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.

Attack of the Vocaloids

Marrying music and mathematics for The Spectator, 3 August 2019

In 1871, the polymath and computer pioneer Charles Babbage died at his home in Marylebone. The encyclopaedias have it that a urinary tract infection got him. In truth, his final hours were spent in an agony brought on by the performances of itinerant hurdy-gurdy players parked underneath his window.

I know how he felt. My flat, too, is drowning in something not quite like music. While my teenage daughter mixes beats using programs like GarageBand and Logic Pro, her younger brother is bopping through Helix Crush and My Singing Monsters — apps that treat composition itself as a kind of e-sport.

It was ever thus: or was once 18th-century Swiss watchmakers twigged that musical snuff-boxes might make them a few bob. And as each new mechanical innovation has emerged to ‘transform’ popular music, so the proponents of earlier technology have gnashed their teeth. This affords the rest of us a frisson of Schadenfreude.

‘We were musicians using computers,’ complained Pete Waterman, of the synthpop hit factory Stock Aitken Waterman in 2008, 20 years past his heyday. ‘Now it’s the whole story. It’s made people lazy. Technology has killed our industry.’ He was wrong, of course. Music and mechanics go together like beans on toast, the consequence of a closer-than-comfortable relation between music and mathematics. Today, a new, much more interesting kind of machine music is emerging to shape my children’s musical world, driven by non-linear algebra, statistics and generative adversarial networks — that slew of complex and specific mathematical tools we lump together under the modish (and inaccurate) label ‘artificial intelligence’.

Some now worry that artificially intelligent music-makers will take even more agency away from human players and listeners. I reckon they won’t, but I realise the burden of proof lies with me. Computers can already come up with pretty convincing melodies. Soon, argues venture capitalist Vinod Khosla, they will be analysing your brain, figuring out your harmonic likes and rhythmic dislikes, and composing songs made-to-measure. There are enough companies attempting to crack it; Popgun, Amper Music, Aiva, WaveAI, Amadeus Code, Humtap, HumOn, AI Music are all closing in on the composer-less composition.

The fear of tech taking over isn’t new. The Musicians’ Union tried to ban synths in the 1980s, anxious that string players would be put out of work. The big disruption came with the arrival of Kyoko Date. Released in 1996, she was the first seriously publicised attempt at a virtual pop idol. Humans still had to provide Date with her singing and speaking voice. But by 2004 Vocaloid software — developed by Kenmochi Hideki at the Pompeu Fabra University in Barcelona — enabled users to synthesise ‘singing’ by typing in lyrics and a melody. In 2016 Hatsune Miku, a Vocaloid-powered 16-year-old artificial girl with long, turquoise twintails, went, via hologram, on her first North American tour. It was a sell-out. Returning to her native Japan, she modelled Givenchy dresses for Vogue.

What kind of music were these idoru performing? Nothing good. While every other component of the music industry was galloping ahead into a brave new virtualised future — and into the arms of games-industry tech — the music itself seemed stuck in the early 1980s which, significantly, was when music synthesizer builder Dave Smith had first come up with MIDI.

MIDI is a way to represent musical notes in a form a computer can understand. MIDI is the reason discrete notes that fit in a grid dominate our contemporary musical experience. That maddenning clockwork-regular beat that all new music obeys is a MIDI artefact: the software becomes unwieldy and glitch-prone if you dare vary the tempo of your project. MIDI is a prime example (and, for that reason, made much of by internet pioneer-turned-apostate Jaron Lanier) of how a computer can take a good idea and throw it back at you as a set of unbreakable commandments.

For all their advances, the powerful software engines wielded by the entertainment industry were, as recently as 2016, hardly more than mechanical players of musical dice games of the sort popular throughout western Europe in the 18th century.

The original games used dice randomly to generate music from precomposed elements. They came with wonderful titles, too — witness C.P.E. Bach’s A method for making six bars of double counterpoint at the octave without knowing the rules (1758). One 1792 game produced by Mozart’s publisher Nikolaus Simrock in Berlin (it may have been Mozart’s work, but we’re not sure) used dice rolls randomly to select beats, producing a potential 46 quadrillion waltzes.

