In 1963, two years after he penned his timeless classic The Phantom Tollbooth, Norton Juster wrote and illustrated The Dot and the Line: A Romance in Lower Mathematics (public library) — the quirky and infinitely wonderful love story that unfolds in a one-dimensional universe called Lineland where women are dots and men are lines; a hopeful straight line falls hopelessly in love with a dot out of his league, who only has eyes for a sleazy squiggle, and sets about wooing her. Inspired by the Victorian novella Flatland: A Romance of Many Dimensions, it’s an endearing and witty fable of persistence and passion, and a creative masterwork at the intersection of mathematics, philosophy, and graphic design.
To woo the dot, the line decides to master the myriad shapes capable of expressing his full potential.
For months he practiced in secret. Soon he was making squares and triangles, hexagons, parallelograms, rhomboids, polyhedrons, trapezoids, parallelepipeds, decagons, tetragrams and an infinite number of other shapes so complex that he had to letter his sides and angles to keep his place.
Before long he had learned to carefully control ellipses, circles and complex curves and to express himself in any shape he wished — “You name it, I’ll play it.”
So he takes the dot out one evening and metamorphoses into a dizzying array of shapes to charm her with his refined versatility.
Juster brings the story to a modern fairy-tale ending, where the dot and the line live “if not happily ever after, at least reasonably so,” and ends with a charming pun for the mathematically tickled:
MORAL: To the vector belongs the spoils.
Juster’s jacket-copy bio is fittingly delightful:
Norton Juster is a dedicated mathematician whose efforts have been focused primarily on the verification of supermarket register receipts and the calculation of restaurant gratuities in a number of foreign currencies. He has also done pioneering work on the psychological effects of mathematical melancholia.
In 1965, the book was adapted into an equally charming, Oscar-winning short film by Chuck Jones, featured here previously and shared again below for our repeated pleasure:
“That lady whom I look upon as a great man… She understands Newton, she despises superstition and in short she makes me happy.”
By Michelle Legro
“I found, in 1733, a young woman who thought as I did, and who decided to spend several years in the country, cultivating her mind.” So begins the description by Voltaire in his memoirs of a relationship that would define the most productive years of his life. The most famous man in Europe had met his match: the twenty-seven-year-old mathematical prodigy Émilie, Marquise du Châtelet.
The pairing was dynamic and productive — together, they would achieve some of the most important Enlightenment writing on science, physics, and philosophy. But as Nancy Mitford explains in her fantastic 1957 biography of the intellectual power couple, Voltaire in Love (public library), they were devoted not just as intellectuals, but as lovers as well as friends. It was an extraordinary bond that lasted for nearly fifteen years.
In his youth, Voltaire enjoyed the education of a minor aristocrat; Émilie could credit her education solely to her father, who recognized her capacity for learning at an early age. She studied Latin, English, Italian, and Greek, translated the Aeneid, read Homer and Cicero, and, most importantly, excelled at math.
She was less successful at the feminine arts, and at the height of her fame would be chastised for having poor teeth, unkempt hair, and messy clothes. Mitford writes:
Elegance, for women, demands undivided attention; Émilie was an intellectual, she had not endless hours to waste with hairdressers and dressmakers.
Émilie and her talents inspired both awe and jealousy among the nobility; she had removed the most charming and witty man in Paris from their dinner tables. While Voltaire remained a bachelor, Émilie had married at nineteen the dull and abiding Marquis du Châtelet, the perfect arrangement for one to conduct a necessary love affair. Mitford explains:
Love, in France, is treated with formality; friends and relations are left in no doubt as to its beginning and its end. Concealment, necessitating confidants and secret meeting places, is only resorted to when there is a jealous husband or wife. The Marquis de Châtelet always behaved perfectly.
Before they met, both Voltaire and Émilie had a parade of lovers: He enjoyed the attentions, though not the intellect, of wealthy aristocrats who would feed and house him, while she entered into passionate affairs and even once drank poison to discourage a lover from leaving. After her third pregnancy at twenty-seven, she renounced the bearing of children and began the serious study of mathematics.
Their meeting was simple: The pair was introduced by another set of aristocratic lovers over a tavern dinner of chicken fricassee. Voltaire had just returned from England and was thrilled to discuss the latest scientific discoveries of the age. He wrote in a letter:
That lady whom I look upon as a great man… She understands Newton, she despises superstition and in short she makes me happy.
