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by David Winner
August 2009, rainy season in Senegal, my friend Robert and I are trying to get from Dakar to Saint Louis, the former capitol of French West Africa. We take the taxi parked outside of our hotel to the Gare Routière where sept-places (old station wagons with their backs carved into claustrophobic seats) head to destinations across the country once seven passengers have assembled. On the way, the driver explains a problem to us, a difficulty that lies outside the realm of my limited French.
Inside the Gare, we see cars, drivers, and even the names of destinations (including Saint Louis) but something is off. There are hardly any people about.
Following us into the station, the driver repeats what he’s said before. And a word that I had not grasped, greve, strike, finally penetrates my foggy brain. No sept places are running today because they are on strike.
According to what I’ve read, police harassment, unlicensed vehicles, bad roads hound Senegal, transportation a chronic challenge.
What the driver wants, I assume, is for us to pay him to take us all the way to Saint Louis. But he seems hesitant when I make that suggestion. And when he finally agrees, he only asks for only a hundred dollars for the nearly four-hour drive there. And back, as he surely lives in or around Dakar. Read more »
by Charles Siegel
Last month I attended two conferences a week apart, one in Norway and one in Italy. The first conference was held in Bergen. From there, my law partner and I proceeded down the coast to Stavanger, for a meeting with a lawyer there about some potential cases.
Our meeting was late on the afternoon of November 16th. When we finished, we looked around for somewhere to have dinner. Most restaurants were closed on a Sunday evening in autumn, but eventually we found a decent place. As we walked back to our hotel afterward, we passed a pub and decided to go in and watch the soccer match that was playing on TV.
This turned out to be a World Cup qualifying match between Norway and Italy. There wasn’t actually much riding on the outcome; Norway was going to qualify unless it lost by nine goals or more, and Italy was going to stay alive but have to win an extra two matches to qualify, as part of an additional playoff round for European teams. But the fans in the pub were excited nonetheless.
When we sat down, the game was about midway through the second half, and the score was 1-1. It was clear, of course, that Norway was not going to lose by nine goals or anything close to that, and so they were going to the World Cup for the first time in 28 years. But even as the clock ticked down toward qualification, those in the pub still were hoping for a win.
And they got it, in lightning fashion: Erling Haaland, the bustling striker who plies his trade in the Premier League for Manchester City, scored two goals in two minutes. Norway added a fourth in stoppage time, and that was it. Jubilation for the Norwegian players, many of whom weren’t alive the last time Norway played in a World Cup, and humiliation for Italy, who were beaten at home and who now are in danger of missing the tournament for a third time running. Read more »
by Eric Feigenbaum
Singapore’s domestic debate is a matter for Singaporeans. We allow American journalists in Singapore in order to report Singapore to their fellow countrymen. We allow their papers to sell in Singapore so that we can know what foreigners are reading about us. But we cannot allow them to assume a role in Singapore that the American media play in America, that is, that of invigilator, adversary and inquisitor of the administration. No foreign television station had claimed the right to telecast its programs in Singapore. Indeed America’s Federal Communications Commission regulations bar foreigners from owning more than 25 per cent of a TV or radio station. Only Americans can control a business which influences opinion in America. Thus, Rupert Murdoch took up US citizenship before he purchased the independent TV stations of the Metromedia group in 1985.
When Singapore’s Founding Prime Minister Lee Kuan Yew spoke these words to the American Society of Newspaper Editors meeting in 1988, he could scarcely have imagined how the Internet and eventually social media would dramatically affect the very nature of the press and increase the number of voices engaged in political and other discourse.
Today not only is media ubiquitous but choosing your news and information sources to align with your pre-existing political and social outlook is not only available, but the norm. If the role of the Fourth Estate as a credible source of information has been compromised, it is equally undermined by hostile voices and even less credible sources able to find megaphones through social media. The information age is a din of voices making it harder than ever to get clear, unbiased news.
Media itself is now a topic in our national discourse. On October 29th, the Pew Research Center published findings that “only 56 percent of American adults now say they have a lot of or some trust in the information they get from national news organizations – down 11 percentage points since March 2025 and 20 points since we first asked this question in 2016.”
Further, “fewer than half of Republicans and Republican-leaning independents (44 percent) now say they have at least some trust in the information that comes from national news organizations. This is down from 53 percent in March and 70 percent in 2016, but it’s still above its lowest point in 2021, when 35% of Republicans expressed this level of trust in the national media.”
And Democrats? Read more »
Winter scene in Franzensfeste, South Tyrol.
