their right, that will remain true forever. There’s no way for them to get into a situation where they’re on the same loop traveling toward each other, since in that situation one beaver has red on its right and one has blue on its right.

That’s not quite the full proof. How do we know the red-blue coloring exists? That is, how do we know it’s self-consistent (unlike, say, a coloring of a Möbius strip in which one side is red and one side is blue, which is impossible)? Mathematicians might realize that there’s something to prove here, but most middle-schoolers will believe it intuitively, especially if you give them crayons and let them try some examples themselves. But we also need to understand why, in the course of the movie, our imaginary blue/red coloring of the rectangle doesn’t undergo drastic large-scale changes when a turntable gets rotated by 90 degrees. So we need one more picture, showing how, when we rotate a turntable on a tile, only the coloring of that tile changes; all the colors elsewhere remain the same, as shown in the picture below.

If neither beaver is on the tile whose turntable rotates, the imaginary colors to the right of each beaver won’t suddenly change. The imaginary blue/red coloring gives our mind’s eye a new way to look upon the scene that clarifies why collisions can’t happen.4

TRAINING THE MIND’S EYE

Remember the picture with the caption “Remember this picture. I’ll come back to it later”? It showed forty-eight beavers (I called them trains back then), and I wrote “Let’s suppose we use just two of those forty-eight trains.” I want you to pick two trains, I mean two beavers, in that picture, and then I want you to mentally apply blue and red paint to the scene, in the manner I described. What color is to the right of the first beaver? What color is to the right of the second beaver? I’ll wait.

What you found (if you painted the scene properly) is that each beaver has red on its left and blue on its right. That’s because in this particular configuration, all the quarter-circles join up to form one big loop, and all forty-eight beavers are circling it clockwise, with red to the left and blue to the right. In fact, the blue/red coloring you made in your head is exactly the one shown at the end of the previous section (the left panel of the the two-panel figure in the last paragraph).

Kids visiting the museum don’t care about the educational value of an exhibit like Beaver Run, but to parents and other adults who wonder “Where’s the math here?”, I’d say: Seeing like a mathematician is about tackling questions through the creative process of removing irrelevant detail and replacing it by relevant detail, often of a very non-obvious kind even when it’s simple.

The coloring idea isn’t just simple; it’s also powerful. Hart initially proposed a number of variants on the scenario described above. What if there were a button that would simultaneously rotate the two turntables that the two beavers were on, or, in the case that the two beavers were passing each other on some Truchet tile, rotate just that one turntable? Once you know how to subtract irrelevant detail (ponds, trees) and add relevant structure (red regions, blue regions) in your imagination, you can show that in this variation on the MoMath exhibit, no collisions will occur. You can also analyze ways to place more than two beavers in the system and to show that with suitable rules no collisions occur.

When a mathematical idea is both simple and powerful, we call it beautiful.

Is the coloring device a mere “trick”? Maybe you’ll want to call it that. In both math and magic, surprise and wonder go hand in hand. Magicians pulling improbable things out of their headgear is such a cliché that someone seeing me introduce the red-blue coloring is likely to complain that I just pulled the idea “out of a hat”. Other magic tricks rely on miraculous disappearances: “Now you see it; now you don’t.” In both cases, knowing how the trick is done spoils most of the fun. Watching a slow-motion video of a production or a vanishing may make you admire the performer’s sleight of hand, but you also realize that it requires years of training to reach that level of skill.

But in math, seeing a trick like the coloring argument empowers you almost immediately to make the trick your own. For instance, what if we made a Truchet tiling in which Smith’s two kinds of marked squares were replaced by the five kinds of marked hexagons shown below, and we put turntables in the middle of each tile? If there were an exhibit similar to Beaver Run but with the beavers replaced by bees, would collisions occur in the honeycomb, or would the laws of math “magically” keep collisions from happening? I’ve given you the tools to figure this out for yourself.5

Mathematical pleasure runs on the same track as the delight we get from a magician’s vanishing effect, but in the opposite direction. Now you don’t see it; and just like that, now you do.

I can’t help but wonder: what if everyone knew that math can be like this?