All these games relied on that unassailable, but frequently disregarded truth, that all music is algorithmic. If music is recognisable as music, then it exhibits a small number of formal structures and aspects that appear in every culture — repetition, expansion, hierarchical nesting, the production of self-similar relations. It’s as Igor Stravinsky said: ‘Musical form is close to mathematics — not perhaps to mathematics itself, but certainly to something like mathematical thinking and relationship.’

As both a musician and a mathematician, Marcus du Sautoy, whose book The Creativity Code was published this year, stands to lose a lot if a new breed of ‘artificially intelligent’ machines live up to their name and start doing his mathematical and musical thinking for him. But the reality of artificial creativity, he has found, is rather more nuanced.

One project that especially engages du Sautoy’s interest is Continuator by François Pachet, a composer, computer scientist and, as of 2017, director of the Spotify Creator Technology Research Lab. Continuator is a musical instrument that learns and interactively plays with musicians in real time. Du Sautoy has seen the system in action: ‘One musician said, I recognise that world, that is my world, but the machine’s doing things that I’ve never done before and I never realised were part of my sound world until now.’

The ability of machine intelligences to reveal what we didn’t know we knew is one of the strangest and most exciting developments du Sautoy detects in AI. ‘I compare it to crouching in the corner of a room because that’s where the light is,’ he explains. ‘That’s where we are on our own. But the room we inhabit is huge, and AI might actually help to illuminate parts of it that haven’t been explored before.’

Du Sautoy dismisses the idea that this new kind of collaborative music will be ‘mechanical’. Behaving mechanically, he points out, isn’t the exclusive preserve of machines. ‘People start behaving like machines when they get stuck in particular ways of doing things. My hope is that the AI might actually stop us behaving like machines, by showing us new areas to explore.’

Du Sautoy is further encouraged by how those much-hyped ‘AIs’ actually work. And let’s be clear: they do not expand our horizons by thinking better than we do. Nor, in fact, do they think at all. They churn.

‘One of the troubles with machine-learning is that you need huge swaths of data,’ he explains. ‘Machine image recognition is hugely impressive, because there are a lot of images on the internet to learn from. The digital environment is full of cats; consequently, machines have got really good at spotting cats. So one thing which might protect great art is the paucity of data. Thanks to his interminable chorales, Bach provides a toe-hold for machine imitators. But there may simply not be enough Bartok or Brahms or Beethoven for them to learn on.’

There is, of course, the possibility that one day the machines will start learning from each other. Channelling Marshall McLuhan, the curator Hans Ulrich Obrist has argued that art is an early-warning system for the moment true machine consciousness arises (if it ever does arise).

Du Sautoy agrees. ‘I think it will be in the world of art, rather than in the world of technology, that we’ll see machines first express themselves in a way that is original and interesting,’ he says. ‘When a machine acquires an internal world, it’ll have something to say for itself. Then music is going to be a very important way for us to understand what’s going on in there.’

The weather forecast: a triumph hiding in plain sight

Reading The Weather Machine by Andrew Blum (Bodley Head) for the Telegraph, 6 July 2019

Reading New York journalist Andrew Blum’s new book has cured me of a foppish and annoying habit. I no longer dangle an umbrella off my arm on sunny days, tripping up my fellow commuters before (inevitably) mislaying the bloody thing on the train to Coulsdon Town. Very late, and to my considerable embarrassment, I have discovered just how reliable the weather forecast is.

My thoroughly English prejudice against the dark art of weather prediction was already set by the time the European Centre for Medium-Range Weather Forecasts opened in Reading in 1979. Then the ECMWF claimed to be able to see three days into the future. Six years later, it could see five days ahead. It knew about Sandy, the deadliest hurricane of 2012, eight days ahead, and it expects to predict high-impact events a fortnight before they happen by the year 2025.

The ECMWF is a world leader, but it’s not an outlier. Look at the figures: weather forecasts have been getting consistently better for 40 straight years. Blum reckons this makes the current global complex of machines, systems, networks and acronyms (and there are lots of acronyms) “a high point of science and technology’s aspirations for society”.

