by Jonathan Kujawa

Mathematics is an inexhaustible subject. Every time you think you’ve mined out a vein, you hit a new gem. There is no clear reason why this should be true, but all evidence shows that it is mathematics all the way down [1].

I once saw a talk by a mathematician near retirement who had made numerous influential discoveries over the years. The recurring theme of the talk was how, time after time, he finally understood everything about his favorite corner of mathematics. The joke, of course, was that each new discovery revealed how his previous “complete” understanding still overlooked interesting and important things.

An excellent example of this phenomenon is knot theory. This is the mathematics of figuring out how to distinguish different knots. Two knots could look very different, but could turn out to be the same after some manipulation. Or not! For example, to the right are two knots created by Conway and by Kinoshita-Terasaka. It turns out to be extraordinarily difficult to figure out if they are the same knot or not.

This is one of those kinds of research that politicians and know-nothings love to sneer at. It sounds stupid, trivial, and of no use to anyone. However, if you’re one of those who care about real-world applications, the mathematics of knot theory plays a role in string theory, protein folding, quantum computing, and more.

But if you’re one of those who find math interesting for its own sake, then you’ll be glad to learn that knot theory is a rich vein of mathematics that continually reveals deep, beautiful, and interesting new discoveries. In this essay, I thought I’d share yet another new discovery in knot theory. Longtime readers of 3QD won’t be surprised: we’ve talked about knots many, many, many times over the years. Read more »

One Thing

Common, simple thoughts are hard to place in
deep context as they rush our minds and hold them fast.

So,
we don’t look in,
but out,
around, and
past.

Too small to be big,
small-thoughts squeeze
between lines
suggesting,
hinting,
asking,
& telling
as the tale unfolds.

To big to be small,
they come off as incidental
while holding a universe
in their grasp.

We wander half-awake through days ignoring
the stupendous humble things each day contains:

—FuturePresentPast & WhatRemains,

the One-Thing

Jim Culleny
7/23/21-Edited 9/28/25

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by Mary Hrovat

I recently went to see a painting that I love but hadn’t seen for about 40 years, Samia Halaby’s Boston Aquarium. It had been important to me when I was an undergrad at Indiana University in the 1980s, and I’d thought I might never see it again in person. When I learned that it was on display at Indiana University’s art museum, the Eskenazi Museum of Art, I was incredulously joyful at the prospect of immersing myself once again in its color and light.

Boston Aquarium is a large abstract painting (72 1/4 × 95 3/4 inches, oil on canvas). Halaby completed it in 1973. I remembered it for years as a dark painting overall, but its darkness is suffused with light and color. When I first saw it in the mid-1980s, I was studying astrophysics. Although the universe is mostly dark, light—ancient light in particular—is a central concern of astronomy, and the sense of rays of light in the painting was one of the things that drew me to it.

I remember visiting it frequently and losing myself in its shapes and colors, but it’s difficult to remember, this many years on, what exactly I saw in it. I think the helicoid shapes reminded me of the curves of the conic sections I learned about in my calculus classes. Some of the arrangements of colors in the painting might have been evocative of stellar spectra. Overall the painting gave me an expansive sense of moving through vast spaces.

∞

At some point I lost track of Boston Aquarium. I moved away from Bloomington briefly after I graduated, and when I returned, I couldn’t find it at the art museum. (I did find another painting by Halaby in a similar style, Red Green Steel, which I also loved.) In the early 2000s, I wrote to the art museum to ask about it and received the disappointing news that there was no record of the university owning it. I assumed it had been on loan when I’d seen it in the 1980s. Read more »

by Thomas Fernandes

A rodent with orange teeth and a paddle tail comes across a river. He tries to build a lodge but the stream is too low so he grunts, “Dam!”.

While the pun captures the beaver in miniature, the real story lies in the web of relationships that shapes it and the ecosystem in return. To see how deeply a species is shaped by these relationships, it helps to look first at its body.

This 20 kg semi-aquatic mammal is an extraordinary swimmer. It uses its tail as a rudder and webbed hind feet to glide efficiently through water. It can hold its breath for up to fifteen minutes thanks to a suite of cardiovascular adjustments known as the diving reflex. This reflex slows the heart and redirects blood flow to vital organs, a trait shared with seals and penguins.

Their fur is equally adapted: a very soft underlayer traps air and provides insulation with an incredibly high hair density (ten times the density of human hair), while a covering layer of longer guard hairs repels water, a property further enhanced by oil secretions that act like wax. Even their eyes are specialized for underwater vision, equipped with a nictitating membrane (a third transparent eyelid) that functions as natural goggles underwater.

