Geometry’s sweet spot

Reading Love Triangle by Matt Parker for the Telegraph

“These are small,” says Father Ted in the eponymous sitcom, and he holds up a pair of toy cows. “But the ones out there,” he explains to Father Dougal, pointing out the window, “are far away.”

It may not sound like much of a compliment to say that Matt Parker’s new popular mathematics book made me feel like Dougal, but fans of Graham Linehan’s masterpiece will understand. I mean that I felt very well looked after, and, in all my ignorance, handled with a saint-like patience.

Calculating the size of an object from its spatial position has tried finer minds than Dougal’s. A long virtuoso passage early on in Love Triangle enumerates the half-dozen stages of inductive reasoning required to establish the distance of the largest object in the universe — a feature within the cosmic web of galaxies called The Giant Ring. Over nine billion light years away, the Giant Ring still occupies 34.5 degrees of the sky: now that’s what I call big and far away.

Measuring it has been no easy task, and yet the first, foundational step in the calculation turns out to be something as simple as triangulating the length of a piece of road.

“Love Triangle”, as no one will be surprised to learn, is about triangles. Triangles were invented (just go along with me here) in ancient Egypt, where the regularly flooding river Nile obliterated boundary markers for miles around and made rural land disputes a tiresome inevitability. Geometry, says the historian Herodotus around 430 BC, was invented to calculate the exact size of a plot of land. We’ve no reason to disbelieve him.

Parker spends a good amount of time demonstrating the practical usefulness of basic geometry, that allows us to extract the shape and volume of triangular space from a single angle and the length of a single side. At one point, on a visit to Tokyo, he uses a transparent ruler and a tourist map to calculate the height of the city’s tallest tower, the SkyTree.

Having shown triangles performing everyday miracles, he then tucks into their secret: “Triangles,” he explains, “are in the sweet spot of having enough sides to be a physical shape, while still having enough limitations that we can say generalised and meaningful things about them.” Shapes with more sides get boring really quickly, not least because they become so unwieldy in higher dimensions, which is where so many of the joys of real mathematics reside.

Adding dimensions to triangles adds just one corner per dimension. A square, on the other hand, explodes, doubling its number of corners with each dimension. (A cube has eight.) This makes triangles the go-to shape for anyone who wants to assemble meshes in higher dimensions. All sorts of complicated paths are brought within computational reach, making possible all manner of civilisational triumphs, including (but not limited to) photorealistic animations.

So many problems can be cracked by reducing them to triangles, there is an entire mathematical discipline, trigonometry, concerned with the relationships between their angles and side lengths. Parker’s adventures on the spplied side of trigonometry become, of necessity, something of a blooming, buzzing confusion, but his anecdotes are well judged and lead the reader seamlessly into quite complex territory. Ever wanted to know how Kathleen Lonsdale applied Fourier transforms to X-ray waves, making possible Rosalind Franklin’s work on DNA structure? Parker starts us off on that journey by wrapping a bit of paper around a cucumber and cutting it at a slant. Half a dozen pages later, we may not have the firmest grasp of what Parker calls the most incredible bit of maths most people have never heard of, but we do have a clear map of what we do not know.

Whether Parker’s garrulousness charms you or grates on you will be a matter of taste. I have a pious aversion to writers who feel the need to cheer their readers through complex material every five minutes. But it’s hard not to tap your foot to cheap music, and what could be cheaper than Parker’s assertion that introducing coordinates early on in a maths lesson “could be considered ‘putting Descartes before the course’”?

Parker has a fine old time with his material, and only a curmudgeon can fail to be charmed by his willingness to call Heron’s two-thousand-year-old formula for finding the area of a triangle “stupid” (he’s not wrong, neither) and the elongated pentagonal gyrocupolarotunda a “dumb shape”.

How many holes has a straw?

Reading Jordan Ellenberg’s Shape for the Telegraph, 7 July 2021

“One can’t help feeling that, in those opening years of the 1900s, something was in the air,” writes mathematician Jordan Ellenburg.

It’s page 90, and he’s launching into the second act of his dramatic, complex history of geometry (think “History of the World in 100 Shapes”, some of them very screwy indeed).
For page after reassuring page, we’ve been introduced to symmetry, to topology, and to the kinds of notation that make sense of knotty-sounding questions like “how many holes has a straw”?

Now, though, the gloves are off, as Ellenburg records the fin de siecle’s “painful recognition of some unavoidable bubbling randomness at the very bottom of things.”
Normally when sentiments of this sort are trotted out, they’re there to introduce readers to the wild world of quantum mechanics (and, incidentally, we can expect a lot of that sort of thing in the next few years: there’s a centenary looming). Quantum’s got such a grip on our imagination, we tend to forget that it was the johnny-come-lately icing on an already fairly indigestible cake.