He knows this is a minority view: “The weather machine is a wonder we treat as a banality,” he writes: “a tool that we haven’t yet learned to trust.” The Weather Machine is his attempt to convey the technical brilliance and political significance of an achievement that hides in plain sight.

The machine’s complexity alone is off all familiar charts, and sets Blum significant challenge. “As a rocket scientist at the Jet Propulsion Laboratory put it to me… landing a spacecraft on Mars requires dealing with hundreds of variables,” he writes; “making a global atmospheric model requires hundreds of thousands.” Blum does an excellent job of describing how meteorological theory and observation were first stitched together, and why even today their relationship is a stormy one.

His story opens in heroic times, with Robert FitzRoy one of his more engaging heroes. Fitzroy is best remembered for captaining the HMS Beagle and weathering the puppyish enthusiasm of a young Charles Darwin. But his real claim to fame is as a meteorologist. He dreamt up the term “forecast”, turned observations into predictions that saved sailors’ lives, and foresaw with clarity what a new generation of naval observers would look like. Distributed in space and capable of communicating instantaneously with each other, they would be “as if an eye in space looked down on the whole North Atlantic”.

You can’t produce an accurate forecast from observation alone, however. You also need a theory of how the weather works. The Norwegian physicist Vilhelm Bjerknes came up with the first mathematical model of the weather: a set of seven interlinked partial differential equations that handled the fact that the atmosphere is a far from ideal fluid. Sadly, Bjerknes’ model couldn’t yet predict anything — as he himself said, solutions to his equations “far exceed the means of today’s mathematical analysis”. As we see our models of the weather evolve, so we see works of individual genius replaced by systems of machine computation. In the observational realm, something similar happens: the heroic efforts of individual observers throw up trickles of insight that are soon subsumed in the torrent of data streaming from the orbiting artefacts of corporate and state engineering.

The American philosopher Timothy Morton dreamt up the term “hyperobject” to describe things that are too complex and numinous to describe in the plain terms. Blum, whose earlier book was Tubes: Behind the Scenes at the Internet (2012), fancies his chances at explaining human-built hyperobjects in solid, clear terms, without recourse to metaphor and poesy. In this book, for example, he recognises the close affinity of military and meteorological infrastructures (the staple of many a modish book on the surveillance state), but resists any suggestion that they are the same system.

His sobriety is impressive, given how easy it is to get drunk on this stuff. In October 1946, technicians at the White Sands Proving Ground in Nevada installed a camera in the nose cone of a captured V2, and by launching it, yielded photographs of a quarter of the US — nearly a million square miles banded by clouds “stretching hundreds of miles in rows like streets”. This wasn’t the first time a bit of weather kit acted as an expendable test in a programme of weapons development, and it certainly wasn’t the last. Today’s global weather system has not only benefited from military advancements in satellite positioning and remote sensing; it has made those systems possible. Blum allows that “we learned to see the whole earth thanks to the technology built to destroy the whole earth”. But he avoids paranoia.

Indeed, he is much more impressed by the way countries at hammer and tongs with each other on the political stage nevertheless collaborated closely and well on a global weather infrastructure. Point four of John F Kennedy’s famous 1961 speech on “Urgent National Needs” called for “a satellite system for worldwide weather observation”, and it wasn’t just militarily useful American satellites he had in mind for the task: in 1962 Harry Wexler of the U.S. Weather Bureau worked with his Soviet counterpart Viktor Bugaev on a report proposing a “World Weather Watch”, and by 1963 there was, Blum finds, “a conscious effort by scientists — on both sides of the Iron Curtain, in all corners of the earth — to design an integrated and coordinated apparatus” — this at a time when weather satellites were so expensive they could be justified only on national security grounds.

Blum’s book comes a little bit unstuck at the end. A final chapter that could easily have filled a third of the book is compressed into just a few pages’ handwaving and special pleading, as he conjures up a vision of a future in which the free and global nature of weather information has ceased to be a given and the weather machine, that “last bastion of international cooperation”, has become just one more atomised ghost of a future the colonial era once promised us.

