True Love by Posy Simmonds (1981)

Guardian cartoonist Posy Simmonds published True Love in 1981. It used characters from her established weekly strip cartoon in the Guardian to create an extended meditation on the nature of love, sex, marriage and adultery in a world saturated by media clichés and, in particular, through the prism of the women’s romance comics read by the book’s young protagonist.

Frontispiece to True Love by Posy Simmonds (1981)

In True Love the plain and mousy young Janice Brady is working in a male-dominated advertising company and mistakenly imagines that tall, handsome, suave Stanhope Wright is in love with her. In reality he is juggling at least two other love affairs which he is trying to keep hidden from his long-suffering wife – but in her naive innocence, Janice dreams that she is trembling on the brink of a Grand Passion.

True Love is often acknowledged to be Britain’s first ‘graphic novel’, although it reads now more as a series of loosely related episodes, and includes interludes with other characters from her established ‘Posy’ strip which are only tangentially related to the plot, such as it is.

Incidents

The fifty or so-page-long book is divided into fourteen or so self-contained strips, each with its own title.

Love (Janice) It is a few days before Christmas and Janice is mooning about the Creative Director of Beazeley and Buffin Advertising, Stanhope Wright, who gave her a tin of stilton cheese at the office party that afternoon. She had gone upstairs to fetch her coat and nearly caught Stanhope in a clinch with a secretary. To cover his confusion, Stanhope reached for the nearest thing – the incongruous tin of stilton – and gave it to her with a dapper flourish. Foolish Janice imagines he was waiting there in the dark for her and her alone. He loves her!

True Love (Janice) That night Janice fantasises about her next meeting with Stanhope and how, if she applies enough make-up and wears the right glamour clothes, she will be transformed into a stereotypical dolly bird and Mr Wright can be hers!

True Love by Posy Simmonds (1981)

She imagines becoming so irresistible that Stanhope embraces her, kisses her and they sink onto the shagpile carpet in his office but, wait! No! He will not go all the way. He will respect her purity! His love will remain a pure flame burning in the cathedral of his heart! And dreaming all this, Janice falls asleep with a smile on her face.

Romance (no Janice) Down the Brass Monk pub Stanhope is chatting up a pretty young thing from the Creative Department. She makes her excuses and leaves Stanhope to daydream an amusing series of images done in an 18th century Rococo manner of him seducing her in a bosquey glade… except that the rude leering comments of the middle-aged codgers at the bar (led by the awful alcoholic Edmund Heep) burst his bubble.

Jealousy (Janice) Janice is waiting in the office after work to talk to Stanhope but hears him coming out of a meeting with a young woman creative director, Vicky. Stanhope is, as usual, leering all over Vicky, pawing her and insinuating at her, while on the surface making plans for the shooting of an advert. The bit Janice hears is Stanhope saying, ‘Let’s do it in the country… we can save money by doing it at my place…’ instantly misinterpreting the conversation to be about them having a date for a shag. But she is then shocked and appalled to hear them discussing the need for sheep. Sheep! This is because they’re talking about hiring suitably farmy animals to be in the background of the shoot, but Janice waits till they’ve left and then goes sadly home, appalled by what she’s heard. Sheep!

Rêves d’amour (Janice) In an extended sequence Janice fantasises about dressing up and being escorted by the tallest, handsomest man in the world to a glittering social occasion when all heads turn to marvel at her and her handsome companion, including Stanhope who comes grovellingly apologising to her.

From True Love by Posy Simmonds (1981)

But then Janice’s fantasy continues on to find her way out in the country where she comes across Stanhope and Vicky in mid-snog on some Lake District hillside when all of a sudden they are set upon by a herd of sheep. Janice scares the attacking sheep off by opening a jar of mint sauce (which they’re scared of because of its associations with Sunday roasts) but in the ensuing stampede Janice is herself stampeded over and killed – prompting Stanhope to fall to his knees in lamentation and to apologise for all the rude things he’d ever said to her and to admit how much he LOVED HER, before the handsomest man in the world Cliff Duff, sweeps her mangled body up in her arms and carries her down off the mountain, tears streaming from her face. All of which Janice imagines, tucked up warm in bed.

A Climate of Implicit Trust (No Janice) shows us Stanhope at home, cleaning teeth, putting on pyjamas and getting into bed with his long-suffering wife Vicky. They have an open marriage which appears to mean he can have as many affairs as he wants so long as he tells her about them. But in practice this makes him feel like a shit or, when Trish complaisantly forgives him, he finds oddly frustrating or, if she gets cross with him, he regrets opening his mouth. The scene is complicated when Trish says one of his secretaries (Janice) rang up blabbering something about sheep. Stanhope explains that just refers to the sheep they’re going to hire for the shoot. Maybe this whole sheep theme is meant to be hilarious, though I found it silly and laboured.