She invited him to her house in the country. He moved in. (The Marquis was often away on military campaigns.) With the essential assistance of Émilie, Voltaire would publish Elémens de la philosophie de Newton in 1738, a simplified guide to the famous scientist, which popularized his most advanced theories, including the gravity of planets, the proof of atoms, the refraction of light, and the uses of telescopes. Voltaire sincerely recognized the intellectual debt he owed his lover. The frontispiece of the work shows the philosopher touched by the divine light light of Newton, reflected down to earth by a heavenly muse, Madame du Châtelet.
Émilie herself sought a more profound goal: the translation into French of Newton’s Mathematica Principia, in which the elements of calculus were first laid out. She not only translated, but also added her own commentary on Newton’s calculations. Her mathematical skills awed her social set. Émilie was a hustler of sorts at the gaming tables in Paris, though she rarely had the luck to win. Mitford writes:
Voltaire said of her that the people she gambled with had no idea she was so learned, though sometimes they were astonished by the speed and accuracy with which she added up the score. He himself once saw her divide nine figures by nine others in her head.
Voltaire and Émilie lived in an intellectual fairyland, punctuated by the occasional need for Voltaire to flee to the country due to an insult or an affront. But as with all French affairs, there was no doubt to the beginning of the love between Voltaire and Émilie, and there was no doubt to its end. In 1744, the Marquis de Saint-Lambert, a poet in the Academie Française and a Byronic figure at court, paid a visit to their country house. Ten years younger than the forty-three-year-old Émilie, Saint-Lambert began a cold seduction of the Marquise, who quickly fell in love. Voltaire, who had been recently ill, was enraged and depressed.
I am here in a beautiful palace… with all of my historical books and my references and with Mme du Châtelet; even so I am one of the most unhappy thinking creatures upon earth.
But in the manner of French love affairs, Voltaire decided that it was better to remain friends with the muse of his life. Instead of challenging the young Saint-Lambert to a duel, he let Émilie go:
No, no, my child, I was in the wrong. You are still in the happy age when one can love and be loved. Make the most of it. An old, ill man like myself can no longer hope for these pleasures.
With Saint-Lambert, Émilie soon found herself pregnant and terrified at the age of forty-four. She threw her attentions on translating Newton from Latin into French. She worked from eight in the morning until coffee at three in the afternoon, then she continued four until ten, and after a few hours with Voltaire, until five in the morning. With her work finished, she died in childbirth surrounded by her husband, her new lover, and Voltaire. (“It is you who has killed me!” he shouted at Saint-Lambert, learning of her death.) Voltaire would help publish her translation of Newton ten years after her death, which remains the standard version of the text in France today.
Math in primary colors and graphic design before there was graphic design.
By Maria Popova
Almost a century before Mondrian made his iconic red, yellow, and blue geometric compositions, and around the time that Edward Livingston Youmans was creating his stunning chemistry diagrams, an eccentric 19th-century civil engineer and mathematician named Oliver Byrne produced a striking series of vibrant diagrams in primary colors for a 1847 edition of the legendary Greek mathematical treatise Euclid’s Elements. Byrne, a vehement opponent of pseudoscience with an especial distaste phrenology, was early to the insight that great design and graphic elegance can powerfully aid learning. He explained that in his edition of Euclid, “coloured diagrams and symbols are used instead of letters for the greater ease of learners.” The book, a masterpiece of Victorian printing and graphic design long before “graphic design” existed as a discipline, is celebrated as one of the most unusual and most beautiful books of the 19th century.
A masterwork of art and science in equal measure, this newly rediscovered treasure mesmerizes the eye with its brightly colored circles, squares, and triangles while it tickles the brain with its mathematical magic.
“Mathematics is the source of timeless profound knowledge, which goes to the heart of all matter and unites us across cultures, continents, and centuries.”
By Maria Popova
French polymath Henri Poincaré saw in mathematics a metaphor for how creativity works, while autistic savant Daniel Tammet believes that math expands our circle of empathy. So how can a field so diverse in its benefits and so rich in human value remain alienating to so many people who subscribe to the toxic cultural mythology that in order to appreciate its beauty, one needs a special kind of “mathematical mind”? That’s precisely what renowned mathematician Edward Frenkel sets out to debunk in Love and Math: The Heart of Hidden Reality (public library) — a quest to unravel the secrets of the “hidden parallel universe of beauty and elegance, intricately intertwined with ours,” premised on the idea that math is just as valuable a part of our cultural heritage as art, music, literature, and the rest of the humanities we so treasure.