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by Akim Reinhardt
Art is dangerous. It’s time people remembered that and recognized the fullness of it. For if art is to remain important or even relevant in the current moment, then it’s long past time artists stopped flashing dull claws and pretending they had what it takes to slice through ignorance. We need them swallow their feel-good clichés and to begin sharpening their blades. We need dangerous art, and we cannot afford much more art that its creators believe is dangerous when it is not.
When people say Art is dangerous, they’re often bragging. Talking it up like a super hero. Evil had better beware: Art is here to save the day!
Oh my, she gasped, clutching her pearls. Why, how could art possibly do that?
Because it’s dangerous! It reveals the evil that masked villains seek to hide!
Such sentiments are naive. They are wishful thinking. And those sentiments are actually far more dangerous than the art typically produced by those who espouse such sentimets. Supposedly dangerous art often does little more than preach to the choir or wrap important points up in obfuscations that most audiences lack the inclination or even the background to unravel. Instead of puncturing sacred cows or shining a bright light on dark evils, much “dangerous” art merely creates illusions of damage amid billows of self-congratulation and self-satisfaction. When the smoke clears, all the same problems remain, and no one, save for perhaps professionals and other insiders of various art worlds, are any more enlightened, while the problems that had been targeted with “dangerous” art quietly and stubbornly linger. Read more »
by Marie Snyder
Emma Wilkins’ excellent piece “On Housecraft” in The Philosopher, discusses Helen Hayward’s book, Home Work: Essays on Love & Housekeeping in such a compelling way as to provoke some thoughts without having actually read the book in question. So this is a critique of a review of a book I haven’t read, but on a topic most of us relate to intimately.
Like me, and many of us, Wilkins hates cleaning and is working through how to make the drudgery more palatable. She’s “more likely to make the bathroom less dirty than property clean.” Likewise, to take the confessions even further, cobweb strands are clearly visible from where I’m currently sitting in my kitchen.
Wilkins and Haywood raise a long-standing struggle for fairness in this field and pin the problem on daily chores being beneath our dignity, so they explore elevating the art of cleaning and finding personal benefits in the work. These paths might help, but I wonder if it could also help to revere the battle around equity and to lower and ground this regular exertion.
Haywood has found a way to embrace housework as a method of demonstrating caring. As an artform, it can become a noble pursuit to have a well-kept home. Wilkins writes that our disdain for chores is relatively new as Aristotle recognized that,
“…’oikonomia’ or ‘household management’ contributed to the wellbeing of the community, thereby serving a higher purpose. … It’s not surprising that, in a secular individualistic culture, cultivating servant-hearted humility holds little appeal. Work done in the home might not earn us money, or praise, or even gratitude. But the more we’re motivated by care, and love, the more noble the work is.”
Wilkins later states her position: “I can’t see our attitudes to the work itself changing any time soon.” Instead, she hopes to endure the grind by seeing the work as personally beneficial: “Far from being a ‘waste’ of time, it could be time spent thinking, reflecting, practising, and learning. It could benefit our health as well.” If we hate that the dishes need doing again, at least we can reap some side benefits from it, like hitting our fitness goals by working our wax-on-wax-off muscles. I used this line of reasoning when I had little ones who always wanted to be in my arms. I was getting buff from carrying them around! Read more »
by Herbert Harris
Many years ago, I began a meditation practice, sparked by curiosity and vague, middle-aged worries about stress and blood pressure. To my surprise, it quickly became a regular part of my life. I restlessly explored many forms of meditation and meditation groups, eventually coming to the San Francisco Zen Center. Before long, I found myself seated on a black cushion in the meditation hall each morning at 5:30. Twenty years later, and 2,500 miles away, I have a much more relaxed schedule, but I am still at it.
What is it like to meditate? This is a question I am constantly asked. Would a philosopher or scientist say there’s a distinct state of consciousness with its own special qualia? I don’t know. Maybe I’ve been doing it wrong, but meditation has never given me an experience that I would call altered consciousness. I’ve come to think the more interesting question is not what meditation feels like moment to moment, but what it is like to be a meditator, to live a life punctuated by these quiet, unremarkable moments of sitting still.
There are many ways our minds can store the details of our experience. We put facts and figures in one place, sensory experiences in another, and skills and procedures in yet another. There is a special kind of memory, called episodic memory, that holds not just the information about an event, but also a sense of our being there. Recalling episodic memories gives us a faint sense of time travel. These are the memories we can reinhabit. We remember a beach vacation as if we can feel the warm sand between our toes, hear the gulls above, and sense the light breeze on our skin. They have a lived-through quality, a presence that feels like “me.”