Lovers’ Tryst (no Janice) Stanhope drives out to the country where he has a rendezvous with Vicky and they have sex in the open air. He kind of ruins this by fussing on about what his wife thinks and fretting about when they can meet again. The whole thing is counterpointed by the lyrics of the Elizabethan song, It was a lover and his lass – which is spelt out in a curly old-fashioned font along the top of the strip, in ironic counterpoint. It’s clever, it wears its learning on its sleeve, but…. I struggled to find it funny. I thought, Oh yes, I see what she’s doing. very clever. Very funny. Without a smile actually crossing my lips.

Cautionary Tales (no Janice) An extended strip: Stanhope is having an argument with Vicky in the street: she’s got fed up of their whole life rotating about when he can get away from his wife, it’s all starting to feel squalid. When along come George and Wendy Weber and a friend of theirs, Nick. they invite a very embarrassed Stanhope to the pub but he and Vicky make their excuses. George and Wendy realise the woman is Stanhope’s latest fling and it prompts them to talk about what it would be like to have an affair with a younger women, which prompts Nick to remember a little comic sequence in which he actually did have an affair with a woman 25 years his junior, and went on a diet and lost weight to be in shape for her, becoming a vegetarian and eating lots of bran and green salad which leads up to the punchline scene where he’s on the sofa with the little popsy when… his stomach begins making epic gurgling noises. Oops. That is quite funny. For his part, George tells them about a spot of bother at the poly where a student, Gabby, is about to be expelled for doing bad work, not attending tutorials etc… but has told George this is because she is having an affair with her tutor who has made her furious by saying he’s not going to support her application to stay at the poly. All this leads up to one of those scenes where Simmonds parodies a famous painting, in this case the famous painting ‘And When Did You Last See Your Father?’ by Victorian artist William Frederick Yeames – a parody in which all the figures are arranged in the same positions and the lead questioner of the polytechnic board is asking poor Gabby – ‘And when did you last see your tutor?’ Ho ho. Very clever.

Married Love (no Janice) Wendy Weber is at the cinema with George watching one of the arty Italian movies he likes when she suddenly realises she is 40, she is never going to have an affair, never have sex with a different man, those days are gone for good. But slowly she talks herself round with by remembering all the drawbacks and inconveniences and ends up snuggling up closer to dear old George.

From True Love by Posy Simmonds (1981)

Tunnel of Love (Janice) On the tube to work Janice gets squashed up against Dave from the office. She’s reading a True Romance magazine and so interprets being squashed up against tall Dave in the crassest true love clichés. Dave, meanwhile, is reading a book titled ‘Exposures of a Beach Photographer’ and is full of tacky double-entendres, so he has something rather more graphic and sexual on his mind. A meeting of two discourses.

True Romance by Posy Simmonds (1981)

Caveat emptor (Janice) Meeting of all the creatives and execs of Beazeley and Buffin advertising to discuss an upcoming commercial for tinned soup. Janice features as the secretary. The only woman exec, Vicky, objects because she finds the whole conception sexist. Chair of the meeting Stanhope gets Janice to read out the minutes. These are very wordy but are designed to show how the seven men in the room do all share sexist stereotypes and preconceptions, in that all of them just see it as right and fitting that the advert shows a man taking his son for a manly trek across the hills, while the wife and mother remains in the kitchen cooking the soup the ad is designed to promote. The final comment Janice reads out was from a Mr Morton-Berry:

‘At the end of the day, when all’s said and done, a kitchen looks an unnatural sort of place without a MOTHER in it, I think we’d all agree’.

By that stage all the men’s faces are red because they have realised what a sexist lot they actually are, and Vicky the Creative Director has a broad smile on her face, having been vindicated.

L’après-midi d’un Fawn Raincoat (Janice) The day of the shoot, which is taking place in the grounds of Stanhope’s 16th century cottage in the country (a location which has featured in earlier Weber strip cartoons). Stanhope has wandered off somewhere and the director of the piece asks Janice to go and find him. Janice discovers Stanhope and Vicky sharing a glass of wine in a bosky glad. In fact they’re having a fight because Vicky is fed up of being squeezed into the gaps in Stanhope’s busy schedule. Stanhope tries to mollify her by opening th eluxury picnic hamper he’s brought with him. Improbably, he exclaims with frustration when he discovers the hamper contains no cheese! This is the farfetched link to Janice rummaging about in her backpack to find the tin of stilton cheese which Stanhope gave her right back at the start of the narrative. Eve more improbably Janice rolls it down the hillside towards the picnicking couple, but it hits a root, bounces into the air and cracks Stanhope on the back of the head knocking him unconscious. Janice runs down the hillside to comfort Vicky who yells, ‘Why the hell did you do that?’ and then, in a neat ironic touch – ‘I was just about to tell him what a swine he is.’ Which is quite funny.