Why end on such a minatory note? The answer, which is by no means obvious, is to be found in Reading. Today 22 nations pay for the ECMWF’s maintenance of a pair of Cray supercomputers. The fastest in the world, these machines must be upgraded every two years. In the US, meanwhile, weather observations rely primarily on the health of four geostationary satellites, at a cost of 11 billion dollars. (America’s whole National Weather Service budget costs only around $1billion.)

Blum leaves open the question, How is an organisation built by nation-states, committed to open data and borne of a global view, supposed to work in a world where information lives on private platforms and travels across private networks — a world in which billions of tiny temperature and barometric sensors, “in smartphones, home devices, attached to buildings, buses or airliners,” are aggregated by the likes of Google, IBM or Amazon?

One thing is disconcertingly clear: Blum’s weather machine, which in one sense is a marvel of continuing modernity, is also, truth be told, a dinosaur. It is ripe for disruption, of a sort that the world, grown so reliant on forecasting, could well do without.

Venice in interesting times

Visiting May You Live In Interesting Times, the 58th International Art Exhibition at the Venice Biennale, for New Scientist, 16 May 2019

BETWEEN now and 24 November, half a million people will visit May You Live in Interesting Times, the main art exhibition of the Venice Biennale. More than 120 years old, the Biennale is the world’s biggest and most venerable art fair. This year’s offering overflows its historical venue in the gardens on Venice’s eastern edge and sprawls across the city.

In a 300-metre-long former rope-making factory in Venice’s Arsenale, a complex of former shipyards and armouries, it is hard to miss data-verse 1 by Japanese DJ and data artist Ryoji Ikeda: the first instalment of a year-long project to realise an entire universe on a gigantic, wall-sized high-definition screen.

Back in Paris, in a studio that consists of hardly more than a few tables and laptops, Ikeda and his programmers have been peeling open huge data sets, using software they have written themselves. From the flood of numbers issuing from CERN, NASA, the Human Genome Project and other open sources, they have fashioned highly detailed abstract animations.

Ikeda is self-taught. He came to visual art from making animations to accompany DJ sets in the squats, clubs and underground parties of Kyoto, Japan. While his own musical taste was eclectic in the extreme, “from classical to voodoo”, Ikeda was drawn to house and dub: forms in which he says “the sound system is the real subject, not the music being played”.

His own “music” reduces sound to sine waves and impulses – and the animations to accompany his sets are equally minimal. “If the sine wave is the simplest expression of sound, what’s the simplest expression of light? For the scientist, that’s a complicated question, but for the artist, the answer is simple: it’s the pixel,” he says.

Ikeda’s project to reduce the world to its essentials continues: “I wondered what would happen if matter were reduced the same way.” Now Ikeda has turned himself into one of art’s curious beasts, the pure “data artist”.

Each of data-verse 1‘s 15-minute-long abstract “dances” explores the universe at a different scale, from the way proteins fold to the pattern of ripples in the cosmic background radiation. However, Ikeda’s aim is not to illustrate or visualise the universe, but to convey the sheer quantity of data we are now gathering in our effort to understand the world.

In the Arsenale, there are glimpses of this new nature. The Milky Way, reduced to wheeling labels. The human body, taken apart and presented as a sequence of what look like archaeological finds. A brain, colour-coded, turned over and over, as if for the inspection of a hyperactive child. A furious blizzard of solar images. And other less-easily identified sequences, where the information has peeled away entirely from the thing it represents, and takes on a life of its own: red pixels move upstream through flowing numbers like so many salmon.

Ikeda differs from his fellow data artists. While a generation has embraced and made art from “big data” – the kind of dynamic information flow that derives from recording a constantly changing world – Ikeda remains wedded to an earlier, more philosophical definition of data as the record of observed facts. Chaos and complexity for their own sake do not interest him. “I never use dynamic data in my work,” he says.

He did try, once. In 2014, he won a residency at the Large Hadron Collider at CERN, Switzerland. But he found the data overwhelming. “They have supercomputers and one experiment takes two years to analyse and compute,” he says, “and still it’s not really enough. They proposed I use this dynamic data, but how could one single artist handle this? We talk of ‘big data’ but no one imagines really how big it is.”

So Ikeda’s data-verse 1 project, which will take a year and two more productions to reach fruition, is founded on that most old-fashioned of ideas, a record of objective truth. It is neither easy nor cheap to realise, and is being supported by watch-makers Audemars Piguet, an increasingly powerful patron of artists who operate on the boundaries between art and science.

