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.

“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

Darren McFarlane, Scarus, Pomacanthus, 2012, oil on canvas. (University of Dundee Museum Services © the artist)

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.

For New Scientist, 1 February 2017

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

Maths into English

One to Nine by Andrew Hodges and The Tiger that Isn’t by Michael Blastland and Andrew Dilnot
reviewed for the Telegraph, 22 September 2007

Twenty-four years have passed since Andrew Hodges published his biography of the mathematician Alan Turing. Hodges, a long-term member of the Mathematical Physics Research Group at Oxford, has spent the years since exploring the “twistor geometry” developed by Roger Penrose, writing music and dabbling with self-promotion.

Follow the link to One to Nine’s web page, and you will soon be stumbling over the furniture of Hodges’s other lives: his music, his sexuality, his ambitions for his self?published novel – the usual spillage. He must be immune to bathos, or blind to it. But why should he care what other people think? He knows full well that, once put in the right order, these base metals will be transformed.

“Writing,” says Hodges, “is the business of turning multi?dimensional facts and ideas into a one?dimensional string of symbols.”

One to Nine – ostensibly a simple snapshot of the mathematical world – is a virtuoso stream of consciousness containing everything important there is to say about numbers (and Vaughan Williams, and climate change, and the Pet Shop Boys) in just over 300 pages. It contains multitudes. It is cogent, charming and deeply personal, all at once.

“Dense” does not begin to describe it. There is extraordinary concision at work. Hodges covers colour space and colour perception in two or three pages. The exponential constant e requires four pages. These examples come from the extreme shallow end of the mathematical pool: there are depths here not everyone will fathom. But this is the point: One to Nine makes the unfathomable enticing and gives the reader tremendous motivation to explore further.

This is a consciously old-fashioned conceit. One to Nine is modelled on Constance Reid’s 1956 classic, From Zero to Infinity. Like Reid’s, each of Hodges’s chapters explores the ideas associated with a given number. Mathematicians are quiet iconoclasts, so this is work that each generation must do for itself.

When Hodges considers his own contributions (in particular, to the mathematics underpinning physical reality), the skin tightens over the skull: “The scientific record of the past century suggests that this chapter will soon look like faded pages from Eddington,” he writes. (Towards the end of his life, Sir Arthur Eddington, who died in 1944, assayed a “theory of everything”. Experimental evidence ran counter to his work, which today generates only intermittent interest.)

But then, mathematics “does not have much to do with optimising personal profit or pleasure as commonly understood”.

The mordant register of his prose serves Hodges as well as it served Turing all those years ago. Like Turing: the Enigma, One to Nine proceeds, by subtle indirection, to express a man through his numbers.

If you think organisations, economies or nations would be more suited to mathematical description, think again. Michael Blastland and Andrew Dilnot’s The Tiger that Isn’t contains this description of the International Passenger Survey, the organisation responsible for producing many of our immigration figures:

The ferry heaves into its journey and, equipped with their passenger vignettes, the survey team members also set off, like Attenboroughs in the undergrowth, to track down their prey, and hope they all speak English. And so the tides of people swilling about the world?… are captured for the record if they travel by sea, when skulking by slot machines, half?way through a croissant, or off to the ladies’ loo.

Their point is this: in the real world, counting is back-breaking labour. Those who sieve the world for numbers – surveyors, clinicians, statisticians and the rest – are engaged in difficult work, and the authors think it nothing short of criminal the way the rest of us misinterpret, misuse or simply ignore their hard-won results. This is a very angry and very funny book.

The authors have worked together before, on the series More or Less – BBC Radio 4’s antidote to the sort of bad mathematics that mars personal decision-making, political debate, most press releases, and not a few items from the corporation’s own news schedule.

Confusion between correlation and cause, wild errors in the estimation of risk, the misuse of averages: Blastland and Dilnot round up and dispatch whole categories of woolly thinking.

They have a positive agenda. A handful of very obvious mathematical ideas – ideas they claim (with a certain insouciance) are entirely intuitive – are all we need to wield the numbers for ourselves; with them, we will be better informed, and will make more realistic decisions.

This is one of those maths books that claims to be self?help, and on the evidence presented here, we are in dire need of it. A late chapter contains the results of a general knowledge quiz given to senior civil servants in 2005.