Home Truths (no Janice) Stanhope is at home on the couch recovering from his concussion and a trip to the hospital, trying to forget the sniggers of the camera crew and the rest of the agency as he was driven off. Now he confesses to his wife Trisha, that he was not hit on the head by a piece of camera equipment as he initially told her; in fact, one of his secretaries threw a cheese at him. Trish puts her hand over her mouth in order not to burst out laughing and says, ‘OK Stanhope… I’ll buy that.’

A Many Splendoured Thing (Janice) It ends oddly. Next morning Stanhope comes into work to find Janice chatting amiably with Dave about  what was on TV last night – it is pretty obvious that he is more her ‘level’ – when Stanhope walks in and Janice gushes her apologies. Stanhope sees a true romance magazine on her desk, picks it up and leafs through it, and the last words belong not to Janice but to the middle-aged philanderer:

‘One is never too old for ROMANCE Janice… Older people have their DREAMS of happiness too, you know…’

And the book ends with Stanhope having a reverie of a True Romance mag for the middle aged (‘Romantic picture stories for MIDDLE-AGED MARRIEDS’) in which an ageing Lothario tells an ageing glamorous woman that he’s not in love with her, doesn’t want to have a heavy affair with her, but just wants to have no-strings, no complications slap and tickle every now and then. And she (Gemma) expresses her relief and thinks: Here at last was the casual fling she had always dreamed of.’

I couldn’t tell if this ending was meant to be satire or mockery or making a feminist point or general social point. Like so many of Simmonds’s strips, I found it attractively drawn, and intelligently expressed, and obviously witty and learnèd and yet somehow, strangely… inconsequential.


A few thoughts

Loose structure

I counted 14 strips or sequences. The ostensible heroine, Janice, is completely absent from six of them, making my point that the thing is not a consecutive novel, but more a string of episodes held together by a very loose narrative about Janice mistakenly falling for Stanhope and, almost on the same day, realising she is deluded – but the loose structure allows Simmonds to give comic or wry meditations on the theme of adultery, open marriages, older men and younger women, and so on, using other, secondary characters.

In other words, contrary to various summaries that I’ve read, this little book is not a sustained parody or pastiche of True Love romance comics. That element is only present in three or four of the strips. It’s about a bit more than that.

The visual style i.e. pink

From a visual point of view, Simmonds enjoys counterpointing the freckly, bong-nosed young heroine with impossibly glamorous images of gorgeous pouting dollybirds from 1950s and 60s romance comics although, as mentioned, this only happens in four or five of the strips.

But the entire book mimics the romance genre’s exaggerated glamour, overblown prose, capital letter fonts, and the liberal use of its tell-tale colour – pink – in a variety of shades from soft lush pink to torrid scarlet.

Intelligence… wasted?

The point is that, even though some of the drawing is actually quite crude (especially seen in hindsight, in the light of how sophisticated Simmonds’s later drawing would become) there is no doubting that a great deal of thought and intelligence have gone into the book’s conception. It shows great ‘learnèd wit’ in the parodies of 18th century rococo nymphs and shepherds, in the parody of the Yeames painting, in the sequence whose main raison d’etre is to counterpoint the Elizabethan song ‘It was a lover and his lass’ with the crude shagging of Stanhope and Vicky on the wet grass of some muddy field.

If you wanted to be critical, you might say that there is an excess of intelligence, sophistication and literary and artistic knowledge on display – expended on a set of pretty trivial subjects (silly office girl gets crush on her boss, boss is having affair with pretty junior, long-suffering wife, tittering friends).

That, although True Love is without doubt clever, wry, amused and mocking – it is rarely actually funny. And I think this is because it all felt too predictable. Middle-aged advertising exec is having an affair while fending off the schoolgirl crush of some secretary, trying to keep his wife onside, and rising above the mockery of his middle-aged friends. The subject matter is not… it’s not very original is it? Maybe the novelty, back in 1981, was treating it in this comic-book style. But that novelty has disappeared over the past 40 years as graphic novels have risen to become commonplace, capable of treating almost any subject, leaving True Love looking more like a historical oddity than a spectacular innovation.

Credit

All images are copyright Posy Simmonds. All images are used under fair play legislation for the purpose of analysis and criticism.


Related links

Other Posy reviews

Alex’s Adventures In Numberland by Alex Bellos (2010)

Alexander Bellos (born in 1969) is a British writer and broadcaster. He is the author of books about Brazil and mathematics, as well as having a column in The Guardian newspaper. After adventures in Brazil (see his Wikipedia page) he returned to England in 2007 and wrote this, his first book. It spent four months in the Sunday Times bestseller list and led on to five more popular maths books.