Last year, the firm helped Brighton-based art duo Semiconductor realise their CERN-inspired kinetic sculpture HALO. Before that, it invited lidar artist Quayola to map the Swiss valley where it has its factory.

While Audemars Piguet has an interest in art that pushes technological boundaries, Ikeda fights shy of talk of technology, or even physics. He is interested in the truth bound up in numbers themselves. In an interview with Japanese art critic Akira Asada in 2009, he remarked: “I cannot help but wonder if there are any artists today that give real consideration to beauty. To me, it is mathematicians, not artists, who epitomise that kind of individual. There is such a freeness to their thinking that it is almost embarrassing to me.”

Other highlights at the Arsenale include Dominique Gonzalez-Foerster’s Endodrome, (above) a purely virtual work, accessed through a HTC Vive Pro headset. The artist envisioned it “as a kind of organic and mental space, a slightly altered state of consciousness”. Manifesting at first as a sort of hyper-intuitive painting app, in which you use your own outpoured breath as a brush, Endodrome’s imagery becomes ever more precise and surreal. In a show that bristles with anxiety, Gonzalez-Foerster offers the festival-goer an oasis of creative contemplation.

Also at the Arsenale, and fresh from her show Power Plants at London’s Serpentine Gallery, the German artist Hito Steyerl presents This Is the Future, (above) a lush, AI-generated garden of the future, all the more tantalising for the fact that you’ll probably die there. Indeed, this being the future, you’re sure to die there. Steyerl mixes up time and risk, hope and fear, in a wonderfully sly send-up of professional future-gazing.

The Giardini, along the city’s eastern edge, are the traditional site of La Biennale Art Exhibitions since they began in 1895. They’re where you’ll find the national pavilions. Hungary possesses one of the 29 permanent structures here, and this year it’s full of imaginary cameras. They’re the work of cartoonist-turned media artist Tamás Waliczky. Some of his Imaginary Cameras and Other Optical Devices (above) are based on real cameras, others on long-forgotten 19th-century machines; still others are entirely fictional (not to mention impossible). Can you tell the difference? In any event, this understated show does a fine job of reminding us that we see the world in many, highly selective ways.

There’s quite as much activity outside the official venues of the Biennale as within them. At the Ca’ Rezzonico palazzo until 6 July, you have a chance to save an internationally celebrated artist from drowning (or not- it’s really up to you). A meticulously rendered volumetric avatar of Marina Abramović beckons from within a glass tank that is slowly filling with water, in a bid to draw attention to rising sea levels in a city which is famously sinking. Don’t knock Rising (above) till you’ve tried it: this ludicrous-sounding jape proved oddly moving.

Back at the Arsenale, Ed Atkins reprises his installation Olde Food, (above) which had its UK outing at London’s Cabinet gallery last year. Atkins has spent much of his career exploring what roboticist Masahiro Mori’s famously dubbed the “uncanny valley” — the gap that is supposed to separate real people from their human-like creations. Mori’s assumption was that the closer our inventions came to resembling us, the creepier they would become.

Using commercially purchased avatars which he animates using facial recognition software, Atkins has created his share of creepy art zombies. In Olde Food, though, he introduces a new element: an almost unbearably intense compassion.

Atkins has created a world populated by uncanny digital avatars who (when they’re not falling from the sky into sandwiches — you’ll have to trust me when I say this does make a sort of sense) quite clearly yearn for the impress of genuine humanity. These near-people pray. They play piano (or try to). They weep. They’re ugly. They’re uncoordinated. They’re quite hopeless, really. I do wish I could have done something for them.

The unfashionable genius of William De Morgan

Visiting Sublime Symmetry at London’s Guildhall Art Gallery for New Scientist, 28 May 2018

William De Morgan was something of a liability. He once used a fireplace as a makeshift kiln and set fire to his rented London home. And as a businessman he was a disaster. The prices he charged for his tiles and ceramics hardly even paid for the materials, never mind his time.