By observing their bodies, it would be tempting to think nutria are close relatives. After all, they share water-repellent fur, webbed feet and even a diving reflex.

But looks can be deceiving: beavers and nutria split from a common ancestor 50–60 million years ago, long before humans and chimps parted ways just 7 million years ago. This is a case of convergent evolution, showing that similar environmental pressures can produce similar forms even in lineages with very different histories. Yet such adaptations are always built on top of existing genetic heritage. To understand what truly set beavers apart, we need to turn away from look-alikes and toward their real cousins in the desert. The closest living relatives of beavers are North American desert-dwelling burrowers. Examples include kangaroo mice and pocket mice. Despite their vastly different appearances, observing their behavior reveals deep evolutionary continuity. Read more »

by Mike Bendzela

You could tell we were on the last leg of our journey to Kentucky when, after we had crossed the Ohio River at Portsmouth and entered the curvaceous roads of the Appalachian Plateau, the puking began. At the time, we were a family of six crammed inside Dad’s 1963 white Ford Fairlane two-door sedan. Mom had one toddler brother up front, and Sis and I watched the other little brother in the backseat. It was exciting to go from the Great Plains cornfields ambience of the highway south of Toledo to the seemingly endless rugged hills below Columbus, but that excitement turned to horror once the back-and-forth, up-and-down, start-and-stop driving commenced on the back roads of Carter County, Kentucky. Mom had brought along a Hefty trash bag for just such an emergency, and as we kids went pale and cold in the backseat one by one, we took turns passing the bag among ourselves and heaving our roadside stop dinners into it.

Thus begins the story of this northern city-slicker’s romance with the American South, a halting, stop-and-go affair that has lasted decades and should, I hope, continue to the end of this life. It was during these trips to Eastern Kentucky in my boyhood that I realized I have a throbbing mass of neurons in my brain that pines for a long lost, agrarian past. This is a self-generated illusion, of course, and I’ve always known that, but it hasn’t stopped me (nor should it stop anyone) from trying to revive, recreate, and reconstruct this remote and vital aspect of North American history. Read more »

by Azadeh Amirsadri

I was an English as a Second Language teacher in a suburban high school in VA from 1997 to 2002. I taught levels 1 and 2, beginners and intermediate levels, and had a total of five classes each year. My students were from all over the world, and my five years with them were some of the best teaching experiences I have had. Every day of the week, we would gather as a family, joined by our otherness, and go through school and life together.

On the first day of school, I would go over their schedule and show them where their classes were, since they were usually grouped together in physical education or electives, and take them to their lockers so they could practice opening and locking them, an activity that would sometimes take the whole period. I would also take them to the cafeteria to show them about getting lunch, avoiding pork products for the Moslem kids, if they chose to, and get an alternative lunch. The county had little pink cards with the picture of a pig on them that were put next to the pork products, so that helped them. They had to memorize their student ID to pay for the food at the end of the line, and I‘d remind them that they didn’t have a lot of time for eating and cleaning up afterwards, 30 minutes from start to finish, and some lines were longer than others, so maybe bringing lunch from home could help them not rush as much. However, the sight of school pizza, fries, and chocolate milk was too much of a competition with lunch from home.

Students would sometimes arrive at any time in the middle of the school year after getting tested at a central registration center in the county, and slide right into any level they tested in. Everyone in class, except two kids, had someone who spoke their language in the school, and when they landed in my class, they had someone who could show them the ropes of an American high school.

There was a girl from Cameroon who didn’t have anyone else in class who spoke French, so I would speak to her when she needed help, and the others would tease her that she was my favorite. She was a bit older than the others, but not very mature, and would get into arguments with them. I was fascinated by how the students argued with their very limited English in level 1, using the profanities they had learned and knowing it wasn’t a nice thing to say, but not having any other words to convey their anger and their dislike of each other during fights. Read more »

by Mindy Clegg

On June 21st, 2025, President Trump ordered a series of strikes on Iran aimed at ending their nuclear program. The Iranians had maintained their legal right to develop a peaceful, domestic nuclear program. After Trump pulled the US out of the JCPOA (the Joint Comprehensive Plan of Action) negotiated during the Obama administration, he claimed he could cut a new deal with the Iranians quickly. After several false starts, he instead joined in on the violence started by the Israelis earlier in June in their obvious effort to distract from the ongoing acts of genocide in Gaza.