Mathematical knowledge is unlike any other knowledge. While our perception of the physical world can always be distorted, our perception of mathematical truths can’t be. They are objective, persistent, necessary truths. A mathematical formula or theorem means the same thing to anyone anywhere — no matter what gender, religion, or skin color; it will mean the same thing to anyone a thousand years from now. And what’s also amazing is that we own all of them. No one can patent a mathematical formula, it’s ours to share. There is nothing in this world that is so deep and exquisite and yet so readily available to all. That such a reservoir of knowledge really exists is nearly unbelievable. It’s too precious to be given away to the “initiated few.” It belongs to all of us.
Math also helps lift our blinders and break the shackles of our own prejudices:
Mathematics is a way to break the barriers of the conventional, an expression of unbounded imagination in the search for truth. Georg Cantor, creator of the theory of infinity, wrote: “The essence of mathematics lies in its freedom.” Mathematics teaches us to rigorously analyze reality, study the facts, follow them wherever they lead. It liberates us from dogmas and prejudice, nurtures the capacity for innovation.
To illustrate why our aversion to math is a product of our culture’s bias rather than of math’s intrinsic whimsy, Frenkel offers an analogy:
What if at school you had to take an “art class” in which you were only taught how to paint a fence? What if you were never shown the paintings of Leonardo da Vinci and Picasso? Would that make you appreciate art? Would you want to learn more about it? I doubt it. You would probably say something like this: “Learning art at school was a waste of my time. If I ever need to have my fence painted, I’ll just hire people to do this for me.” Of course, this sounds ridiculous, but this is how math is taught, and so in the eyes of most of us it becomes the equivalent of watching paint dry. While the paintings of the great masters are readily available, the math of the great masters is locked away.
Countering these conventional attitudes toward math, Frenkel argues that it isn’t necessary to immerse yourself in the field for years of rigorous study in order to appreciate its far-reaching power and beauty:
Mathematics directs the flow of the universe, lurks behind its shapes and curves, holds the reins of everything from tiny atoms to the biggest stars.
There is a common fallacy that one has to study mathematics for years to appreciate it. Some even think that most people have an innate learning disability when it comes to math. I disagree: most of us have heard of and have at least a rudimentary understanding of such concepts as the solar system, atoms and elementary particles, the double helix of DNA, and much more, without taking courses in physics and biology. And nobody is surprised that these sophisticated ideas are part of our culture, our collective consciousness. Likewise, everybody can grasp key mathematical concepts and ideas, if they are explained in the right way. . . .
The problem is: while the world at large is always talking about planets, atoms, and DNA, chances are no one has ever talked to you about the fascinating ideas of modern math, such as symmetry groups, novel numerical systems in which 2 and 2 isn’t always 4, and beautiful geometric shapes like Riemann surfaces. It’s like they keep showing you a little cat and telling you that this is what a tiger looks like. But actually the tiger is an entirely different animal. I’ll show it to you in all of its splendor, and you’ll be able to appreciate its “fearful symmetry,” as William Blake eloquently said.
And as if a mathematician quoting Blake weren’t already an embodiment that boldly counters our cultural stereotypes, Frenkel adds even more compelling evidence from his own journey: Born in Soviet Russia where mathematics had become “an outpost of freedom in the face of an oppressive regime,” discriminatory policies denied him entrance into Moscow State University. But already enamored with math, he secretly snuck into lectures and seminars, read books well into the night, and gave himself the education the system had attempted to bar him from. A young self-taught mathematician, he began publishing provocative papers, one of which was smuggled abroad and gained international acclaim. Soon, he was invited as a visiting professor at Harvard. He was only twenty-one.
The point of this biographical anecdote, of course, isn’t that Frenkel is brilliant, though he certainly is — it’s that the love math ignites in those willing to surrender to its siren call can stir hearts, move minds, and change lives. Frenkel puts it beautifully, returning to math’s equalizing quality:
Mathematics is the source of timeless profound knowledge, which goes to the heart of all matter and unites us across cultures, continents, and centuries. My dream is that all of us will be able to see, appreciate, and marvel at the magic beauty and exquisite harmony of these ideas, formulas, and equations, for this will give so much more meaning to our love for this world and for each other.