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Posts Tagged ‘Beauty

“Beauty is the first test: there is no permanent place in this world for ugly mathematics”*…

Is mathematical beauty real? Or is it just a subjective, human ‘wow’ that is becoming redundant in an AI age? Rita Ahmadi explores…

It is a hot July day in London and I take the bus to Bloomsbury. I often come here for the British Library, the British Museum or the London Review Bookshop. More than a location, Bloomsbury feels like stepping into a work of art – maybe one of Virginia Woolf’s stories, or Duncan Grant’s paintings.

This time, I am here for mathematics: the Hardy Lecture at the London Mathematical Society (LMS), named after G H Hardy, a professor of mathematics at the University of Cambridge, a member of the Bloomsbury Group, and a president of the LMS. You may know him from the film The Man Who Knew Infinity (2015), in which he’s played by Jeremy Irons.

The 2025 lecture is by Emily Riehl of Johns Hopkins University in Baltimore, who is talking about a complex mathematical ‘language’ known as infinity category theory: could we teach it to computers so that they could understand it? If successful, computer programs could verify proofs and construct complex structures in this area.

A few seats to my left, I recognise Kevin Buzzard, wearing the multi-coloured, patterned trousers he’s known for among mathematicians. Based at Imperial College London, Buzzard is working on a computer proof assistant called Lean. His interest is personal: after long disputes with a colleague over a flawed proof, he lost trust, as he often puts it, in ‘human mathematicians’. His mission now is to convince all mathematicians to write their proofs in Lean. In the Q&A after one of his talks, he said of the debate between truth and beauty in mathematics: ‘I reject beauty, I want rigour’ – though his vibrant sense of fashion suggests otherwise.

Interest in an AI-driven approach to mathematics has been exponential, and many mathematicians have left traditional academic research to explore its potential. Recently, one group of distinguished mathematicians designed 10 active, research-level questions for AI to tackle. At the time of writing, various AI companies and researchers had claimed to find solutions, which were under evaluation by the community.

Sitting in the room in Bloomsbury, I stared at the Hardy plaque and wondered: would Hardy find proofs generated by AI beautiful? I wasn’t sure. He believed there should be a strong aesthetic judgment in mathematics, drawing parallels with poetry, and argued that beauty is the first test of good mathematics. He went as far as to say that there is no permanent place in the world for ugly mathematics.

If asked, many mathematicians today still talk about the aesthetic appeal of one approach over another.

Yet we live in a different century to Hardy and his Bloomsbury peers, with different technologies and techniques, so perhaps we need a clearer definition of what mathematical beauty actually is. Over the history of mathematics, we can find examples where both rigour and the pursuit of beauty have shaped mathematics itself. So, if we’re completely replacing this with a computer-assisted quest for truth and rigour, we ought to know what we’d be abandoning, if anything. Is mathematical beauty like the beauty in literature and art – or is it something else?…

[Ahmadi explores the idea of “beauty,” generally and in mathematics; traces the rise of AI as a tool, and concludes…]

… my own definition of beauty in mathematics would be as follows:

“Asimplemathematical structure that surprises even the most experienced mathematicians and transfers a sense of vitality.”

But is an AI-assisted proof simple or surprising? How do we define vitality in a machine? On these questions, the jury is out. Myself, I am torn. Maybe models just need more training to match our creativity. But I also wonder whether our limbic system is required. Can we write proofs without emotional kicks? I am also unsure if perfectly efficient brains can come up with novel revolutionary ideas.

Ultimately, this debate is about more than aesthetics; it is closely tied to the development of AI-assisted mathematics. If AI models can produce novel mathematical structures, how should we direct them? Is it a search for beautiful or truthful structures? A question that possibly guides the years to come.

Some mathematicians say they prefer the ‘truth’ and only the ‘truth’. However, my recent discussions with mathematicians showed me that most immediately recognise, enjoy, and even wholeheartedly smile at a beautiful piece of maths. In fact, they spend their whole lives in search of one…

Fascinating: “The eye of the mathematician,” from @ritaahmadi.bsky.social in @aeon.co.