It’s a hugely enjoyable read for three reasons:

  1. Bellos immediately establishes a candid, open, good bloke persona, sharing stories from his early job as a reporter on the Brighton Argus, telling some colourful anecdotes about his time in Brazil and then being surprisingly open about the way that, when he moved back to Britain, he had no idea what to do. The tone of the book is immediately modern, accessible and friendly.
  2. However this doesn’t mean he is verbose. The opposite. The book is packed with fascinating information. Every single paragraph, almost every sentence contains a fact or insight which makes you sit up and marvel. It is stufffed with good things.
  3. Lastly, although its central theme is mathematics, it approaches this through a wealth of information from the humanities. There is as much history and psychology and anthropology and cultural studies and philosophy as there is actual maths, and these are all subjects which the average humanities graduate can immediately relate to and assimilate.

Chapter Zero – A Head for Numbers

Alex meets Pierre Pica, a linguist who’s studied the Munduruku people of the Amazon and discovered they have little or no sense of numbers. They only have names for numbers up to five. Also, they cluster numbers together logarithmically i.e. the higher the number, the closer together they clustered them. Same thing is done by kindergarten children who only slowly learn that numbers are evenly spaced, in a linear way.

This may be because small children and the Munduruku don’t count so much as estimate using the ratios between numbers.

It may also be because above a certain number (five) Stone Age man needed to make quick estimates along the lines of, Are there more wild animals / members of the other gang, than us?

Another possibility is that distance appears to us to be logarithmic due to perspective: the first fifty yards we see in close detail, the next fifty yards not so detailed, beyond 100 yards looking smaller, and so on.

It appears that we have to be actively taught when young to overcome our logarithmic instincts, and to apply the rule that each successive whole number is an equal distance from its predecessor and successor i.e. the rational numbers lies along a straight line at regular intervals.

More proof that the logarithmic approach is the deep, hard-wired one is the way most of us revert to its perspective when considering big numbers. As John Allen Paulos laments, people make no end of fuss about discrepancies between 2 or 3 or 4 – but are often merrily oblivious to the difference between a million or a billion, let alone a trillion. For most of us these numbers are just ‘big’.

He goes on to describe experiments done on chimpanzees, monkeys and lions which appear to show that animals have the ability to estimate numbers. And then onto experiments with small babies which appear to show that as soon as they can focus on the outside world, babies can detect changes in number of objects.

And it appears that we also have a further number skill, that guesstimating things – the journey takes 30 or 40 minutes, there were twenty or thirty people at the party, you get a hundred, maybe hundred and fifty peas in a sack. When it comes to these figures almost all of us give rough estimates.

To summarise:

  • we are sensitive to small numbers, acutely so of 1, 2, 3, 4, less so of 5, 6, 7, 8, 9
  • left to our own devices we think logarithmically about larger numbers i.e lose the sense of distinction between them, clump them together
  • we have a good ability to guesstimate medium size numbers – 30, 40, 100

But it was only with the invention of notation, a way of writing numbers down, that we were able to create the linear system of counting (where every number is 1 larger than its predecessor, laid out in a straight line, at regular intervals).

And that this cultural invention enabled human beings to transcend our vague guesstimating abilities, and laid the basis for the systematic manipulation of the world which followed

Chapter One – The Counter Culture

The probable origins of counting lie in stock taking in the early agricultural revolution some 8,000 years ago.

We nowadays count using a number base 10 i.e. the decimal system. But other bases have their virtues, especially base 12. It has more factors i.e. is easier to divide: 12 can be divided neatly by 2, 3, 4 and 6. A quarter of 10 is 2.5 but of 12 is 3. A third of 10 is 3.333 but of 12 is 4. Striking that a version of the duodecimal system (pounds, shillings and pence) hung on in Britain till we finally went metric in the 1970s. There is even a Duodecimal Society of America which still actively campaigns for the superiority of a base 12 counting scheme.

Bellos describes a bewildering variety of other counting systems and bases. In 1716 King Charles XII of Sweden asked Emmanuel Swedenborg to devise a new counting system with a base of 64. The Arara in the Amazon count in pairs, the Renaissance author Luca Paccioli was just one of hundreds who have devised finger-based systems of counting – indeed, the widespread use of base 10 probably stems from the fact that we have ten fingers and toes.

He describes a complicated Chinese system where every part of the hand and fingers has a value which allows you to count up to nearly a billion – on one hand!

The Yupno system which attributes a different value for parts of the body up to its highest number, 33, represented by the penis.

Diagram showing numbers attributed to parts of the body by the Yupno tribe

Diagram showing numbers attributed to parts of the body by the Yupno tribe

There’s another point to make about his whole approach which comes out if we compare him with the popular maths books by John Allen Paulos which I’ve just read.