The group probably most excited by Trump’s actions are his far right white evangelical followers. They know that this very well could cause a chain reaction of violence across the region and possibly the world. They are hoping for and counting on it with glee in their hearts. They view it through a cult-like belief in the End Times. In their fevered imaginary, the End Times are a kind of ultimate victory that will prove their righteousness. They want nothing more than to watch their enemies burn from their clouds on high. But what if they are wrong about this being the start of the apocalypse? I argue, that, in fact we already live in a post-apocalyptic world and have done since 1945.

Definitions matter of course, but definitions can change with the historical context. We like to pretend words’ meanings are stable across human history and experience, but it’s just not so. Today, the term apocalypse has become a bit secularized and unmoored from its religious origins. The original meaning connected to the modern understanding of the term comes from Judaism. The term apocalypse was a genre of literature that followed from their Babylonian Exile of the 6th century BCE. The term means “revelation” in the Greek. It also denoted a change in circumstances and context, a historical break. The dreams or visions experienced by the early prophets of Judaism represented the meaning of the word during that era. The term that relates to God destroying the world, but saving his followers is more accurately apocalyptic eschatology. Revelation was not just about the world inevitably ending but it can refer to prophetic revelations about the “end of days.” That could denote an end or a new beginning. Read more »

by Alizah Holstein

Today an electrician came to visit. He was tall and broad-shouldered and had arms like sausage links that were fairly covered in tattoos. One of the tattoos was a date: January something-or-other. I tried to read it as he walked through my front door, but he looked me in the eyes and so I glanced away quickly without having absorbed any of the details. He had come to inspect my attic wiring, for which he had to get on his hands and knees and crawl around the attic floorboards. It was a short but dirty job. When he came downstairs his palms were blackened and so he asked if he could wash up somewhere. I pointed him to my kitchen sink and to a small bar of soap on one side of it. While he was washing his hands (very thoroughly, I noted), he turned to me and starting cheerfully recounting how important it was to him to be clean. He had a pink, friendly face, sort of like a big baby. He had shaved blond hair that had grown out ever so slightly and a twinge of orange in his beard stubble. I told him I was accustomed to dirt, having two sons and a male dog, although upon saying that I realized I wasn’t sure whether my dog’s sex was much of a factor in how dirty or clean he tended to be. The electrician nodded when I spoke but seemed eager to get back to his own story. He went on to tell me that he had a child but that he was no longer together with the mother. It’s not like me to have a one-night stand though, he said, it’s not a hygienic thing to do. And anyway, he went on, I could never have stayed with her—she was a slob, an unbel-IEV-able slob. She couldn’t focus, couldn’t pay attention to me or anyone else, and certainly not her surroundings. Keep your eye on the ball, I told her, but she didn’t know what I meant. Believe me, he said, that girl and all her stuff was all over the place.

I was startled by the unanticipated intimacy of this conversation (his conversation, really, seeing as I said little) and I was perhaps a little aware of being alone in my home and of having tied my dog outside. But if my hackles were up it was only very slightly, and in their place curiosity had begun doing its work. Why was he telling me all about his private life? The writerly instinct had kicked in; the man had leaped cleanly from contractor to material.

I started to think about where we get material. Sometimes, material is the byproduct of effort. But other times we run into it as if by chance. In the last few years, I have started taking notes on the way people talk. There’s a woman I knew who threw the expression “madder than a wet hen” into every conversation; and a man who repeated the same few sentences so many times that I wrote them down out of the sheer need to occupy myself as he spoke. Read more »

Two ceramic swans my wife bought cheaply at a flea market somewhere. They might be tea-light holders. Or something like that. I like them.

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by Steve Szilagyi

Henry James once observed that Robert Louis Stevenson “wrote with a kind of gallantry — as if language were a pretty woman.”

C.P. Snow (1905–1980) did not write like that. Pretty did not seem to enter any area of his work. The author of the famous essay The Two Cultures and the Scientific Revolution (1959) and the eleven-book cycle of novels known as Strangers and Brothers was, in his physical person, emphatically not pretty. As he approached the end of his life, he came more and more to resemble a pink, pointy-headed Patrick Star from SpongeBob SquarePants—decidedly dour, with National Health panto glasses somehow attached to his face.

If Snow wrote with any kind of gallantry, it was the solemn courtesy you would show a group of serious men seated around a table in an oak-paneled boardroom. His prose was careful, deliberate, and capable of discovering fine distinctions in faces, morals, and motives. His essays and novels were best sellers in Great Britain and the United States, and he won awards and honors for his writing. The BBC even made Strangers and Brothers into a thirteen-part series that no one liked.