G. H. Hardy

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As we embrace elegance, we might send garcefully-calculated birthday greetings to Eduard Heine; he was born on this date in 1821. A mathematician, he is best remembered for his introduction of the concept of uniform continuity, for the Mehler–Heine formula, and for the Heine–Cantor theorem… all of them, quite beautiful.

source

Written by (Roughly) Daily

March 16, 2026 at 1:00 am

“These gems have life in them: their colors speak, say what words fail of”*…

Ryan McManus on geologic byproducts, American automotive lore, and the hidden beauty of the industrial age…

Let’s be real for a second: gems are basically very pretty garbage. A byproduct of a geologic or biological process we hang on our bodies for some weird reason. Diamonds are just charcoal that has been squeezed a little longer than usual. Geodes are dried lava that got a little wet. Pearls? Oyster mucus, secreted to smooth out the bivalve equivalent of a stone in one’s shoe. They are unintentional, accidental and, due to the peculiarities of our brains, unquestionably beautiful.

So, if nature can accidentally create works of art while going about its business, why not us?

In the 1940s and 50s, Detroit’s automotive factories had a beautifully messy problem: paint. Workers hand-sprayed enamel paint onto cars on assembly lines, with excess overspray gradually building up on the tracks, skids, and walls of painting bays. This oversprayed paint accumulated over years, layer by layer—creating thick, multicolored deposits that factory workers would periodically chip away and discard.

What they were throwing out, however, would eventually become one of the most sought-after materials in contemporary jewelry making: Fordite, also known as Detroit agate.

Fordite is a man-made gemstone (technically a non-faceted gemstone, called a cabochan), but not like the rubies and diamonds grown in labs. The material represents a fascinating intersection of industrial inefficiency and geological mimicry: As cars were spray painted by hand, the hardened enamel paint built up and baked, creating sedimentary-like layers that mirror how natural agates form over millennia, except compressed into decades of automotive production.

When cut and polished, the visual result is striking—especially in the most valuable samples from the late 1960s and early 1970s, the period of bright “high impact” colors like Ford’s Grabber Blue or Mopar’s Plum Crazy purple. Fordite reveals psychedelic swirls and bands that chronicle automotive color trends year by year, strata by strata: It’s essentially a cross-section of American car culture, with each stratum representing a different model year’s palette. Like the proverbial Jurassic Age mosquito trapped in the stone of amber, the DNA of a bygone epoch is hidden inside.

But Fordite’s deeper appeal lies in its accidental scarcity and temporal specificity. By the 1980s, car manufacturers had moved away from hand-spray painting, adopting an electrostatic process that magnetizes enamels to car bodies, leaving little to no overspray. Efficient, yes—but this electrostatic process ended Fordite production in abundance by the late 1970s. The material can never be recreated—it’s a finite byproduct of a particular moment in industrial history.

This creates a curious value proposition: Fordite is valuable precisely because modern manufacturing has become more efficient. It is industrial waste transformed into luxury material through the simple passage of time and technological progress. With the old factories long gone and today’s automated systems eliminating waste, each piece of Fordite becomes increasingly rare (and expensive), turning Detroit’s former inefficiency into today’s artisanal treasure.

In an age of planned obsolescence and disposable manufacturing, Fordite stands as an accidental monument to the beauty that emerges from industrial imperfection—proof that sometimes the most interesting materials come from processes we’ve intentionally left behind.

A better poet might even see Fordite as metaphor for the City of Detroit itself—forever tied to the automobile, often written off as worthless; unrecognized as a thing of curious quality and glorious beauty waiting for its moment…

More beautiful examples at “Fordite,” from @ryantomorrow.bsky.social in the always-illuminating Why is This Interesting?

* George Eliot (Mary Ann Evans)

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As we rethink remnants, we might recall that it was on this date in 1899 that America’s first fatal automobile accident occurred. At  West 74th Street and Central Park West in New York City, Henry Hale Bliss, a 69-year-old local real estate dealer, stepped off of a south bound 8th Avenue trolley car and was struck by the driver of an electric-powered taxicab (Automobile No. 43). Bliss hit the pavement, crushing his head and chest. He was taken by ambulance to Roosevelt Hospital, but upon arrival the house surgeon, Dr. Marny, judged his injuries too severe to survive. Bliss died the next morning.

Bliss in 1873 (source)

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September 13, 2025 at 1:00 am

“Study Bach. There you will find everything.”*…

Portrait of Johann Sebastian Bach beside a sheet of his handwritten musical notation.

When eminent biologist and author Lewis Thomas was asked what message he would choose to send from Earth into outer space in the Voyager spacecraft, he answered, “I would send the complete works of Johann Sebastian Bach.” After a pause, he added, “But that would be boasting.” (Indeed.)