Paulos clearly sees the need to leaven his explanations of comparative probability and Arrow’s Theorem and so on with lighter material and so his strategy is to chuck into his text things which interest him: corny jokes, anecdotes about baseball, casual random digressions which occur to him in mid-flow. But al his examples clearly 1. emanate from Paulos’s own interests and hobby horses (especially baseball) and 2. they are tacked onto the subjects being discussed.

Bellos, also, has grasped that the general reader needs to be spoonfed maths via generous helpings of other, more easily digestible material. But Bellos’s choice of material arises naturally from the topic under discussion. The humour emerges naturally and easily from the subject matter instead of being tacked on in the form of bad jokes.

You feel yourself in the hands of a master storyteller who has all sorts of wonderful things to explain to you.

In fourth millennium BC, an early counting system was created by pressing a reed into soft clay. By 2700 BC the Sumerians were using cuneiform. And they had number symbols for 1, 10, 60 and 3,600 – a mix of decimal and sexagesimal systems.

Why the Sumerians grouped their numbers in 60s has been described as one of the greatest unresolved mysteries in the history of arithmetic. (p.58)

Measuring in 60s was inherited by the Babylonians, the Egyptians and the Greeks and is why we still measure hours in 60 minutes and the divisions of a circle by 360 degrees.

I didn’t know that after the French Revolution, when the National Convention introduced the decimal system of weights and measures, it also tried to decimalise time, introducing a new system whereby every day would be divided into ten hours, each of a hundred minutes, each divided into 100 seconds. Thus there were a very neat 10 x 100 x 100 = 100,000 seconds in a day. But it failed. An hour of 60 minutes turns out to be a deeply useful division of time, intuitively measurable, and a reasonable amount of time to spend on tasks. The reform was quietly dropped after six months, although revolutionary decimal clocks still exist.

Studies consistently show that Chinese children find it easier to count than European children. This may be because of our system of notation, or the structure of number names. Instead of eleven or twelve, Chinese, Japanese and Koreans say the equivalent of ten one, ten two. 21 and 22 become two ten one and two ten two. It has been shown that this makes it a lot simpler and more intuitive to do basic addition and subtraction.

Bellos goes on to describe the various systems of abacuses which have developed in different cultures, before explaining the phenomenal popularity of abacus counting, abacus clubs, and abacus championships in Japan which helps kids develop the ability to perform anzan, using the mental image of an abacus to help its practitioners to sums at phenomenal speed.

Chapter Two – Behold!

The mystical sense of the deep meaning of numbers, from Pythagoras with his vegetarian religious cult of numbers in 4th century BC Athens to Jerome Carter who advises leading rap stars about the numerological significance of their names.

Euclid and the elegant and pure way he deduced mathematical theorems from a handful of basic axioms.

A description of the basic Platonic shapes leads into the nature of tessalating tiles, and the Arab pioneering of abstract design. The complex designs of the Sierpinski carpet and the Menger sponge. And then the complex and sophisticated world of origami, which has its traditionalists, its pioneers and surprising applications to various fields of advanced science, introducing us to the American guru of modern origami, Robert Lang, and the Japanese rebel, Kazuo Haga, father of Haga’s Theorem.

Chapter Three – Something About Nothing

A bombardment of information about the counting systems of ancient Hindus, Buddhists, about number symbols in Sanskrit, Hebrew, Greek and Latin. How the concept of zero was slowly evolved in India and moved to the Muslim world with the result that the symbols we use nowadays are known as the Arabic numerals.

A digression into ‘a set of arithmetical tricks known as Vedic Mathematics ‘ devised by a young Indian swami at the start of the twentieth century, Bharati Krishna Tirthaji, based on a series of 16 aphorisms which he found in the ancient holy texts known as the Vedas.

Shankaracharya is a commonly used title of heads of monasteries called mathas in the Advaita Vedanta tradition. Tirthaji was the Shankaracharya of the monastery at Puri. Bellos goes to visit the current Shankaracharya who explains the closeness, in fact the identity, of mathematics and Hindu spirituality.

Chapter Four – Life of Pi

An entire chapter about pi which turns out not only to be a fundamental aspect of calculating radiuses and diameters and volumes of circles and cubes, but also to have a long history of mathematicians vying with each other to work out its value to as many decimal places as possible (we currently know the value of pi to 2.7 trillion decimal places) and the surprising history of people who have set records reciting the value if pi.

Thus, in 2006, retired Japanese engineer Akira Haraguchi set a world record for reciting the value of pi to the first 100,000 decimal places from memory! It took 16 hours with five minute beaks every two hours to eat rice balls and drink some water.

There are several types or classes of numbers:

  • natural numbers – 1, 2, 3, 4, 5, 6, 7…
  • integers – all the natural numbers, but including the negative ones as well – …-3, -2, -1, 0, 1, 2, 3…
  • fractions
  • which are also called rational numbers
  • numbers which cannot be written as fractions are called irrational numbers
  • transcendent numbers – ‘a transcendental number is an irrational number that cannot be described by an equation with a finite number of terms’

The qualities of the heptagonal 50p coin and the related qualities of the Reuleux triangle.