Thanks to George Hart, Cindy Lawrence, Tim Nissen, Beverly Tomov, and Glen Whitney. Much of the information on which this essay is based came from past and present personnel of MoMath, and many of the illustrations were either provided by MoMath or are based on materials provided by MoMath. Thanks also to Sandi Gubin.

ENDNOTES

#1. I also have a weakness for Comic Sans.

#2. See Further Travels with My Ant by David Gale, Jim Propp, Scott Sutherland, and Serge Troubetzkoy.

#3. Hint: Pivot your head by 45 degrees so that your right ear is close to your right shoulder.

#4. Here I’m omitting some technicalities that don’t get fully explained until one takes an introductory course in topology. To get a sense of the subtleties that I’m not treating, imagine a third kind of tile in the train network: an overcrossing/undercrossing junction in which one beaver goes under the other in the middle of the tile without the two critters colliding. For this kind of network, the blue/red coloring argument falls apart, and in fact the non-colliding property falls apart too.

#5. The same argument goes through using the same sort of blue/red coloring of the scene. At work here is a general principle called the Jordan Curve Theorem which asserts that any simple closed curve in the plane divides the plane into two regions. This may seem obvious, but it becomes tricky to prove when the curve is really twisty like some fractals.

REFERENCES

Kenneth Chang (New York Times), At Museum of Mathematics, Meet 2 Beavers That’ll Never Meet.

MoMath, A Truchet Tale.

If you ask a person on the street whether 1 is a prime number, they’ll probably pause, try to remember what they were taught, and say “no” (or “yes” or “I don’t remember”). Or maybe they’ll cross the street in a hurry. On the other hand, if you ask a mathematician, there’s a good chance they’ll say “That’s an excellent question” or “It’s kind of an interesting story…”

Some people treat the non-primeness of 1 as a mathematical fact and nothing more, but those people are missing out on something important about the nature of mathematics.

THE PREHISTORY OF PRIMES

In early days, 1 wasn’t universally regarded as a number at all. For the Pythagoreans, the first counting number was 2; 1 was the Unit from which all the numbers (2, 3, etc.) were built. So 1, not being a number, was certainly not a prime number. Euclid, although not a member of the Pythagorean Order, agreed that the first prime number was 2.

But Greek thought wasn’t homogeneous. Some, such as Plato’s nephew Speussippus, thought that 1 was not only a number, but a prime number at that. So controversy about the status of 1 has a respectable pedigree.

Nor can the practice of calling 1 a prime be complacently relegated to the midden of ancient, long-discarded mistakes. The great Leonhard Euler, the pre-eminent mathematician of the eighteenth century, treated 1 as a prime in his correspondence with number-theorist Christian Goldbach. Even in the twentieth century, the mathematician G. H. Hardy, coauthor of the first great work on number theory written in the English language, classified 1 as a prime in his early writings.

Were Euler and Hardy being stupid or careless? Far from it. They were doing what good mathematicians always do: maintaining a flexible attitude toward terminology, and keeping in mind that sometimes the right way to define things only comes into focus when you’ve played with several variants.

So if your attitude toward my title was “Yeah, why does it matter?” you’re asking a question that Euler and Hardy – who both sometimes included 1 among the primes and sometimes didn’t – would have endorsed. After all, the number 1 has many properties in common with the primes.1

But you shouldn’t get the idea that in the modern era there’s disagreement about the status of 1; by universal consensus, 1 isn’t a prime.2 Does that mean we’re forced to classify 1 as a composite number, i.e., a factorable number like 4, 6, 8, and 9? Or is there a third possibility?

THE LONELIEST NUMBER

In the preface to his 1914 table of primes, the number theorist D. N. Lehmer, by way of justifying his decision to include 1 in the table, pointed out that “the number 1 is certainly not composite in the same sense as the number 6,” adding “if it is ruled out of the list of primes it is necessary to create a particular class for this number alone.” For Lehmer, that was sufficient reason to list 1 as a prime; leaving 1 out in the cold, calling it neither prime nor composite, didn’t seem like an option.