Evan Goldfine agreed– and decided to devote a year to listening to Bach… all of Bach…

… Take your favorite author and read all of it! The early exploratory stuff, the later material you’ll throw against the wall, the angry letters to publishers and daffy love letters, every word. (This strategy also works for visual artists and filmmakers, etc.)

I’d listened to only about a third of Bach’s work prior to Year of Bach. Why so little? I loved what I’d heard…

How much of Bach do you know? You’ve tasted only a morsel of the world’s biggest cake.

That said, of all the great artists, Bach has the lowest variance in style and quality of output from youth to old age…

My Year of Bach ended in December, but I’m still listening to plenty of JSB. I’m happy whenever my algo serves it up. The music is fundamental and infinite…

… My gamble last January was that I’d enjoy listening to all 200 hours of Bach’s compositions. My goodness, did Bach live up to his part of the deal. The music was never less than excellent…

See also here

A year of majesty and beauty: “37 takeaways from 200 hours with Bach.”

(Image at top: source)

* Johannes Brahms

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As we celebrate completists, we might recall that it was on the date in 1749 that George Frideric Handel’s “Music For The Royal Fireworks” debuted in London’s Green Park. Bach and Handel were contemporaries; they (and Domenico Scarlatti) were born in 1685.

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April 27, 2025 at 1:00 am

“Mathematics, rightly viewed, possesses not only truth, but supreme beauty”*…

Mark Frauenfelder at Boing Boing with a glorious memory…

This cover from the July 1965 issue of Scientific American illustrates the “Four Bugs Problem” featured in Martin Gardner’s “Mathematical Games” column about op art [see here].

The setup: Four bugs are placed at the corners of a square. They start crawling clockwise (or counterclockwise) at a constant rate, with each bug moving directly toward its neighbor. As the bugs move, they always form the corners of a square that both diminishes in size and rotates. Each bug’s path forms a logarithmic spiral.

Gardner said this can be generalized to any number of bugs starting at the corners of a regular polygon with n sides. In these cases, the bugs will always form the corners of a similar polygon that shrinks and rotates as they move.

Here’s an animated version of the Four Bugs Problem you can try out. If you want to try it with a different number of bugs, go here.

Your correspondent still has his copy of that issue. “The beautiful ‘Four Bugs Problem’” from @Frauenfelder in @BoingBoing.

* Bertrand Russell, A History of Western Philosophy

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As we marvel, we might send carefully-calculated birthday greetings to Ian Stewart; he was born on this date in 1945. As a teenager, he was an avid reader of Gardner’s “Mathematical Games,” from which he developed a love of the subject that led him to become a mathematician who has gone on to make important contributions to the field, especially in catastrophe theory.

But Stewart is more widely known as a popularizer of math– who credits Gardner with modeling the skills needed to be an entertaining communicator. Indeed, from 1991 to 2001 Stewart took over the Scientific American column (which had been renamed “Mathematical Recreations”).

For a list of his (remarkable) books on math and science, see here.

source

“Lack of ornamentation is a sign of spiritual strength”*…

(Left) Ranganathaswamy Temple, Srirangam. (Middle) Crossness Pumping Station, London. (Right) Lever House, New York.

Why are buildings today drab and simple, while buildings of the past were ornate and elaborately ornamented? Samuel Hughes proposes an answer…

One of the unifying features of architectural styles before the twentieth century is the presence of ornament. We speak of architectural elements as ornamental inasmuch as they are shaped by aesthetic considerations rather than structural or functional ones. Pilasters, column capitals, sculptural reliefs, finials, brickwork patterns, and window tracery are straightforward examples. Other elements like columns, cornices, brackets, and pinnacles often do have practical functions, but their form is so heavily determined by aesthetic considerations that it generally makes sense to count them as ornament too.

Ornament is amazingly pervasive across time and space. To the best of my knowledge, every premodern architectural culture normally applied ornament to high-status structures like temples, palaces, and public buildings. Although vernacular buildings like barns and cottages were sometimes unornamented, what is striking is how far down the prestige spectrum ornament reached: our ancestors ornamented bridges, power stations, factories, warehouses, sewage works, fortresses, and office blocks. From Chichen Itza to Bradford, from Kyiv to Lalibela, from Toronto to Tiruvannamalai, ornament was everywhere.