Chapter Five – The x-factor

The origin of algebra (in Arab mathematicians).

Bellos makes the big historical point that for the Greeks (Pythagoras, Plato, Euclid) maths was geometric. They thought of maths as being about shapes – circles, triangles, squares and so on. These shapes had hidden properties which maths revealed, thus giving – the Pythagoreans thought – insight into the secret deeper values of the world.

It is only with the introduction of algebra in the 17th century (Bellos attributes its widespread adoption to Descartes’s Method in the 1640s) that it is possible to fly free of shapes into whole new worlds of abstract numbers and formulae.

Logarithms turn the difficult operation of multiplication into the simpler operation of addition. If X x Y = Z, then log X + log Y = log Z. They were invented by a Scottish laird John Napier, and publicised in a huge book of logarithmic tables published in 1614. Englishman Henry Briggs established logarithms to base 10 in 1628. In 1620 Englishman Edmund Gunter marked logarithms on a ruler. Later in the 1620s Englishman William Oughtred placed two logarithmic rulers next to each other to create the slide rule.

Three hundred years of dominance by the slide rule was brought to a screeching halt by the launch of the first pocket calculator in 1972.

Quadratic equations are equations with an x and an x², e.g. 3x² + 2x – 4 = 0. ‘Quadratics have become so crucial to the understanding of the world, that it is no exaggeration to say that they underpin modern science’ (p.200).

Chapter Six – Playtime

Number games. The origin of Sudoku, which is Japanese for ‘the number must appear only once’. There are some 5 billion ways for numbers to be arranged in a table of nine cells so that the sum of any row or column is the same.

There have, apparently, only been four international puzzle crazes with a mathematical slant – the tangram, the Fifteen puzzle, Rubik’s cube and Sudoku – and Bellos describes the origin and nature and solutions to all four. More than 300 million cubes have seen sold since Ernö Rubik came up with the idea in 1974. Bellos gives us the latest records set in the hyper-competitive sport of speedcubing: the current record of restoring a copletely scrambled cube to order (i.e. all the faces of one colour) is 7.08 seconds, a record held by Erik Akkersdijk, a 19-year-old Dutch student.

A visit to the annual Gathering for Gardner, honouring Martin Gardner, one of the greatest popularisers of mathematical games and puzzles who Bellos visits. The origin of the ambigram, and the computer game Tetris.

Chapter Seven – Secrets of Succession

The joy of sequences. Prime numbers.

The fundamental theorem of arithmetic – In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers.

The Goldbach conjecture – one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that, Every even integer greater than 2 can be expressed as the sum of two primes. The conjecture has been shown to hold for all integers less than 4 × 1018, but remains unproven despite considerable effort.

Neil Sloane’s idea of persistence – The number of steps it takes to get to a single digit by multiplying all the digits of the preceding number to obtain a second number, then multiplying all the digits of that number to get a third number, and so on until you get down to a single digit. 88 has a persistence of three.

88 → 8 x 8 = 64 → 6 x 4 = 24 → 2 x 4 = 8

John Horton Conway’s idea of the powertrain – For any number abcd its powertrain goes to abcd, in the case of numbers with an odd number of digits the final one has no power, abcde’s powertrain is abcde.

The Recamán sequence Subtract if you can, unless a) it would result in a negative number or b) the number is already in the sequence. The result is:

0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11….

Gijswijt’s sequence a self-describing sequence where each term counts the maximum number of repeated blocks of numbers in the sequence immediately preceding that term.

1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 1, …

Perfect number A perfect number is any number that is equal to the sum of its factors. Thus 6 – its factors (the numbers which divided into it) are 1, 2 and 3. Which also add up to (are the sum of) 6. The next perfect number is 28 because its factors – 1, 2, 4, 7, 14 – add up to 28. And so on.

Amicable numbers A number is amicable if the sum of the factors of the first number equals the second number, and if the sum of the factors of the second number equals the first. The factors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110. Added together these make 284. The factors of 284 are 1, 2, 4, 71 and 142. Added together they make 220!

Sociable numbers In 1918 Paul Poulet invented the term sociable numbers. ‘The members of aliquot cycles of length greater than 2 are often called sociable numbers. The smallest two such cycles have length 5 and 28’

Mersenne’s prime A prime number which can be written in the form 2n – 1 a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. The exponents n which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, … and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, …

These and every other sequence ever created by humankind are documented on The On-Line Encyclopedia of Integer Sequences (OEIS), also cited simply as Sloane’s. This is an online database of integer sequences, created and maintained by Neil Sloane while a researcher at AT&T Labs.

Chapter Eight – Gold Finger

The golden section a number found by dividing a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part.