At the turn of the 20th century, when serious financial problems loomed, only a man of De Morgan’s impractical stripe would resort to writing fiction. But the tactic paid off. No one remembers them these days, but the autobiographical Joseph Vance (1903) and subsequent novels were well regarded at the time, and hugely popular.

Sublime Symmetry at London’s Guildhall Art Gallery wants to tell the story of this polymathic artist but (like De Morgan himself, one suspects) it keeps disappearing down intellectual rabbit holes. De Morgan’s father was the freethinking mathematician Augustus De Morgan, whose student Francis Guthrie came up with the four-colour hypothesis (whereby designing a map, so that countries with a common boundary are differently shaded, requires only four colours). His whimsical tiled fire surround for his friend Charles Dodgson (Lewis Carroll) might have inspired that author’s nonsense verses. Other ceramic projects included the tiles on a dozen P&O liners. Ada Lovelace was a family friend.

On and on like this, until it dawns on you that none of this is an accident, the show’s endless rabbit holes are its point, and fashioning a man like William de Morgan – a mathematically inventive painter of pots, for heaven’s sake – would today be an impossibility.

With all our talk of STEAM and “Sci Art”, the sciences and the humanities are more isolated and defended against each other (“siloed” is the current term of art) than they ever were in De Morgan’s day. And the world itself, as a consequence, is a little less capable of sustaining wonder.

Fusion and freedom
Like Maurits Escher, half a century later, the ceramicist De Morgan drew inspiration from natural forms, and rendered them with a rigor learned from studying classical Arabic design. This fusion of the animate and the geometrical was best expressed on plates and bowls, the best of them made, not in a fireplace, but in the rather more sensible setting of Sand’s End Pottery in Fulham.

De Morgan’s skills as a draftsman were extraordinary. He could draw, free-hand, any pattern around a central line that would have perfect mirror symmetry. Becoming expert in lustreware, he painted his designs directly onto the ceramic surface of his pots and plates, manipulating his original sketches to fit every curve of an object.

It fits De Morgan’s somewhat disorganised reputation that lustreware should have become unfashionable by the end of the century, just as he perfected it.

Even now, it takes a few minutes’ wandering around the Guildhall Gallery for the visitor’s eye to accommodate itself to these objects: so very Victorian, so very hand-done and apparently quotidian. Make the time. This show is a gem, and De Morgan’s achievement is extraordinary. Among these tiles and pots and plates are some of the most natural and apparently effortless fusions of artistic proportion and mathematical rigor ever committed to any medium.

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.

A kind of “symbol knitting”

Reviewing new books by Paul Lockhart and Ian Stewart for The Spectator 

It’s odd, when you think about it, that mathematics ever got going. We have no innate genius for numbers. Drop five stones on the ground, and most of us will see five stones without counting. Six stones are a challenge. Presented with seven stones, we will have to start grouping, tallying and making patterns.

This is arithmetic, ‘a kind of “symbol knitting”’ according to the maths researcher and sometime teacher Paul Lockhart, whose Arithmetic explains how counting systems evolved to facilitate communication and trade, and ended up watering (by no very obvious route) the metaphysical gardens of mathematics.

Lockhart shamelessly (and successfully) supplements the archeological record with invented number systems of his own. His three fictitious early peoples have decided to group numbers differently: in fours, in fives, and in sevens. Now watch as they try to communicate. It’s a charming conceit.

Arithmetic is supposed to be easy, acquired through play and practice rather than through the kind of pseudo-theoretical ponderings that blighted my 1970s-era state education. Lockhart has a lot of time for Roman numerals, an effortlessly simple base-ten system which features subgroup symbols like V (5), L (50) and D (500) to smooth things along. From glorified tallying systems like this, it’s but a short leap to the abacus.

It took an eye-watering six centuries for Hindu-Arabic numbers to catch on in Europe (via Fibonacci’s Liber Abaci of 1202). For most of us, abandoning intuitive tally marks and bead positions for a set of nine exotic squiggles and a dot (the forerunner of zero) is a lot of cost for an impossibly distant benefit. ‘You can get good at it if you want to,’ says Lockhart, in a fit of under-selling, ‘but it is no big deal either way.’