The questions were simple enough. Among them: what share of UK income tax is paid by the top one per cent of earners? For the record, in 2005 it was 21 per cent. Our policy?makers didn’t have a clue.

“The deepest pitfall with numbers owes nothing to the numbers themselves and much to the slack way they are treated, with carelessness all the way to contempt.”

This jolly airport read will not change all that. But it should stir things up a bit.

Unknown Quantity: a Real and Imagined History of Algebra by John Derbyshire

Unknown Quantity: a Real and Imagined History of Algebra by John Derbyshire
reviewed for the Telegraph,  17 May 2007

In 1572, the civil engineer Rafael Bombelli published a book of algebra, which, he said, would enable a novice to master the subject. It became a classic of mathematical literature. Four centuries later, John Derbyshire has written another complete account. It is not, and does not try to be, a classic. Derbyshire’s task is harder than Bombelli’s. A lot has happened to algebra in the intervening years, and so our expectations of the author – and his expectations of his readers – cannot be quite as demanding. Nothing will be mastered by a casual reading of Unknown Quantity, but much will be glimpsed of this alien, counter-intuitive, yet extremely versatile technique.

Derbyshire is a virtuoso at simplifying mathematics; he is best known for Prime Obsession (2003), an account of the Riemann hypothesis that very nearly avoided mentioning calculus. But if Prime Obsession was written in the genre of mathematical micro-histories established by Simon Singh’s Fermat’s Last Theorem, Derbyshire’s new work is more ambitious, more rigorous and less cute.

It embraces a history as long as the written record and its stories stand or fall to the degree that they contribute to a picture of the discipline. Gone are Prime Obsession’s optional maths chapters; in Unknown Quantity, six “maths primers” preface key events in the narrative. The reader gains a sketchy understanding of an abstract territory, then reads about its discovery. This is ugly but effective, much like the book itself, whose overall tone is reminiscent of Melvyn Bragg’s Radio 4 programme In Our Time: rushed, likeable and impossibly ambitious.

A history of mathematicians as well as mathematics, Unknown Quantity, like all books of its kind, labours under the shadow of E T Bell, whose Men of Mathematics (1937) set a high bar for readability. How can one compete with a description of 19th-century expansions of Abel’s Theorem as “a Gothic cathedral smothered in Irish lace, Italian confetti and French pastry”?

If subsequent historians are not quite left to mopping-up operations, it often reads as though they are. In Unknown Quantity, you can almost feel the author’s frustration as he works counter to his writerly instinct (he is also a novelist), applying the latest thinking to his biography of the 19th-century algebraist Évariste Galois – and draining much colour from Bell’s original.

Derbyshire makes amends, however, with a few flourishes of his own. Also, he places himself in his own account – a cultured, sardonic, sometimes self-deprecating researcher. This is not a chatty book, thank goodness, but it does possess a winning personality.

Sometimes, personality is all there is. The history of algebra is one of stops and starts. Derbyshire declares that for 269 years (during the 13th, 14th and early 15th centuries) little happened. Algebra is the language of abstraction, an unnatural way of thinking: “The wonder, to borrow a trope from Dr Johnson, is not that it took us so long to learn how to do this stuff; the wonder is that we can do it at all.”

The reason for algebra’s complex notation is that, in Leibniz’s phrase, it “relieves the imagination”, allowing us to handle abstract concepts by manipulating symbols. The idea that it might be applicable to things other than numbers – such as sets, and propositions in logic – dawned with tantalising slowness. By far the greater part of Derbyshire’s book tells this tale: how mathematicians learned to let go of number, and trust the terrifying fecundity of their notation.

Then, as we enter the 20th century, and algebra’s union with geometry, something odd happens: the mathematics gets harder to do but easier to imagine. Maths, of the basic sort, is a lousy subject to learn. Advanced mathematics is rich enough to sustain metaphor, so it is in some ways simpler to grasp.

Derbyshire’s parting vision of contemporary algebra – conveyed through easy visual analogies, judged by its applicability to physics, realised in glib computer graphics – is almost a let-down. The epic is over. The branches of mathematics have so interpenetrated each other, it seems unlikely that algebra, as an independent discipline, will survive.

This is not a prospect Derbyshire savours, which lends his book a mordant note. This is more than an engaging history; it records an entire, perhaps endangered, way of thinking.