1 is certainly an exceptional number for many reasons. One distinctive property of the number 1 is that it’s its own reciprocal. No other positive integer has this property. When we enlarge our number system to include zero and the negative integers, 1 acquires a buddy in the person of its negative, the number −1, which, like 1, is its own reciprocal. Further enlarging our scope to include the rational numbers and the real numbers brings us no new numbers with this property. But when we enlarge yet again, to the complex numbers, although we don’t get any new numbers that are their own reciprocals, we get two numbers that are simultaneously each other’s negatives and each other’s reciprocals: i and −i.

Just as the integers form an interesting subsystem of the real numbers, the Gaussian integers — complex numbers of the form a + bi where a and b are ordinary integers — form an interesting subsystem of the complex numbers. The Gaussian integers taken in aggregate form what mathematicians call an integral domain (in this essay I’ll use the shorter term “domain” for brevity) in which numbers can safely be added, subtracted, or multiplied without ever leaving the domain. Notice that I left division off the list of safe operations; in a domain, you usually can’t divide one element by another. But when a special element of a domain — call it u — has the property that the reciprocal of u also belongs to that domain, then every element of the domain can be divided by u: just multiply that element by the reciprocal of u. In the domain of integers, the only such elements are u = 1 and u = −1, but in the domain of Gaussian integers, there are four of them: 1, −1, i and −i.

An even more interesting example is the domain consisting of all numbers of the form a + b sqrt(2) where again a and b are ordinary integers. In this domain there are infinitely many numbers whose reciprocals belong to that same domain: for instance, 1 + sqrt(2) and −1 + sqrt(2) are each other’s reciprocals, 3 + 2 sqrt(2) and 3 − 2 sqrt(2) are each other’s reciprocals, and so on.

The study of such number systems, pioneered by Carl-Friedrich Gauss and now a thriving specialty in its own right, is called algebraic number theory. In this subject, numbers in the domain whose reciprocals also belong to the domain are called units. 1 is no longer lonely; it has a hip-and-happening club to belong to.3

So those Pythagoreans from the start of this essay were onto something. From a modern perspective, they were right in singling out 1 for special treatment and insisting that we pay deference to 1 as a Unit; but whereas they viewed being-a-Unit as incompatible with being-a-number, we regard 1 as both a unit and a number.

AVOIDING AWKWARDNESS

I suspect one reason Lehmer persisted in calling 1 prime is etymological. The Greeks called the primes the protoi arithmoi or “first numbers”, and the Latin word “primus”, from which we derive the words “prime” and “primary”, has similar connotations. How can 1 be the first number we say when we count, and yet not be counted as one of the First Numbers?

But even before Lehmer classified 1 as a prime, most modern mathematicians had quietly decided it didn’t deserve that designation. This consensus arose not because of any one thing, but because of dozens of different ways in which treating 1 as a prime led to awkwardness.

A case in point is the sieve of Eratosthenes, mentioned by Lehmer on the same page as his argument for calling 1 a prime. Lehmer writes: “Eratosthenes, a contemporary of Euclid, was the inventor of a ‘sieve’ process for removing the composite numbers from the series of natural numbers. He first wrote the numbers in order, and then removed the multiples of 2 by erasing every other number after 2. He then erased every third number after 3, then every fifth after 5, and so on. In this way, by rejecting the multiples of the successive unerased numbers, he obtained the series of primes.”

Let’s break this down. First I’ll erase, or rather shade out, the multiples of 2 (not including 2 itself, of course) between 1 and 25:

Then I’ll get rid of the multiples of 3:

Then the multiples of 5:

And so on.4

But hang on a minute. If we’re supposed to remove the multiples of each successive number that hasn’t been removed yet, shouldn’t we start the game by removing all the multiples of 1? Of course, then the game would end very quickly, and 1 would be declared the only prime.

Of course that’s not how the sieve works. We treat 1 in a different way than 2, 3, 5, etc.; specifically, we don’t cross out all its multiples. If we insist on calling it a prime anyway, we must admit it’s a very special prime.