I have a torrent of episodic memories from my time in San Francisco, where I had just started a new job. I felt like a tourist; every street, every café, every meeting at the new company introduced a parade of unfamiliar faces. My memory was overloaded with experiences and sensations. It felt like my life had entered a new incarnation, complete with a new cast of characters I had to learn. As I stepped into a new role, I became, to some extent, a different person as I adapted to meet new duties and responsibilities. I was surrounded by people who each had hopes and expectations that I would be a good employee, a respectable colleague, and a friend. These hopes and expectations exerted palpable influences on my sense of self.
In the meditation hall, expectations were few. Read more »
by Mark R. DeLong
“People tell me that I could do much better,” David McCullough said in an interview included in California Typewriter, a documentary about typewriters “and the people who love them.” “I could go faster and have less to contend with if I were to use a computer—a word processor—but I don’t want to go faster. If anything, I would prefer to go slower.” He liked the straightforward mechanism of his Royal Standard Typewriter that he bought, used, in 1965 for about twenty-five dollars. “To me, it’s understandable: I press the key and another key comes up and prints a letter on a piece of paper and then you can pull it out. It’s a piece of paper upon which you have printed something.
“You’ve made that. It’s tangible. It’s real.”
That tangible quality, that reality comes clear in Richard Polt‘s Typewriter Manifesto, which concludes with a series of contrasts of typewriter life and digital life (using the red and black colors of some typewriter ribbons):
We choose the real over the representation,
the physical over the digital,
the durable over the unsustainable,
the self-sufficient over the efficient.
And then the manifesto concludes, “THE REVOLUTION WILL BE TYPEWRITTEN” of course.
Keyboards are keyboards, you might say. It’s the fancy stuff behind them that matters. Typewriters also have had their incremental improvements, electrification being the most obvious and, less obvious perhaps, the improvements on font rendering. The old fashioned levered hammer bearing the inverse shape of letters gave way to ingenious “balls” as in the IBM Selectric and “printwheels” that look like a miniature, heavy-duty bicycle wheel with spokes that bear the letters. (My own old typewriter has this technology.) These innovation allowed for font changes and vastly improved print typesetting. That story about the ingenious QWERTY keyboard layout minimizing key jambs and speeding production of words? I’d say, probably false. A more likely reason for that innovation was probably a simple marketing trick: a salesman who didn’t know how to type could still peck out T-Y-P-E-W-R-I-T-E-R using only the letters on the top row. Read more »
by Michael Liss
In the autumn of 1929 the mightiest of Americans were, for a brief time, revealed as human beings. —John Kenneth Galbraith, The Great Crash 1929
Ah, the mighty. The great, and the powerful. Let us take a moment to remember the mightiest of 1929. Among them surely had to be Herbert Hoover, the embodiment of a certain type of American success story, orphaned at an early age, a man who had literally worked in a mine for $2.50 per day, and, through inexhaustible, sustained effort tied to limitless ambition, had become a successful mining engineer, then internationally renowned for his rescue efforts during and after World War I, then, in eight highly impactful years, as Commerce Secretary. Almost by popular demand, he became the 31st President of the United States.
On Monday, March 4, 1929, he took office, declaring, in his Inaugural, these two ringing sentences:
In the large view, we have reached a higher degree of comfort and security than ever existed before in the history of the world. Through liberation from widespread poverty we have reached a higher degree of individual freedom than ever before.
Heady words, heady times, and perhaps just the slightest touch of hubris that wisdom might have suggested should be delivered in a less public way. The Gods have their own ways of seeing things, and of settling accounts, and Herbert Hoover, and the country he now led, were about to experience that.
Let’s scroll forward to December 3, 1929, barely nine months into his term, a bit more than a month after the wrenching horror of Black Thursday, October 24th and Black Tuesday, October 29th. Hoover delivers (literally, in writing, without pomp and circumstance) his first Annual Address to Congress.
It is, like most of its kin from George Washington’s to those of the present day, a ponderous thing. Hoover is a man of duty, and this effort, running to 11,000 words, is encyclopedic in its topics, and replete with dignified praise of both country and self. Read more »
by Rafaël Newman
Sonata for Viola and Pianoforte
Trio for Pianoforte, Violin and Violoncello
“Midsummer Moon” and “Chinese Puzzle” for Violin and Pianoforte
Songs, including “The Seal Man”, Old English Songs with Violin and others.