Phi The number is often symbolized using phi, after the 21st letter of the Greek alphabet. In an equation form:

a/b = (a+b)/a = 1.6180339887498948420 …

As with pi (the ratio of the circumference of a circle to its diameter), the digits go on and on, theoretically into infinity. Phi is usually rounded off to 1.618.

The Fibonnaci sequence Each number in the sequence is the sum of the two numbers that precede it. So the sequence goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. The mathematical equation describing it is Xn+2= Xn+1 + Xn.

as the basis of seeds in flowerheads, arrangement of leaves round a stem, design of nautilus shell and much more.

Chapter Nine – Chance Is A Fine Thing

A chapter about probability and gambling.

Impossibility has a value 0, certainty a value 1, everything else is in between. Probabilities can be expressed as fractions e.g. 1/6 chance of rolling a 6 on a die, or as percentages, 16.6%, or as decimals, 0.16…

The probability is something not happening is 1 minus the probability of that thing happening.

Probability was defined and given mathematical form in 17th century. One contribution was the questions the Chevalier de Méré asked the mathematical prodigy Blaise Pascal. Pascal corresponded with his friend, Pierre de Fermat, and they worked out the bases of probability theory.

Expected value is what you can expect to get out of a bet. Bellos takes us on a tour of the usual suspects – rolling dice, tossing coins, and roulette (invented in France).

Payback percentage if you bet £10 at craps, you can expect – over time – to receive an average of about £9.86 back. In other words craps has a payback percentage of 98.6 percent. European roulette has a payback percentage of 97.3 percent. American roulette, 94.7 percent. On other words, gambling is a fancy way of giving your money away. A miserly slot machine has a payback percentage of 85%. The National Lottery has a payback percentage of 50%.

The law of large numbers The more you play a game of chance, the more likely the results will approach the statistical probability. Toss a coin three times, you might get three heads. Toss a coin a thousand times, the chances are you will get very close the statistical probability of 50% heads.

The law of very large numbers With a large enough sample, outrageous coincidences become likely.

The gambler’s fallacy The mistaken belief that, if something happens more frequently than normal during a given period, it will happen less frequently in the future (or vice versa). In other words, that a random process becomes less random, and more predictable, the more it is repeated.

The birthday paradox The probability that, in a set of n randomly chosen people, some pair of them will have the same birthday. By the pigeonhole principle, the probability reaches 100% when the number of people reaches 367 (since there are only 366 possible birthdays, including February 29). However, 99.9% probability is reached with just 70 people, and 50% probability with 23 people. (These conclusions are based on the assumption that each day of the year (excluding February 29) is equally probable for a birthday.) In other words you only need a group of 23 people to have an evens chance that two of them share a birthday.

The drunkard’s walk

The difficulty of attaining true randomness and the human addiction to finding meaning in anything.

The distinction between playing strategy (best strategy to win a game) and betting strategy (best strategy to maximise your winnings), not always the same.

Chapter Ten – Situation Normal

Carl Friedrich Gauss, the bell curve, normal distribution aka Gaussian distribution. Normal or Gaurrian distribution results in a bell curve. Bellos describes the invention and refinement of the bell curve (he explains that ‘the long tail’ results from a mathematician who envisioned a thin bell curve as looking like two kangaroos facing each other with their long tails heading off in opposite directions). And why

Regression to the mean – if the outcome of an event is determined at least in part by random factors, then an extreme event will probably be followed by one that is less extreme. And recent devastating analyses which show how startlingly random sports achievements are, from leading baseball hitters to Simon Kuper and Stefan Szymanski’s analysis of the form of the England soccer team.

Chapter Eleven – The End of the Line

Two breakthroughs which paved the way for modern i.e. 20th century, maths: the invention of non-Euclidean geometry, specifically the concept of hyperbolic geometry. To picture this draw a triangle on a Pringle. it is recognisably a triangle but all its angles do not add up to 180°, therefore it defies, escapes, eludes all the rule of Euclidean geometry, which were designed for flat 2D surfaces.

Bellos introduces us to Daina Taimina, a maths prof at Cornell University, who invented a way of crocheting hyperbolic surfaces. The result looks curly, like curly kale or the surface of coral.

Anyway, the breakaway from flat 2-D Euclidean space led to theories about curved geometry, either convex like a sphere, or hyperbolic like the pringle. It was this notion of curved space, which paved the way for Einstein’s breakthrough ideas in the early 20th century.

The second big breakthrough was Georg Cantor’s discovery that you can have many different types of infinity. Until Cantor the mathematical tradition from the ancient Greeks to Galileo and Newton had fought shy of infinity which threatened to disrupt so many formulae.

Cantor’s breakthrough was to stop thinking about numbers, and instead think of sets. This is demonstrated through the paradoxes of Hilbert’s Hotel. You need to buckle your safety belt to understand it.