Another case in point is the Fundamental Theorem of Arithmetic, otherwise known as the uniqueness-of-factorization theorem.5 Every composite number can be written as a product of primes, and moreover, there’s only one way to do it, if we agree to ignore the order in which the factors appear, so that for instance 2 × 3 and 3 × 2 count as the same factorization of 6.6 If 1 were classified as a prime, then there’d be more than one way — infinitely many ways, in fact — to write 6 as a product of primes: 2×3 and 1×2×3 and 1×1×2×3 and so on.

Of course a fervent 1-is-prime holdout could stand his ground and rephrase the Fundamental Theorem of Arithmetic to allow for this, so that two factorizations that differ only in the ordering of the factors, or in the inclusion of a different number of factors equal to 1, would still count as the same. Then the deviant definition of primeness that includes 1 as a prime would still permit him to formulate the uniqueness of factorization theorem, but at the cost of some awkwardness.

PUTTING THEOREMS FIRST

I don’t know of any modern-day 1-is-prime holdouts, but I imagine that Christian Goldbach, the correspondent of Euler whom I mentioned before, would have held onto the idea longer than most of his contemporaries. Goldbach is mostly known nowadays for coming up with the celebrated and still-unproved conjecture that every even number bigger than 2 can be written as a sum of two primes. Or at least, that’s how we phrase it nowadays. Goldbach himself conjectured that every even number (meaning, every even positive integer) can be written as a sum of two primes, including the even number 2, because 2 can be written as 1+1, and for him, 1 was prime.

If I were at a party with Goldbach and we were debating the proper definition of “prime“, I’d be forced to admit that his conjecture is more easily stated using his more inclusive definition of the word, but I’d tell him that his conjecture is one of the few cases in which treating 1 as a prime makes things simpler; more often, it makes things more complicated. “You couldn’t have known this,” I tell him, “because the theorems that make it more natural not to call 1 prime still lay in the future when you did your work.”

“What theorems?” asks Goldbach, and I proceed to tell him about a few, taking special pleasure in introducing him to Gauss’ Law of Quadratic Reciprocity. Quadratic reciprocity is a beautiful fact that relates mod p arithmetic with mod q arithmetic whenever p and q are two different odd primes, that is, two different primes bigger than 2.7 In telling Goldbach the story behind this theorem, I’m careful to use the phrase “odd prime bigger than 1” to talk about the things that I would call (more simply) “odd primes”, so as not to confuse him.

At this point, Nicomachus of Gerasa, who is at the same party and has been eavesdropping on our conversation, pipes up and says “I couldn’t help overhearing the last bit of your conversation about that theorem of Gauss, and I was struck by your use of the phrase ‘odd prime’. Surely you must know that the phrase is redundant; only an odd number can be prime!” The historical Nicomachus defined a prime as an odd number that can’t be expressed as a product of two smaller odd numbers, so for him, 2 wasn’t prime. My imaginary Nicomachus thinks me slow-witted for failing to notice that, even as I fault Goldbach for having an over-generous definition of the word “prime”, I myself am guilty of the same fault, by failing to notice how different 2 is from the true primes. Goldbach says that the first prime is 1, and I say that the first prime is 2, but Nicomachus says that the first prime is 3, and he takes the Law of Quadratic Reciprocity as clinching evidence: “This Law is simpler to state if the number 2 is treated separately, and you yourself have called this Law the most beautiful proposition of number theory; so you must admit that 2 isn’t truly prime!”

A lively argument ensues about the merits of our respective definitions, but it’s important to notice what isn’t at stake in this argument: the three of us agree on the facts of math, such as the uniqueness of factorization into primes or the Law of Quadratic Reciprocity. We just talk about those facts using different words. The observation that people use different words to describe a shared reality is banal when multiple languages are involved, but we somehow forget this fundamental observation about language in the context of mathematical discourse.

Definitions in math are not eternal truths. They’re human choices, shaped by our need for coherence and our desire for beauty. We as a species get to choose how we define our words. Of course, we should think hard before we define a word, and think harder before we try to uproot an established definition. But we should never forget that humans invented the words to begin with.