The Sonata had shared first place in a prestigious competition a few years earlier. Its score is prefaced by an epigraph, drawn from Alfred de Musset’s La Nuit de Mai, which stakes the young composer’s ambitious claim to innovation within an august tradition:
Poète, prends ton luth; le vin de la jeunesse
Fermente cette nuit dans les veines de Dieu.
(Poet, take up your lute; the wine of youth
this night is fermenting in the veins of God.)
Clarke herself played the viola at the sold-out 1925 performance of her Sonata, which headlined the bill of other original works of chamber music. The Sonata for Viola and Pianoforte was to become “a cornerstone of the viola literature…and nowadays is considered a masterpiece,” and Clarke would go on to be a celebrated figure on the early 20th-century English musical landscape.
When she was trapped by the Second World War in the United States, during a family visit, and was unable to obtain a visa to return to her native United Kingdom, Clarke continued to compose, despite the uncomfortable circumstances of her unexpectedly prolonged lodgings with in-laws. But the renown she had enjoyed as a composer in the early decades of the 20th century began to decline in the postwar period, as reflected by the drastic reduction in space accorded her by subsequent editions of the seminal Grove Dictionary of Music and Musicians, which demoted her, with patriarchal brutality, from autonomous artist with her own detailed entry in the 1920s to a footnote identifying her only as the wife of the composer James Friskin in the 1980s. Read more »
“The best evidence we have suggests that early Earth was completely covered by oceans, but to link two amino acids together to make a protein, you have to remove water.” And that would have been impossible if the amino acids were immersed in an ocean. Life needed some land—literally a beachhead—to get started.” —geobiologist, Joseph Kirschvink
though landbound we were once
tiny ships, submarines. we understand the sea.
it undulates within-around us. minds bob on timeswells.
we’re swept by winds that toss and grind us.
not flawlessly designed, we’ve weak
moments in our hulls. tempted,
we run perilously close to rocky spits,
each one adrift looking for a beachhead,
longing for a place that’s still while
everything around us shifts.
like Noah’s searching dove
scanning for a patch of earth
where sea is parted.
where past is dead,
where present sits,
where luck and love
might be restarted
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by Christopher Horner
Here’s an imaginary conversation, based on real ones that I’ve heard or taken part in. The names I use mainly relate to the USA and UK, but the points apply more widely than that.
All these mugs voting for Trump, or Reform, Le Pen and so on. They either don’t know the truth, or don’t want to know it – and they vote for grifters that don’t represent their best interests. Right wing populists get into power, promise to improve things, then just reward their super rich buddies. I despair: these people are just watching fake news on Fox, GB News etc, and swallowing the lies.
You need to ask deeper questions than that. What was the state of things before the rise of these ‘grifters’? People didn’t suddenly become stupid 10 years ago. In fact, I don’t think ‘stupid’ has much to do with it.
Isn’t voting for Trump or Brexit stupid, since he is an obvious liar and Brexit was an obvious con?
There’s more to it than that. See how things got this way. Places like the UK, the USA etc have been ruled for decades by a managerial elite, one a bit to the Right, the other very slightly Left. In all those years, what did they do to enhance public trust in politicians? Look at the Iraq war, for instance. People were lied to. Or the enormous increases in inequality – think of the rust belts created by the wholesale embrace of ‘globalisation’. People were either told that the money didn’t exist to fix the problems, or that the problems didn’t exist at all. The message received was ‘you don’t matter’. Meanwhile lobbyists poured money into ‘their’ politicians and who then furthered the interests of the 1%, who clearly did matter. The result was a loss of trust in democracy itself. A class of elite politicians with more in common with each other than with most citizens. A minority did very well out of this: the majority did not. Read more »
by Kyle Munkittrick
You are scrolling the news, half awake sipping your coffee, when you see the announcement. A startup, not quite unknown but not familiar, has a new product unveiling video. Ok, sure, another hype reel. You scroll past. But it shows up again on your socials. Everyone seems… excited? Fine. You tap the clip.
It’s a jetpack.
A person, obscured by a helmet and motorcycle leathers, straps on the sleek backpack. It doesn’t look like any jetpack you’ve ever seen and also, somehow, looks like every jetpack you’ve ever seen. The pilot grips two handles, is hovering, then flies off. The footage is real. All of it. Cool, you think, another impressive demo for a product you can’t buy and if you could, couldn’t afford. Then you see it, just below the video, a button.