Since the Second World War, this has changed profoundly. For the first time in history, many high-status buildings have little or no ornament. Although a trained eye will recognize more ornamental features in modern architecture than laypeople do, as a broad generalization it is obviously true that we ornament major buildings far less than most architectural cultures did historically. This has been celebrated by some and lamented by others. But it is inarguable that it has greatly changed the face of all modern settlements. To the extent that we care about how our towns and cities look, it is of enormous importance.

The naive explanation for the decline of ornament is that the people commissioning and designing buildings stopped wanting it, influenced by modernist ideas in art and design. In the language of economists, this is a demand-side explanation: it has to do with how buyers and designers want buildings to be. The demand-side explanation comes in many variants and with many different emotional overlays. But some version of it is what most people, both pro-ornament and anti-ornament, naturally assume.

However, there is also a sophisticated explanation. The sophisticated explanation says that ornament declined because of the rising cost of labor. Ornament, it is said, is labor-intensive: it is made up of small, fiddly things that require far more bespoke attention than other architectural elements do. Until the nineteenth century, this was not a problem, because labor was cheap. But in the twentieth century, technology transformed this situation. Technology did not make us worse at, say, hand-carving stone ornament, but it made us much better at other things, including virtually all kinds of manufacturing and many kinds of services. So the opportunity cost of hand-carving ornament rose. This effect was famously described by the economist William J Baumol in the 1960s, and in economics it is known as Baumol’s cost disease [see here].

To put this another way: since the labor of stone carvers was now far more productive if it was redirected to other activities, stone carvers could get higher wages by switching to other occupations, and could only be retained as stone carvers by raising their wages so much that stone carving became prohibitively expensive for most buyers. So although we didn’t get worse at stone carving, that wasn’t enough: we had to get better at it if it was to survive against stiffer competition from other productive activities. And so the labor-intensive ornament-rich styles faded away, to be replaced by sparser modern styles that could easily be produced with the help of modern technology. Styles suited to the age of handicrafts were superseded by the styles suited to the age of the machine. So, at least, goes the story.

This is what economists might call a supply-side explanation: it says that desire for ornament may have remained constant, but that output fell anyway because it became costlier to supply. One of the attractive features of the supply-side explanation is that it makes the stylistic transformation of the twentieth century seem much less mysterious. We do not have to claim that – somehow, astonishingly – a young Swiss trained as a clockmaker and a small group of radical German artists managed to convince every government and every corporation on Earth to adopt a radically novel and often unpopular architectural style through sheer force of ideas. In fact, the theory goes, cultural change was downstream of fairly obvious technical and economic forces. Something more or less like modern architecture was the inevitable result of the development of modern technology.

I like the supply-side theory, and I think it is elegant and clever. But my argument here will be that it is largely wrong. It is just not true that twentieth-century technology made ornament more expensive: in fact, new methods of production made many kinds of ornament much cheaper than they had ever been. Absent changes in demand, technology would have changed the dominant methods and materials for producing ornament, and it would have had some effect on ornament’s design. But it would not have resulted in an overall decline. In fact, it would almost certainly have continued the nineteenth-century tendency toward the democratization of ornament, as it became affordable to a progressively wider market. Like furniture, clothes, pictures, shoes, holidays, carpets, and exotic fruit, ornament would have become abundantly available to ordinary people for the first time in history.

In other words, something like the naive demand-side theory has been true all along: to exaggerate a little, it really did happen that every government and every corporation on Earth was persuaded by the wild architectural theory of a Swiss clockmaker and a clique of German socialists, so that they started wanting something different from what they had wanted in all previous ages. It may well be said that this is mysterious. But the mystery is real, and if we want to understand reality, it is what we must face…

And face it Hughes does: “The beauty of concrete,” from @SCP_Hughes in @WorksInProgMag.

Adolf Loos (architect and polemicist of modern architecture)

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As we ponder plainness, we might send ornate birthday greetings to Sir Bertram Clough Williams-Ellis; he was born on this date in 1883. An architect who resisted the modernist trends of his time, he is best remembered as the creator of the Italianate village of Portmeirion in North Wales– the setting of the wonderful televisions series The Prisoner (and the Doctor Who arc The Masque of Mandragora).

Clough Williams-Ellis at Portmeirion in 1969 (source)

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May 28, 2024 at 1:00 am