It’s natural for math teachers to stress precision in speech and to insist on adherence to shared conventions about the meanings of words. But an unintended consequence of teacherly fussiness can be the misimpression that the definitions of mathematical terms are revelations from on high.

This misimpression hides from students something important: although we need conventions in order to communicate, mathematical truth is deeper than mere convention. The numerical facts asserted by the Law of Quadratic Reciprocity are equally true for Christian Goldbach and Carl Friedrich Gauss and Nicomachus of Gerasa, and you and me, even if we might initially use different definitions of the word “prime” when we’re talking about them.

So, does it matter whether 1 is prime? Maybe not. But the specific way in which it doesn’t matter matters very much.

Thanks to Sandi Gubin.

This essay is a supplement to chapter 1 (“The Infinite Stairway”) of a book I’m writing, tentatively called “What Can Numbers Be?: The Further, Stranger Adventures of Plus and Times”. If you think this sounds cool and want to help me make the book better, check out http://jamespropp.org/readers.pdf. And as always, feel free to submit comments on this essay at the Mathematical Enchantments WordPress site!

NOTES

#1. One way in which 1 “quacks” like a prime is the way it accords with Euclid’s Lemma, the principle that asserts that if p is a prime, then whenever the product of two integers is divisible by p, one of the two numbers or both must be divisible by p. The numbers 2, 3, 5, 7, … all have this property, and the non-prime numbers 4, 6, 8, 9, … all lack it. On which side of this dichotomy does 1 stand? Well, the proposition “Whenever the product of two integers is divisible by 1, one of the two numbers or both must be divisible by 1” is as true as “If 2+2 = 4 then 2+2 = 4” – it’s not an interesting assertion, but it’s certainly not false. So Euclid’s Lemma seems to counsel us to lump 1 together with the primes.

#2. I had a middle school classmate who’d been taught back in elementary school that “a prime is any positive integer that is divisible only by 1 and itself,” which would seem to make 1 prime, and he was taken aback in middle school to learn that, no, 1 isn’t prime after all. He felt he’d been misinformed by his earlier teachers, but our middle school teacher insisted he hadn’t been, and explained via a kind of Talmudic reasoning that the word “and” in the phrase “1 and itself” requires that the words “1” and “itself” refer to different numbers. I think that’s disingenuous; what’s more, it sets students up to make a basic conceptual mistake in algebra, namely, the mistake of thinking that when there are two or more variables sitting around, they can’t be equal to each other because “If x and y referred to the same number we wouldn’t have given them different names.” It’s best to teach kids from the start that mathematicians define a prime as a number greater than 1 with no (positive) divisors other than 1 and itself.

#3. Among the numbers of the form a + b sqrt(2), the units are the ones that satisfy a2 − 2b2 = ±1. This is sometimes called Pell’s equation after the 17th century mathematician John Pell, but it has interested mathematicians since the time of Pythagoras. The Indian mathematician Brahmagupta discovered a way to combine two old solutions to get a new one; for instance, by combining (a,b) = (3,2) with (a,b) = (7,5), Brahmagupta derived the solution (a,b) = (41,29). Brahmagupta probably didn’t know it, but his way of combining solutions was a disguised way of multiplying algebraic numbers: 3 + 2 sqrt(2) times 7 + 5 sqrt(2) equals 41 + 29 sqrt(2). The fruitfulness of Brahmagupta’s approach arises from the fact that the product of two units is always a unit.

#4. Our sieving procedure is making faster progress than you might think; all the composite numbers up to 25 have now been sifted out, so 2, 3, 5, 7, 11, 13, 17, 19, and 23 are all the primes up to 25.

#5. Although it’s usually credited to Gauss, the theorem was first stated and possibly proved by the Islamic mathematician Kamal al-din Al-Farisi around the year 1300.

#6: You may think that the uniqueness of prime factorizations is obvious. If so, I ask you to check that 209 × 221 equals 187 × 247, and then to tell me why you’re so sure that those four three-digit numbers aren’t prime. Or, let’s say we restrict ourselves to the number system that contains only even integers; in that restricted number system 2 × 18 equals 6 × 6, but I defy you to break down 2, 6, or 18 as a product of two even integers.