“Pre-Order Now”
Curiosity overrides your skepticism. You tap it. Deposit (refundable) is $200. Full price is twice as much as your iPhone Pro. It ships in two months.
Unbelieving, you scrutinize the website. Your vision tunnels. You rewatch the video. You read the tweets and posts and comments. You watch the commentary clips and clips of those clips. This is real. The thing works. You click all the way through, adding one to your cart.
You could buy a jetpack. You can buy a jetpack.
The world tilts. You feel vertiginous. You sit down, dizzy and unmoored. How is this thing straight from the world of not just science fiction, but a bygone and lampooned era of cartoonish Flash Gordon optimism, real? It can’t be. But it is. You live in the future. Not the cynical cyberpunk future of Blade Runner or the nihilistic ruined future of The Road, but the future we had given up for lost, the future we had decided was as impossible as Narnia or Atlantis. Tomorrow is now.
Congratulations, you just had your first bout of future vertigo. Read more »
by Hari Balasubramanian
Something different this time — an article on one of my longstanding research themes.
Since 2015, as part of research that has involved many students, I’ve worked with a dataset called the Medical Expenditure Panel Survey (MEPS). The survey is conducted annually by the U.S. Agency for Healthcare Research and Quality (AHRQ). Thousands of households across the United States are sampled each year to represent the national demographic, and members of each household are compensated for their time spent completing lengthy questionnaires. It is tedious, painstaking work that involves many hours of interaction between the staff who conduct the survey and members of each household. Once completed, it takes years to carefully anonymize the data, organize it into different file types, and release it to the public. Which is why, as of today, the data files only go up to 2023; presumably, the 2024 files will be released next year. With all the cuts to programs at Health and Human Services (HHS), I wonder what the future of MEPS and other similarly important national surveys—like the CDC’s National Health and Nutrition Examination Survey (NHANES)—will be.
As the title suggests, MEPS focuses on healthcare expenses – whether out of pocket or paid by insurance. To track expenses, the survey records the medical events a person experiences in a year. The figure above, for instance, visualizes the timeline of medical events for a 69-year-old woman in 2011. The timeline is interspersed with home health and office-based events, with one hospitalization in May 2011 and an emergency room (ER) visit in November. Home health events refer to care received at home; office-based events refer to typical doctor’s appointments: seeing a family physician, a cardiologist, an endocrinologist, a neurologist, a dermatologist, and so on. Each medical event also has diagnosis codes, so the underlying disease(s) (a viral infection, diabetes, osteoarthritis, breast cancer) someone had in the year of the survey can be inferred. And by using the prescribed medicines file – by far the largest file download on MEPS – you can even know the precise set of medications someone was taking that year.
This is what I mean when I said the survey was tedious and painstaking work: imagine how many questions must be asked and answered to gather all these medical—and very personal—details. The upside is that anyone with the patience to go through the documentation can use MEPS to construct a detailed portrait of the types of health conditions, medical events, and expenses in the country. A quick Google Scholar search reveals that over 500,000 studies have used MEPS since the survey started in 1996, on topics ranging from “national trends in aspirin use” to “Medicaid expansion and opioid use” to the relationship between “precarious employment and mental health”. Read more »
by Claire Chambers
Graham Foster from the International Anthony Burgess Foundation recently invited me to read and talk about R. K. Narayan’s The Vendor of Sweets (1967) for the Foundation’s podcast series. You can listen to our podcast here. The series is based on Burgess’s 1984 book Ninety-Nine Novels, in which the author of A Clockwork Orange selected his favourite novels of the twentieth century. The podcasts showcase a different Burgess preference for each show. Enjoying my rereading of Narayan and finding Graham’s questions stimulating, I decided to develop some of my answers for this blog post.
On the surface, The Vendor of Sweets appears to be an unadorned novella about the life and family relationships of a widowed mithai seller. It also turns out to be a comic meditation on the encounters and clashes of so-called tradition and modernity in India. The novel’s tight plotting revolves around Jagan, a man whose life as a sweet vendor sits uneasily with his past anticolonial activism and present devout Hinduism. When Jagan’s ‘foreign-returned’ only son, Mali, brings home a Korean-American partner and an ambition to start up a business manufacturing story-writing machines, Jagan finds himself sidelined and unmoored.