It took another four centuries for calculation to become a career, as sea-going powers of the late 18th century wrestled with the problems of navigation. In an effort to improve the accuracy of their logarithmic tables, French mathematicians broke the necessary calculations down into simple steps involving only addition and subtraction, assigning each step to human ‘computers’.

What was there about navigation that involved such effortful calculation? Blame a round earth: the moment we pass from figures bounded by straight lines or flat surfaces we run slap into all the problems of continuity and the mazes of irrational numbers. Pi, the ratio of a circle’s circumference to its diameter, is ugly enough in base 10 (3.1419…). But calculate pi in any base, and it churns out numbers forever. It cannot be expressed as a fraction of any whole number. Mathematics began when practical thinkers like Archimedes decided to ignore naysayers like Zeno (whose paradoxes were meant to bury mathematics, not to praise it) and deal with nonsenses like pi and the square root of 1.

How do such monstrosities yield such sensible results? Because mathematics is magical. Deal with it.

Ian Stewart deals with it rather well in Significant Figures, his hagiographical compendium of 25 great mathematicians’ lives. It’s easy to quibble. One of the criteria for Stewart’s selection was, he tells us, diversity. Like everybody else, he wants to have written Tom Stoppard’s Arcadia, championing (if necessary, inventing) some unsung heroine to enliven a male-dominated field. So he relegates Charles Babbage to Ada King’s little helper, then repents by quoting the opinion of Babbage’s biographer Anthony Hyman (perfectly justified, so far as I know) that ‘there is not a scrap of evidence that Ada ever attempted original mathematical work’. Well, that’s fashion for you.

In general, Stewart is the least modish of writers, delivering new scholarship on ancient Chinese and Indian mathematics to supplement a well-rehearsed body of knowledge about the western tradition. A prolific writer himself, Stewart is good at identifying the audiences for mathematics at different periods. The first recognisable algebra book, by Al-Khwarizmi, written in the first half of the 9th century, was commissioned for a popular audience. Western examples of popular form include Cardano’s Book on Games of Chance, published 1663. It was the discipline’s first foray into probability.

As a subject for writers, mathematics sits somewhere between physics and classical music. Like physics, it requires that readers acquire a theoretical minimum, without which nothing will make much sense. (Unmathematical readers should not start withSignificant Figures; it is far too compressed.) At the same time, like classical music, mathematics will not stand too much radical reinterpretation, so that biography ends up playing a disconcertingly large role in the scholarship.

In his potted biographies Stewart supplements but makes no attempt to supersede Eric Temple Bell, whose history Men of Mathematics of 1937 remains canonical. This is wise: you wouldn’t remake Civilisation by ignoring Kenneth Clark. At the same time, one can’t help regretting the degree to which a Scottish-born mathematician and science fiction writer born in 1945 has had his limits set by the work of a Scottish-born mathematician and science fiction writer born in 1883. It can’t be helped. Mathematical results are not superseded. When the ancient Babylonians worked out how to solve quadratic equations, their result never became obsolete.

This is, I suspect, why both Lockhart and Stewart have each ended up writing good books about territories adjacent to the meat of mathematics. The difference is that Lockhart did this deliberately. Stewart simply ran out of room.

D’Arcy Wentworth Thompson, the man who shaped biology and art

Biomorphic portrait of D'Arcy Thompson

For New Scientist, 1 February 2017

In a small, windowless corner of the University of Dundee, UK, Caroline Erolin of the Centre for Anatomy & Human Identification is ironing a fossilised pterodactyl.

At least, that’s what she appears to be doing. In fact, Erolin’s “iron” is a handheld 3D scanner, and her digitised animals are now being used as teaching aids worldwide. Her enthusiasm for the work (which she has to squeeze between research into medical visualisation and haptics) is palpable. She is not just bringing animals back from the dead, but helping to bring a great collection back to life.

In 1884, the biologist and classicist D’Arcy Wentworth Thompson began assembling a teaching and research museum in Dundee. An energetic philanthropist and a natural diplomat, Thompson had a broad network of friends and contacts – among them members of Dundee’s own whaling community, who provided him with extraordinary, then-unique specimens of Arctic fauna.