#7: If we define the special symbol (p|q) to be +1 when p has a square root in mod q arithmetic and to be −1 when p doesn’t (and to be 0 when p = q, if you’re going to be fussy), then the Law can be summarized by the equation

(p|q) (q|p) = (−1)(p−1)(q−1)/4

REFERENCES

Chris K. Caldwell and Yeng Xiong, What is the Smallest Prime?

MathOverflow, “When did the career of 1 as a prime number begin and when did it end?“.

Wikipedia, Prime number.

Since you’re reading this essay, you probably already know about the mathematical holiday called Pi Day held on March 14th of each year in honor of the mystical quantity π = 3.14…. Pi isn’t just a universal constant; it’s trans-universal in the sense that, even in an alternate universe with a different geometry than ours, conscious beings who wondered1 about the integral of sqrt(1–x2) from x = –1 to x = 1 would still get — well, not 3.14…, but exactly half of it, or 1.57…. Therein lies a catch in the universality of pi: why should 3.14… be deemed more fundamental than 1.57… or other naturally-occurring2 pi-related quantities?

I suspect that, even if we limit attention to planets in our own universe harboring intelligent beings that divide their years into something like months and their months into something like days, many of those worlds won’t celebrate the number pi on the fourteenth day of the third month. It’s not just because 3.14 is a very decimal-centric approximation to pi (is there a reason to think that intelligent beings tend to have exactly ten fingers or tentacles or pseudopods or whatever?). Nor is it just because interpreting the “3” as a month-count and the “14” as a day-count is a bit sweaty. And it’s not just because holding a holiday to celebrate a number is a weird thing to do in the first place. It’s also because, in our own world, we came close to having a different multiple of pi serve as our fundamental bridge between measuring straight things and measuring round things.

BECOMING A NUMBER

Pi is sometimes called Archimedes’ constant3 because Archimedes was the first mathematician to describe a procedure for estimating pi to any desired level of approximation. Archimedes applied his method to show that pi was between 223/71 and 22/7. Incidentally, Archimedes didn’t think of 22/7, 223/71, or pi as numbers; to him, they were ratios of magnitudes, and the last of them had to be treated geometrically rather than numerically.4 But the key thing is that Archimedes did not show that pi lies between 314/100 and 315/100 (corresponding to the modern assertion 3.14 < π < 3.15) or between any other pair of fractions with powers of ten as their denominator. There was no reason for him to do so; the decimal system, with its built-in fixation on power of ten, was far off in the future.

Two millennia after Archimedes, Simon Stevin’s manuals De Thiende (The Tenth) and L’Arithmétique (Arithmetic), published in 1585, brought Europe into the decimal age. Stevin explicitly renounced the old distinction between numbers and ratios, proclaiming emphatically if obscurely “There are no numbers that are not comprehended under number.” In Stevin’s system, integers, fractions, and irrational numbers could all dine together at the same table and participate in the operations of arithmetic by way of their decimal representations.

Stevin didn’t specifically discuss pi, but his framework encouraged mathematicians to think about pi as a number. One such mathematician was Ludolph van Ceulen, who spent decades applying Archimedes’ method to analyze polygons with enormous numbers of sides, obtaining 35 decimal digits of pi. After van Ceulen’s death in 1610, the digits he’d calculated were inscribed on his tombstone and pi was dubbed “the Ludolphine number” in some parts of Germany and the Netherlands.5

So now 3.14… was acknowledged as a bona fide number. (Originally I was going to write a blog post called “The Velveteen Ratio; or, How Numbers Become Real”, but that’ll have to wait for some other occasion.)

WRONG NUMBER?

Yes, 3.14… was finally a number, but was it the right number to look at?