A good twenty years before physical reality was shown to be unreliable at small scales, mathematicians were pretzeling our very ideas of space. They had no choice: at the Louisiana Purchase Exposition in 1904, Henri Poincarre, by then the world’s most famous geometer, described how he was trying to keep reality stuck together in light of Maxwell’s famous equations of electromagnetism (Maxwell’s work absolutely refused to play nicely with space). In that talk, he came startlingly close to gazumping Einstein to a theory of relativity.
Also at the same exposition was Sir Ronald Ross, who had discovered that malaria was carried by the bite of the anopheles mosquito. He baffled and disappointed many with his presentation of an entirely mathematical model of disease transmission — the one we use today to predict, well, just about everything, from pandemics to political elections.
It’s hard to imagine two mathematical talks less alike than those of Poincarre and Ross. And yet they had something vital in common: both shook their audiences out of mere three-dimensional thinking.

And thank goodness for it: Ellenburg takes time to explain just how restrictive Euclidean thinking is. For Euclid, the first geometer, living in the 4th century BC, everything was geometry. When he multiplied two numbers, he thought of the result as the area of a rectangle. When he multiplied three numbers, he called the result a “solid’. Euclid’s geometric imagination gave us number theory; but tying mathematical values to physical experience locked him out of more or less everything else. Multiplying four numbers? Now how are you supposed to imagine that in three-dimensional space?

For the longest time, geometry seemed exhausted: a mental gym; sometimes a branch of rhetoric. (There’s a reason Lincoln’s Gettysburg Address characterises the United States as “dedicated to the proposition that all men are created equal”. A proposition is a Euclidean term, meaning a fact that follows logically from self-evident axioms.)

The more dimensions you add, however, the more capable and surprising geometry becomes. And this, thanks to runaway advances in our calculating ability, is why geometry has become our go-to manner of explanation for, well, everything. For games, for example: and extrapolating from games, for the sorts of algorithmical processes we saddle with that profoundly unhelpful label “artificial intelligence” (“artificial alternatives to intelligence” would be better).

All game-playing machines (from the chess player on my phone to DeepMind’s AlphaGo) share the same ghost, the “Markov chain”, formulated by Andrei Markov to map the probabilistic landscape generated by sequences of likely choices. An atheist before the Russian revolution, and treated with predictable shoddiness after it, Markov used his eponymous chain, rhetorically, to strangle religiose notions of free will in their cradle.

From isosceles triangles to free will is quite a leap, and by now you will surely have gathered that Shape is anything but a straight story. That’s the thing about mathematics: it does not advance; it proliferates. It’s the intellectual equivalent of Stephen Leacock’s Lord Ronald, who “flung himself upon his horse and rode madly off in all directions”.

Containing multitudes as he must, Ellenberg’s eyes grow wider and wider, his prose more and more energetic, as he moves from what geometry means to what geometry does in the modern world.

I mean no complaint (quite the contrary, actually) when I say that, by about two-thirds the way in, Ellenberg comes to resemble his friend John Horton Conway. Of this game-playing, toy-building celebrity of the maths world, who died from COVID last year, Ellenburg writes, “He wasn’t being wilfully difficult; it was just the way his mind worked, more associative than deductive. You asked him something and he told you what your question reminded him of.”
This is why Ellenberg took the trouble to draw out a mind map at the start of his book. This and the index offer the interested reader (and who could possibly be left indifferent?) a whole new way (“more associative than deductive”) of re-reading the book. And believe me, you will want to. Writing with passion for a nonmathematical audience, Ellenberg is a popular educator at the top of his game.

Tyrants and geometers

Reading Proof!: How the World Became Geometrical by Amir Alexander (Scientific American) for the Telegraph, 7 November 2019

The fall from grace of Nicolas Fouquet, Louis XIV’s superintendant of finances, was spectacular and swift. In 1661 he held a fete to welcome the king to his gardens at Vaux-le-Vicomte. The affair was meant to flatter, but its sumptuousness only served to convince the absolutist monarch that Fouquet was angling for power. “On 17 August, at six in the evening Fouquet was the King of France,” Voltaire observed; “at two in the morning he was nobody.”

Soon afterwards, Fouquet’s gardens were grubbed up in an act, not of vandalism, but of expropriation: “The king’s men carefully packed the objects into crates and hauled them away to a marshy town where Louis was intent on building his own dream palace,” the Israeli-born US historian Amir Alexander tells us. “It was called Versailles.”

Proof! explains how French formal gardens reflected, maintained and even disseminated the political ideologies of French monarchs. from “the Affable” Charles VIII in the 15th century to poor doomed Louis XVI, destined for the guillotine in 1793. Alexander claims these gardens were the concrete and eloquent expression of the idea that “geometry was everywhere and structured everything — from physical nature to human society, the state, and the world.”