In 1956, the building that housed the University of Dundee’s natural history department was scheduled for demolition and Thompson’s collection, created as part of his work there, was dispersed. Scholars have been scrambling to recover its treasures ever since. Asked whether it can in fact be reassembled, Erolin laughs and gestures at the confines in which the surviving items are (rather artfully) squeezed. “It’s a question of space. We’re already sitting on an entire elephant skeleton. Where on earth would we put that?”

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It’s largely not a genuine problem though, in part because advances in digitisation are changing the priorities of collections worldwide. Even more importantly, it is generally acknowledged that Thompson has outgrown Dundee: he belongs to the world. Together with Charles Darwin, Thompson, who died in 1948, is the most culturally influential English-speaking biologist in history.

We have one book to thank for that: On Growth and Form, first published in 1917 – an event commemorated by an exhibition, A Sketch of the Universe: Art, science and the influence of D’Arcy Thompson, at the Edinburgh City Art Centre.

“Thompson described his landmark book as all preface”

In neither the first edition nor the revised and expanded 1942 version does Thompson talk much about Darwin, and even in the 1940s he considered genetics hardly more than a distraction. Thompson was pursuing an entirely different line: the way in which physical constraints and the initial conditions of life shape the development of plants and animals.

Thompson was fascinated by tiny, single-celled shelled organisms such as foraminifera and radiolaria. He was convinced (rightly) that their wildly diverse shell shapes play no evolutionary role: they arise at random, their beauty emerging from the self-organising properties of matter, not from any biological code.

Even as geneticists like Ernst Mayr and Theodosius Dobzhansky were revealing the genetic mechanisms that constrain how living things evolve, Thompson was revealing the constraints and opportunities afforded to living things by physics and chemistry. Crudely put, genetics explains why dogs, say, look like other dogs. Thompson did something different: he glimpsed why dogs look the way they do.

Most of Thompson’s contemporaries were caught up in a genetic revolution, synthesising the seemingly incompatible demands of chromosomal genetics and Darwinian selection theory. No one ever seriously doubted Thompson’s importance – his book has always been a classic text – but at the same time, few have ever quite known what to do with him.

Portrait of D'Arcy Thompson by Darren McFarlane
Darren McFarlane, Scarus, Pomacanthus, 2012, oil on canvas. (University of Dundee Museum Services © the artist)

Thompson himself (pictured above as morphed by artist Darren McFarlane) understood the problem; he described his landmark book as “all preface”: the sketch of a territory he lacked the mathematical skill to penetrate. What the arguments in On Growth and Form really needed is a computer, and a big one at that (which makes Thompson a character who might have dropped straight out of the pages of Tom Stoppard’s play Arcadia).

Artists, on the other hand, from Henry Moore to Richard Hamilton to Eduardo Paolozzi, knew exactly what to do – and the Edinburgh exhibition combines the University of Dundee’s own collection of biomorphic, Thompsonesque art with new commissions. Several stand-out pieces are by artists who were students at Dundee’s own Duncan of Jordanstone College of Art and Design.

To its credit, Thompson’s alma mater has not been slow to exploit the way his meticulous and beautiful work straddles art and science: it supports a dedicated art-science crossover gallery called LifeSpace, as well as offering degrees in animation, medical art and medical imaging, connecting digital processes with traditional illustration. They are making the most of On Growth and Form‘s centenary, but the influence of Thompson on the university is deep and abiding.

That is as well. For all our anxious predictions about genetic engineering, for all the hype surrounding synthetic biology, and all the many hundreds of graduate design shows stuffed with “imaginary animals”, we have barely begun to explore let alone exploit the spaces Thompson’s vision revealed to us.

Read more: https://www.newscientist.com/article/2120057-darcy-wentworth-thompson-the-man-who-shaped-biology-and-art/#ixzz63gNDj1gc

Let maths illuminate your life!

Thanks to the review I wrote of Thinking in Numbers, an excellent collection of essays about the psychology and culture of numbers, the RSA has invited me along to talk with the author Daniel Tammett on 27 Nov 2012 at 13:00. Follow this link for details of venue etc; you can also follow the event remotely through the following links

http://www.thersa.org/events/listen-live

http://www.thersa.org/events/watch-live