William Oughtred, in his 1631 work Clavis Mathematicae (The Key of Mathematics), used the notation “π/δ” where π is the circumference of a circle and δ its diameter; that quotient is our friend 3.14…. On the other hand, James Gregory, in his own 1668 book Geometriae Pars Universalis (The Universal Part of Geometry), focused on “π/ρ” instead, where ρ is the radius of the circle; this quotient is 3.14…’s relative 6.28…. Although the two mathematicians focused on two different ratios, for both Oughtred and Gregory, “π” did not denote a number; it denoted the circumference of a circle of arbitrary size. (I obtained much of the information for this section from Jeff Miller’s document Earliest Uses of Symbols for Constants.)

In 1671, Gregory went on to derive a formula6 for one-fourth of the modern pi constant as 1 − 1/3 + 1/5 − 1/7 + …. This formula implies that pi itself is 4 − 4/3 + 4/5 − 4/7 + …. I don’t know if you’ll agree, but I find 1 − 1/3 + 1/5 − 1/7 + … a much prettier sum than 4 − 4/3 + 4/5 − 4/7 + …. Perhaps Gregory felt that way too, and maybe he even wondered from time to time whether 1 − 1/3 + 1/5 − 1/7 + …, or .79…, might be the truly fundamental numerical characteristic of circles.

In 1706 William Jones, in his Synopsis Palmariorum Matheseos (later published in English as A New Introduction to the Mathematics), followed in Oughtred’s footsteps but with a notational twist: he proposed that “π” should denote not the circumference of a circle but the ratio of the circumference to the diameter.

The greatest mathematician of the 18th century, Leonhard Euler, followed Jones in adopting “π” as a dimensionless constant, but he took his time figuring out which constant it should be. In 1729, he used “π” to denote the number 6.28… (using the symbol “p” to denote 3.14…). In the late 1730s, Euler switched to using “π” for 3.14…, but he switched back to using “π” for 6.28… again in 1747. Along the way he wrote the formula for the area of the circle as “A = C r / 2” instead of “A = πr2”, skirting the use of the symbol “π”. Then in 1748, in his deeply influential Introductio in analysin infinitorum (Introduction to the Analysis of the Infinite) he switched again to using “π” to denote 3.14… (and “∆” to denote 6.28…) and he never looked back.

The world took up Euler’s (eventual) choice, and has forever since used “π” to signify 3.14….

So Jones’ convention prevailed. But did it deserve to? Here according to ChatGPT are the ten most widely-used mathematical equations involving π:

Of these, (1), (3), (4), (6), and (7) become simpler when expressed in terms of 2π and (2), (5), (8), (9), and (10) are simpler when expressed in terms of π. I’d call that a dead heat.

Missing from that list is a very important formula involving pi that is not an equation but an approximation:

This formula, due to James Stirling in the 1730s, was expressed by him verbally rather than symbolically, including the Latin words for “the circumference of a circle whose radius is 1”, which is to say, in modern terms, 2π. So Stirling, too, would probably have favored 6.28… over 3.14….

SO WHAT HAPPENED?

Why did Euler eventually stick with 3.14…? We can’t ask him, but it’s natural to think that he wanted to follow precedent. Archimedes wasn’t the only ancient to focus on 3.14… rather than 6.28…. Many civilizations studied circles and found approximate ways to compute their circumferences, but as far as I’m aware, none of them looked at the ratio of circumference to radius. That’s probably because of practical considerations: if you’re holding a disk in your hand or trying to measure a circular plot of land, it’s easier to get an accurate estimate of the diameter than to get an accurate estimate of the radius (other than by estimating the diameter and then dividing by two). And Archimedes was a practical fellow, being not just a mathematician but an engineer as well.

On the other hand, Archimedes was a Greek mathematician following in the tradition of Euclid, and that tradition venerated construction almost as much as it venerated proof. The Greek definition of a circle was in terms of its radius – which makes sense since the classic Greek way to construct a circle was via a compass. So if Archimedes had been feeling more like a disciple of Euclid when he did his work on circles, he might’ve favored measuring the circumference of a circle in terms of its radius rather than its diameter. And then we might well have adopted 6.28… as the circle constant, and even adopted the Greek letter “π” to represent it.