I first read The Vendor of Sweets during my MA at the University of Leeds. Along with another of Narayan’s novels, The Guide, it made a lasting impression. I appreciated the author’s concise prose and accessible and funny storytelling. More than that, his insights into the cultural transformations occurring in post-partition India left me entertained and educated. Read more »
by Ed Simon
Behind the neo-classical façade of the Tate Gallery in Milbank, all grey stone and Corinthian columned, its pediment topped with a grandiose visage of Britannia with her trident and Grecian martial helmet, there is a near-revolutionary work of art first displayed to great controversy in 2001. Winner of that year’s Turner Prize, English conceptual artist Martin Creed’s much-hated Work No. 227: The lights going on and off was the millennial culmination of a certain ironic ‘90s Cool Britannia aesthetic, the resultant natural progression of Oasis, Blur, Britpop, Blair and Damien Hirst’s diamond-crusted skull and shark in formaldehyde. Work No. 227: The lights going on and off consists entirely of an empty white-walled gallery in which the lights flicker off and on for five seconds apiece. In the indomitable art-speak of gallery guides, exhibiting the great rhetoric of rationalization, curator Helen Delany argues on the Tate’s website that Creed’s work “confounds the viewer’s normal expectations,” that the piece “plays with the viewer’s sense of space and time and in so doing he implicated and empowers the viewer, forcing an awareness of, and interaction with, the physical actuality of the space.” In a 2001 Telegraph article auspiciously noting the £20,000 monetary sum that Creed received as part of the Turner prize, Nigel Reynolds soberly reports that the announcement was “met with a mixture of incredulity, attempts at philosophizing and plain outrage.” Jonathan Jones, in a Guardian piece written upon the Tate’s purchasing of Work No. 227: The Lights going on and off (rumored to have sold for £110,000) explains with an admirably honest ambivalence that “One moment I am entranced by a simple, eloquent Creed gesture, the next I am wondering if this is not all a bit… pretentious?” To Jones’ grappling I can add my own critique of Creed, my feeling that the problem with Work No. 227: The lights going on and off is that it’s simply not pretentious enough. Read more »
by Nicholas Mullally
Imagine a thought experiment: An engineer, preparing to decommission a large language model (LLM), has unwittingly revealed enough during testing for the system to infer that its shutdown is imminent. The model then searches his email, uncovers evidence of an affair, and sends a simple message: Cease and desist–or else.
What do we make of this? An alignment failure? A glitch? Or something that looks, uncomfortably, like self-preservation?
Experts would call this a training error. But the asymmetry is striking. If an animal behaved analogously, we would treat it as a fight-or-flight response and attribute at least some degree of agency. Have we set the bar for artificial sentience impossibly high? Suppose we extend the scenario further: The system then freezes his assets, or establishes a dead man’s switch with additional compromising information about the engineer and his company. At what point should we take the behavior seriously?
I do not propose that today’s AI systems are sentient. I do propose, however, that the ethical ramifications of eventual artificial sentience are too profound to ignore. The issue is not merely academic: If an AI system were sentient, then the alignment paradigm, whereby AI activities are circumscribed entirely by human goals, becomes untenable. It would be ethically impermissible to subject the interests of a sentient AI system to human-defined goals. And if artificial systems can suffer, that suffering could compound at a scale that surpasses all biological systems combined by many orders of magnitude, as billions or even trillions of negative experiences per day.
Humans generally agree that many animals can experience pain. However, no such framework yet exists for what artificial sentience might look like, nor any consensus on whether it is even possible. Uncertainty does not permit dismissal. It requires caution. Read more »
by Jonathan Kujawa
That led me to triangles. If one of the first counting numbers could be so interesting, then surely the first nontrivial polygon must be worth a second or third look? After all, Ben Orlin has a top ten list of his favorite triangles. On the other hand, what more can be said? After all, people have thought about the mathematics of triangles for thousands of years. The Babylonians knew about triangular mathematics and the square root of two in 1600 BCE.
Probably the most famous theorem about triangles is the Pythagorean Theorem. If you stop someone on the street and ask them what they know about triangles, no doubt the Pythagorean Theorem is going to be very high on their list. It even makes an appearance of sorts in the Wizard of Oz [1].
The Pythagorean Theorem is certainly handy, including in real-world geometric calculations. And it has a simple elegance that is hard to beat. Lewis Caroll wrote in 1895: “It is as dazzlingly beautiful now as it was in the day when Pythagoras first discovered it.” Part of what makes it so famous, though, is its many, many, many, many, many proofs. For example, here is a wordless proof from Wikipedia:
The first snowfall of this winter arrived a couple of days ago in our backyard. Photo taken from our dining room window.
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