Thoughts

This is easily the best book about maths I’ve ever read. It gives you a panoramic history of the subject which starts with innumerate cavemen and takes us to the edge of Einstein’s great discoveries. But Bellos adds to it all kinds of levels and abilities.

He is engaging and candid and funny. He is fantastically authoritative, taking us gently into forests of daunting mathematical theory without placing a foot wrong. He’s a great explainer. He knows a good story when he sees one, and how to tell it engagingly. And in every chapter there is a ‘human angle’ as he describes his own personal meetings and interviews with many of the (living) key players in the world of contemporary maths, games and puzzles.

Like the Ian Stewart book but on a vastly bigger scale, Bellos makes you feel what it is like to be a mathematician, not just interested in nature’s patterns (the basis of Stewart’s book, Nature’s Numbers) but in the beauty of mathematical theories and discoveries for their own sakes. (This comes over very strongly in chapter seven with its description of some of the weirdest and wackiest number sequences dreamed up by the human mind.) I’ve often read scientists describing the beauty of mathematical theories, but Bellos’s book really helps you develop a feel for this kind of beauty.

For me, I think three broad conclusions emerged:

1. Most mathematicians are in it for the fun. Setting yourself, and solving, mathematical puzzles is obviously extremely rewarding. Maths includes the vast territory of puzzles and games, such as the Sudoku and so on he describes in chapter six. Obviously it has all sorts of real-world application in physics, engineering and so on, but Bellos’s book really brings over that a true understanding of maths begins in puzzles, games and patterns, and often remains there for a lifetime. Like everything else maths is no highly professionalised the property of tenured professors in universities; and yet even to this day – as throughout its history – contributions can be made by enthusiastic amateurs.

2. As he points out repeatedly, many insights which started out as the hobby horses of obsessives, or arcane breakthroughs on the borders of our understanding, and which have been airily dismissed by the professionals, often end up being useful, having applications no-one dreamed of. Either they help unravel aspects of the physical universe undreamed of when they were discovered, or have been useful to human artificers. Thus the development of random number sequences seemed utterly pointless in the 19th century, but now underlies much internet security.

On a profounder note, Bellos expresses the eerie, mystical sense many mathematicians have that it seems so strange, so pregnant with meaning, that so many of these arcane numbers end up explaining aspects of the world their inventors knew nothing of. Ian Stewart has an admirably pragmatic explanation for this: he speculates that nature uses everything it can find in order to build efficient life forms. Or, to be less teleological, over the past 3 and a half billion years, every combination of useful patterns has been tried out. Given this length of time, and the incalculable variety of life forms which have evolved on this planet, it would be strange if every number system conceivable by one of those life forms – humankind – had not been tried out at one time or another.

3. My third conclusion is that, despite John Allen Paulos’s and Bellos’s insistence, I do not live in a world ever-more bombarded by maths. I don’t gamble on anything, and I don’t follow sports – the two biggest popular areas where maths is important – and the third is the twin areas of surveys and opinion polls (55% of Americans believe in alien abductions etc etc) and the daily blizzard of reports (for example, I see in today’s paper that the ‘Number of primary school children at referral units soars’).

I register their existence but they don’t impact on me for the simple reason that I don’t believe any of them. In 1992 every opinion poll said John Major would lose the general election, but he won with a thumping majority. Since then I haven’t believed any poll about anything. For example almost all the opinion polls predicted a win for Remain in the Brexit vote. Why does any sane person believe opinion polls?

And ‘new and shocking’ reports come out at the rate of a dozen a day and, on closer examination, lots of them turn out to be recycled information, or much much more mundane releases of data sets from which journalists are paid to draw the most shocking and extreme conclusions. Some may be of fleeting interest but once you really grasp that the people reporting them to you are paid to exaggerate and horrify, you soon learn to ignore them.

If you reject or ignore these areas – sport, gambling and the news (made up of rehashed opinion polls, surveys and reports) – then unless you’re in a profession which actively requires the sophisticated manipulation of figures, I’d speculate that most of the rest of us barely come into contact with numbers from one day to the next.

I think that’s the answer to Paulos and Bellos when they are in ‘why aren’t more people mathematically numerate?’ mode – maths is difficult, and counter-intuitive, and hard to understand and follow, it is a lot of work, it does make your head ache. Even trying to solve a simple binomial equation hurt my brain. But I think the biggest reason that ‘we’ are so innumerate is simply that – beautiful, elegant, satisfying and thought-provoking though it may be to the professionals – maths is more or less irrelevant to most of our day to day lives, most of the time.


Related links

Reviews of other science books

Cosmology

Environment / human impact

Genetics

  • The Double Helix by James Watson (1968)

Maths

Particle physics

Psychology

  • Irrationality: The Enemy Within by Stuart Sutherland (1992)
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