In recent years, it’s been suggested that math would be simpler if we’d all adopted 2π as the fundamental quantity to begin with. The suggestion was first made by Bob Palais in his article π is Wrong!, and others after him have argued for the adoption of the Greek letter tau to represent 6.28…: see Michael Hartl’s Tau Manifesto. As I found above, roughly half of the commonest formulas involving pi become simpler when expressed in terms of tau. Nobody expects τ to prevail over π, but its advocates have a kind of cheerfulness one often finds among the partisans of inconsequential lost causes.7 Tau Day is celebrated each June 28, but that date falls between when most school years end and when most math camps start, so the upstart holiday has had trouble gaining traction.

I have two modest proposals of my own. One is that, while we continue to accept 3.14… as the “right” pi, we celebrate Good-Enough Pi Day in honor of the approximation 3.1, which is close enough for most purposes. It could be celebrated on either the 3rd day of the 1st month, which happens to coincide with the perihelion of Earth’s orbit around the Sun, or on the 1st day of the 3rd month, which happens to coincide with the day on which the Earth has traveled one radian past the perihelion. Okay, I’m fudging those dates a bit, but only by a day or two. Could these astronomical coincidences be nudging us in the direction of Good-Enough Pi Day?

But an even more important approximation to pi that deserves wide acclaim is the number three. Three is, after all, the Bible’s approximation to pi – a circumstance which has led one Prof. Beauregard G. Bogusian to contend that the circumference of a circle is exactly equal to thrice its diameter, and that any seeming deviations from equality are due to humanity’s fallen state, as described in his video The Truth About Pi. Three is also the truly trans-universal approximation to pi, not beholden to any particular choice of base and hence likely to appeal to creatures with any number of appendages. As a way of celebrating π ≈ 3, I propose that the third repast of each day be hereafter dubbed Pi Meal, and that we honor the number 3 at each such meal by consuming exactly 3 slices of pie. If we consistently perform this ritual, we will in the course of time approximate the mystical rotundity of the circle to any desired level of approximation.

ENDNOTES

#1. Of course one could argue that in a deeply different universe no one would invent integration or even square roots, let alone try to evaluate that integral. It’s hard to argue with such a position, but it’s also hard to have much fun in a conversation in which any attempt at logical deduction could be countered with “Yes, that’s only logical, but what if the rules of logic themselves were different there?”

#2. 1.57… has fewer fans than 3.14… and 6.28…, but fans do exist. One such is K. W. Regan, who, responding to Lance Fortnow and Bill Gasarch’s blog post on reasons why 6.28… is better than 3.14…, wrote: “I am agreed that the ‘true’ value of ‘pi’ is off by a factor of 2 – but in the opposite direction!” I sympathize with fans of 1.57…. For one thing, 1.57… is the universal coefficient of annoyance that measures how much further you have to walk when there’s a circular obstacle in front of you that you have to walk around instead of through.

#3. Attempts to approximate pi long predate Archimedes so the term is a bit of a misnomer, and you might guess that it was European Grecophiles who introduced the term but you’d be wrong: the term “Archimedes’ constant” was introduced by Japanese Grecophiles in the early 20th century.

#4. The difference in point of view is subtle, and can be hard for modern readers to understand since it requires unlearning familiar ways of thinking about numbers and measurement that are drilled into us from an early age. I explain more about the Greek theory of proportions in my essay Dedekind’s Subtle Knife from last month.

#5. Another term that was used was “van Ceulen’s number”, though this sometimes denoted van Ceulen’s approximation to pi rather than pi itself.

#6. Gregory’s formula was discovered independently two years later by Gottfried Wilhelm Leibniz. Neither one knew it, but the formula had been found by Madhava of Sangamagrama more than two centuries earlier. All three mathematicians derived the formula in the same way: by finding the power series expansion of the function arctan x and plugging in x = 1. To this day, when I want to know the first dozen digits of π, I often type “4*a(1)” into the UNIX program bc, causing my laptop to compute 4 times the arctangent of 1. For more about pi and bc, see John D. Cook’s essay Computing pi with bc.

#7. I understand this kind of fatalistic partisanship; I was a Red Sox fan until they actually won the World Series, at which point being a Red Sox fan lost much of its appeal.