In the 1960s, The New York Times referred to him as “the most eminent living English author” at a time when no less than Evelyn Waugh, Iris Murdoch, and Graham Greene were all busily writing away. Yet no author of similar stature in the 20th century has been more thoroughly and publicly excoriated for being slow, dull, and unreadable. Not even Gertrude Stein! Read more »

by TJ Price

F-1: blueprint

The house, then, and its rooms. Viewed from the outside, it is nothing extraordinary: situated at the top of a hill, its single-level structure is unassuming. The front lawn is studded with the acorns of oaks and maples, themselves with none of their lowest branches reachable from the ground. There is a small flower garden in the center, ringed with stones yanked from the surrounding woods like teeth, and the façade of the house is guarded by unkempt rhododendrons even on the brittlest of winter days. The long driveway bristles with forsythia on one side, golden and inviting in the early spring. 

It is not a place I visit anymore, except for in memory and in dream, both of which are unreliable navigators, though I imagine it cannot help that the map I’ve given them is outdated. When I left the house and its rooms for the last time, I did not know I would not be returning, otherwise I would have taken more careful note of its territory.

A house has multiple forms of ingress and egress, though only some are doors. This house had a front door—rarely used, except for Halloween trick-or-treat (and even then, only a handful of times before we started leaving the porch light off)—a back door, which was reached via a small porch, and a sliding-glass door through which was the kitchen, gained access to by means of a deck. There are more doors than there are rooms in the house, but there are more windows than doors, and some rooms have neither. The most common entry-point, however, was through the garage, past the stairs to the cellar, and into the kitchen.

The garage also held a secret door: a mouth which was heralded only by its dangling, uvula-like pullcord. Once tugged, out unfolded a set of creaky, hinged steps, leading up to a squarish hole in the ceiling. This, the attic, whose floor was more like a ribcage, spanned with planks that sprouted off in either direction from the main beam. Between them billowed pink, filamentous clouds of fiberglass insulation. I remember thinking the first time I saw up into that space how visceral it seemed, and from a very young age internalized the image, believing my own chest—and perhaps that attic of my own body, the skull—to be crammed with the same stuffing. Besides that, there wasn’t much in the attic. It was a hollow space, fit only for adventurous rodents and the odd avian inhabitant. Perhaps it is thematically appropriate that the place hovering over all of our heads for so long—the skull of our house’s body—was largely empty. Read more »

by Dilip D’Souza

The Hanle Dark Sky Reserve is a spectacular spot in Ladakh, in the north of India. It’s surrounded by snow-capped mountains, and at 14000 feet, it’s well above the treeline. So the mountains and the surroundings are utterly barren. Yet that barrenness seems only to enhance the beauty of the Reserve.

But as the name might suggest, it is a place where you see some incredible night skies, and that’s really why it is spectacular. Last night was partly cloudy, but I saw the Milky Way arcing high above my head, as well as plenty of stars I would never see at home in Bombay. The dark sky is the reason there are two large optical telescopes here, operated by the Indian Institute of Astrophysics (IIA), in Bangalore.

And under that dark sky last night, taking several photographs of the stars that I’m inordinately pleased with, I also had plenty of time to ruminate. At one point, it was probably being at this IIA facility that had me thinking briefly – of all things – of a temple in Ayodhya. That’s the temple to Lord Ram, consecrated by the Prime Minister in January last year.

One day a few months later, a shaft of sunlight fell directly on the idol of Lord Ram in that temple, making for a gorgeous sight. At the time, I was reminded of a photograph I have somewhere, that I took in a cavern at the ancient fort of Masada, in Israel. A man stands there, reading a book, and there’s a shaft of sunlight directly on him, just like Lord Ram in Ayodhya. While that photograph owed everything to a hole in the roof of the cavern, the sunbeam in Ayodhya was a little more complex. Read more »

by Brooks Riley

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by John Allen Paulos

A bit of information is common knowledge among a group of people if all parties know it, know that the others know it, know that the others know they know it, and so on. It is much more than “mutual knowledge,” which requires only that the parties know a particular bit of information, not that they be aware of others’ knowledge of it. This distinction between mutual and common knowledge has a long philosophical history and has long been well-understood by gossips and inside traders. In modern times the notion of common knowledge has been formalized by David Lewis, Robert Aumann, and others in various ways and its relevance to everyday life has been explored, most recently by Steven Pinker in his book When Everyone Knows That Everyone Knows.

I discussed common knowledge in my books Once Upon a Number and A Mathematician Plays the Stock Market and illustrated it with the following parable, which suggests its power.