If you think geometrical figures are abstract artefacts of the human mind, think again. Their regularities turn up in the natural world time and again, leading classical thinkers to hope that “underlying the boisterous chaos and variety that we see around us there may yet be a rational order, which humans can comprehend and even imitate.”

It is hard for us now to read celebrations of nature into the rigid designs of 16th century Fontainebleau or the Tuileries, but we have no problem reading them as expressions of political power. Geometers are a tyrant’s natural darlings. Euclid spent many a happy year in Ptolemaic Egypt. King Hiero II of Syracuse looked out for Archimedes. Geometers were ideologically useful figures, since the truths they uncovered were static and hierarchical. In the Republic, Plato extols the virtues of geometry and advocates for rigid class politics in practically the same breath.

It is not entirely clear, however, how effective these patterns actually were as political symbols. Even as Thomas Hobbes was modishly emulating the logical structure of Euclid’s (geometrical) Elements in the composition of his (political) Leviathan (demonstrating, from first principles, the need for monarchy), the Duc de Saint-Simon, a courtier and diarist, was having a thoroughly miserable time of it in the gardens of Louis XIV’s Versailles: “the violence everywhere done to nature repels and wearies us despite ourselves,” he wrote in his diary.

So not everyone was convinced that Versailles, and gardens of that ilk, revealed the inner secrets of nature.

Of the strictures of classical architecture and design, Alexander comments that today, “these prescriptions seem entirely arbitrary”. I’m not sure that’s right. Classical art and architecture is beautiful, not merely for its antiquity, but for the provoking way it toys with the mechanics of visual perception. The golden mean isn’t “arbitrary”.

It was fetishized, though: Alexander’s dead right about that. For centuries, Versailles was the ideal to which Europe’s grand urban projects aspired, and colonial new-builds could and did out-do Versailles, at least in scale. Of the work of Lutyens and Baker in their plans for the creation of New Delhi, Alexander writes: “The rigid triangles, hexagons, and octagons created a fixed, unalterable and permanent order that could not be tampered with.”

He’s setting colonialist Europe up for a fall: that much is obvious. Even as New Delhi and Saigon’s Boulevard Norodom and all the rest were being erected, back in Europe mathematicians Janos Bolyai, Carl Friedrich Gauss and Bernhard Riemann were uncovering new kinds of geometry to describe any curved surface, and higher dimensions of any order. Suddenly the rigid, hierarchical order of the Euclidean universe was just one system among many, and Versailles and its forerunners went from being diagrams of cosmic order to being grand days out with the kids.

Well, Alexander needs an ending, and this is as good a place as any to conclude his entertaining, enlightening, and admirably well-focused introduction to a field of study that, quite frankly, is more rabbit-hole than grass.

I was in Washington the other day, sweating my way up to the Lincoln Memorial. From the top I measured the distance, past the needle of the Washington Monument, to Capitol Hill. Major Pierre Charles L’Enfant built all this: it’s a quintessential product of the Versailles tradition. Alexander calls it “nothing less than the Constitutional power structure of the United States set in stone, pavement, trees, and shrubs.”

For nigh-on 250 years tourists have been slogging from one end of the National Mall to the other, re-enacting the passion of the poor Duc de Saint-Simon in Versailles, who complained that “you are introduced to the freshness of the shade only by a vast torrid zone, at the end of which there is nothing for you but to mount or descend.”

Not any more, though. Skipping down the steps, I boarded a bright red electric Uber scooter and sailed electrically east toward Capitol Hill. The whole dignity-dissolving charade was made possible (and cheap) by map-making algorithms performing geometrical calculations that Euclid himself would have recognised. Because the ancient geometer’s influence on our streets and buildings hasn’t really vanished. It’s been virtualised. Algorithmized. Turned into a utility.

Now geometry’s back where it started: just one more invisible natural good.

M C Escher: “Indulging in imaginary thoughts”

Beating piteously at the windows for New Scientist, 25 May 2018

Leeuwarden-Fryslan, one of the less populated parts of the Netherlands, has been designated this year’s European Capital of Culture. It’s a hub of social and technological and cultural innovation and yet hardly anyone has heard of the place. It makes batteries that the makers claim run circles around Tesla’s current technology, there are advanced plans for the region to go fossil free by 2025, it has one of the highest (and happiest) immigrant populations in Europe, and yet all we can see from the minibus, from horizon to horizon, is cows.

When you’re invited to write about an area you know nothing about, a good place to start is the heritage. But even that can’t help us here. The tiny city of Leeuwarden boasts three hugely famous children: spy and exotic dancer Mata Hari, astrophysicist Jan Hendrik Oort (he of the Oort Cloud) and puzzle-minded artist Maurits Cornelis Escher. The trouble is, all three are famous for being maddening eccentrics.