The parable takes place in a benightedly sexist village of uncertain location. In this village there are many married couples and each woman immediately knows when another woman’s husband has been unfaithful but not when her own has. The very strict feminist statutes of the village require that if a woman can prove her husband has been unfaithful, she must kill him that very day. Assume that the women are statute-abiding, intelligent, aware of the intelligence of the other women, and, mercifully, that they never inform other women of their philandering husbands. As it happens, 20 of the village men have been unfaithful, but since no woman can prove her husband has been so, village life proceeds merrily and warily along. Then one morning the tribal matriarch comes to visit from the far side of the forest. Her honesty is acknowledged by all and her word is taken as truth. She warns the assembled villagers that there is at least one philandering husband among them. Once this fact, already known to everyone, becomes common knowledge, what happens? Read more »

by David J. Lobina

One of the most disappointing aspects of modern life is seeing peers in academia, and intellectuals in general, share their personal and private selves on social media.

Now, it is not the case that prior to social media one didn’t know anything about the personalities of many academics or intellectuals, but it was all a bit more circumspect, and salutary. The issue of whether there was a conflict between a person’s work and their personal lives or beliefs would rarely arise, for instance. Most academics are not public figures anyway, and before social media the vast majority of them received no wide exposure anywhere (let’s put public intellectuals to one side for the time being).

The problem now is that anyone can self-publish and self-expose on the internet, and so we suddenly have to face the fact that many clever academics and intellectuals are really out there and can also behave like, well, idiots.[i] There is something about social media, and it only takes a single look at Twitter/stupid-name-of-X to see what a cesspool social media can be. Umberto Eco may have been right when he pointed out that social media had brought loads of stupid conversations, and stupid people, out of the bar and moved them into the wide open world, but to see academics behaving badly in the public sphere is too close to home for comfort.[ii] Read more »

Sughra Raza. Departure. December 2024.

Digital photograph.

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by Richard Farr

Terrified people from ethnic minorities being threatened, rounded up, and beaten by heavily armed men in uniforms or not-quite-uniforms: we have seen these images before. In the US, the theory endlessly parroted to us over the decades has been that the Founders’ exceptional wisdom meant American institutions would protect us from such uncivilized nastiness. (Such characteristically European nastiness: Jefferson himself put it this way, looking down his nose at the old systems he thought he had transcended.) Well, well. Now that the shiny new system has failed — now that Blackshirt Theater is playing in our streets and on our farms and in our parking lots, to entertain the Führer — we have to make the best of one small silver lining, which is that you hear the customary smug nonsense about exceptionalism less and less. America’s institutions are not in danger of failing, as they were in 1972 or 2016 for example. Now, under the weight of 2025, they have failed, and the only question is whether they can be rebuilt one day. We are in the midst of existential catastrophe, waking up to the fact that the checks and the balances never were uniquely wise, or uniquely well-protected against failure, and that for now they are part of history.

The upper echelons of the chatterati, paid to wear ties and sound sober, will scoff at this. The air of finality is grossly premature, they will say. We’ve weathered crises before — and maybe the Dems will win the mid-terms and “restore democracy.” 

Piffle. 

Consider Gavin Newsom’s current brinksmanship over gerrymandering — and bear in mind that this practice, to which we have become inured, is impossible in most actually functioning democracies because independent commissions draw electoral boundaries and interference with that process by political parties is scarcely imaginable and anyway illegal.  Read more »

by Peter Topolewski

Imagine you could you travel back in time with your phone. Imagine you presented your phone to the first stranger you encountered. Didn’t dilly dally at all but straightaway showed this lucky stranger the Google Home app. How far back in time would you have to travel to ensure the app didn’t make one bit of sense to the stranger?

The 1940s?

Doubtful. The TV screen, though not yet in wide use, existed and set a high bar in the imaginations of who saw or even heard of such a thing.

The 1840s? Most certainly. Sure, Google Home is mostly colored icons, but to a citizen of the world of 1840—where and when electric power is not yet present—a glass-faced notepad with moving, colored icons would look like magic.

For those not in the know, the colored icons of Google Home exist in service of “setting up, managing, and controlling compatible Google and third-party smart home devices.” Apple users undoubtedly roll their eyes at this point, but they have their own version, smartly called Home. It works with Apple and Apple-adjacent products.

The purpose of setting up this Google (or Apple) app is to create a home partially or fully stuffed with lights, thermostats, speakers, and cameras you control from anywhere, anytime. Using your phone.

If all goes well, the results will be magical, even to present-day folks, never mind those of the 1840s.

Too bad the app—and the entire enterprise of setting up, managing, and controlling devices connected to it—is awful. Read more »