All Leeuwarden’s poor publicists can do then, having brought us here, is throw everything at us and hope something sticks. And so it happens that, somewhere between the (world-leading) Princessehof ceramics museum and Lan Fan Taal, a permanent pavilion celebrating world languages, someone somewhere makes a small logistical error and locks me inside an M C Escher exhibition.

Escher, who died in 1972, is famous for using mathematical ideas in his art, drawing on concepts from symmetry and hyperbolic geometry to create complex tessellated images. And the Fries Museum in Leeuwarden has gathered more than 80 original prints for me to explore, along with drawings, photographs and memorabilia, so there is no possibility of my getting bored.

Nor is the current exhibition, Escher’s Journey, the usual, chilly celebration of the man’s puzzle-making ability and mathematical sixth sense. Escher was a pleasant, passionate man with a taste for travel, and this show reveals how his personal experiences shaped his art.

Escher’s childhood was by his own account a happy one. His parents took a good deal of interest in his education without ever restricting his intellectual freedom. This was as well, since he was useless at school. Towards the end of his studies, he and his parents traveled through France to Italy, and in Florence he wrote to a friend: “I wallow in it, but so greedily that I fear that my stomach will not be able to withstand it.”

The cultural feast afforded by the city was the least of it. The Leeuwarden native was equally staggered by the surrounding hills – the sheer, three-dimensional fact of them; the rocky coasts and craggy defiles; the huddled mountain villages with squares, towers and houses with sloping roofs. Escher’s love of the Italian landscape consumed him and, much to his mother’s dismay, he was soon permanently settled in the country.

For visitors familiar to the point of satiety and beyond with Escher’s endlessly reproduced and commodified architectural puzzles and animal tessellations, the sketches he made in Italy during the 1920s and 1930s are the highlight of this show. Escher’s favored medium was the engraving. It’s a time-consuming art, and one that affords the artist time to think and to tinker. Inevitably, Escher began merging his sketches into new, realistic wholes. Soon he was trying out unusual perspectives and image compilations. In Still Life with Mirror (1934), he crossed the threshold, creating a reflected world that proves on close inspection to be physically and mathematically impossible.

The usual charge against Escher as an artist – that he was too caught up in the toils of his own visual imagination to express much humanity – is hard to rebuff. There’s a gap here it’s not so easy to bridge: between Escher the approachable and warm-hearted family man and Escher the grumpy Parnassian (he once sent Mick Jagger away with a flea in his ear for asking him for an album cover).

The second world war had a lot to answer for, of course, not least because it drove Escher out of his beloved Italian hills and back, via Switzerland, to the flat old, dull old Netherlands. “Italy, the landscape, the people, they speak to me.” he explained in 1968. “Switzerland doesn’t and Holland even less so.”

Without the landscape to inform his art, other influences came to dominate. Among the places he had visited as war gathered was the Alhambra in Granada. The complex geometric patterns covering its every surface, and their timeless, endless repetition, fascinated him. For days on end he copied the Arab motifs in the palace. Back in the Netherlands, their influence, and Escher’s growing fascination with the mathematics of tessellation, would draw him away from landscapes toward an art consisting entirely of “visualised thoughts”.

By the time his images were based on periodic tilings (meaning that you can slide a pattern in a certain direction and have it exactly overlay the original), his commentaries suggest that Escher had come to embrace his own, somewhat sterile reputation. “I played a game,” he recalled, “indulged in imaginary thoughts, with no other intention than to explore the possibilities of representation. In my work I give a report on these discoveries.”

In the end Escher’s designs became so fiendishly complex, his output dropped almost to zero, and much of his time was taken up lecturing and corresponding about his unique way of working. He corresponded with mathematicians, though he never considered himself one. He knew Roger Penrose. He lived to see the first fractal shapes evolve out of the mathematical studies of Koch and Mandelbrot, though it wasn’t until after his death that Benoît Mandelbrot coined the word “fractal” and popularised the concept.

Eventually, I am missed. At any rate, someone thinks to open the gallery door. I don’t know how long I was in there, locked in close proximity to my childhood hero. (Yes, as a child I did those jigsaw puzzles; yes, as a student I had those posters on my wall) I can’t have been left inside Escher’s Journey for more than a few minutes. But I exited a wreck.

The Fries Museum has lit Escher’s works using some very subtle and precise spot projection; this and the trompe-l’œil monochrome paintwork on the walls of the gallery form a modestly Escherine puzzle all by themselves. Purely from the perspective of exhibition design, this charming, illuminating, and comprehensive show is well worth a visit.

You wouldn’t want to live there, though.