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Posts Tagged ‘math’

14 FEBRUARY, 2014

The Dot and the Line: A Quirky Vintage Love Story in Lower Mathematics by Norton Juster, Animated by Chuck Jones

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“Moral: The vector belongs to the spoils.”

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: The vector belongs to 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:

Thankfully, The Dot and the Line didn’t suffer the fate of so many vintage gems that now rest in the out-of-print cemetery — it was salvaged in 2001 with a shiny new edition.

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22 NOVEMBER, 2013

Love and Math: Equations as an Equalizer for Humanity

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“Mathematics is the source of timeless profound knowledge, which goes to the heart of all matter and unites us across cultures, continents, and centuries.”

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.

Frenkel makes the same case for math that philosopher Judith Butler made for reading and the humanities, arguing for it as a powerful equalizer of humanity:

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.

BEAUTY OF MATHEMATICS by Yann Pineill & Nicolas Lefaucheux

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.

Drawing from Soviet artist and mathematician Anatolii Fomenko’s 'Mathematical Impressions.' Click image for more.

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.

Love and Math goes on to explore the alchemy of that magic through its various facets, including one of the biggest ideas that ever came from mathematics — the Langlands Program, launched in the 1960s by Robert Langlands, the mathematician who currently occupies Einstein’s office at Princeton, and considered by many the Grand Unified Theory of mathematics. Complement it with Paul Lockhart’s exploration of the whimsy of math and Daniel Tammet on the poetry of numbers.

Thanks, Kirstin

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05 AUGUST, 2013

Synesthesia and the Poetry of Numbers: Autistic Savant Daniel Tammet on Literature, Math, and Empathy, by Way of Borges

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“Like works of literature, mathematical ideas help expand our circle of empathy, liberating us from the tyranny of a single, parochial point of view.”

Daniel Tammet was born with an unusual mind — he was diagnosed with high-functioning autistic savant syndrome, which meant his brain’s uniquely wired circuits made possible such extraordinary feats of computation and memory as learning Icelandic in a single week and reciting the number pi up to the 22,514th digit. He is also among the tiny fraction of people diagnosed with synesthesia — that curious crossing of the senses that causes one to “hear” colors, “smell” sounds, or perceive words and numbers in different hues, shapes, and textures. Synesthesia is incredibly rare — Vladimir Nabokov was among its few famous sufferers — which makes it overwhelmingly hard for the majority of us to imagine precisely what it’s like to experience the world through this sensory lens. Luckily, Tammet offers a fascinating first-hand account in Thinking In Numbers: On Life, Love, Meaning, and Math (public library) — a magnificent collection of 25 essays on “the math of life,” celebrating the magic of possibility in all its dimensions. In the process, he also invites us to appreciate the poetics of numbers, particularly of ordered sets — in other words, the very lists that dominate everything from our productivity tools to our creative inventories to the cheapened headlines flooding the internet.

Reflecting on his second book, Embracing the Wide Sky: A Tour Across the Horizons of the Mind, and the overwhelming response from fascinated readers seeking to know what it’s really like to experience words and numbers as colors and textures — to experience the beauty that a poem and a prime number exert on a synesthete in equal measure — Tammet offers an absorbing simulation of the synesthetic mind:

Imagine.

Close your eyes and imagine a space without limits, or the infinitesimal events that can stir up a country’s revolution. Imagine how the perfect game of chess might start and end: a win for white, or black, or a draw? Imagine numbers so vast that they exceed every atom in the universe, counting with eleven or twelve fingers instead of ten, reading a single book in an infinite number of ways.

Such imagination belongs to everyone. It even possesses its own science: mathematics. Ricardo Nemirovsky and Francesca Ferrara, who specialize in the study of mathematical cognition, write that “like literary fiction, mathematical imagination entertains pure possibilities.” This is the distillation of what I take to be interesting and important about the way in which mathematics informs our imaginative life. Often we are barely aware of it, but the play between numerical concepts saturates the way we experience the world.

Sketches from synesthetic artist and musician Michal Levy's animated visualization of John Coltrane's 'Giant Steps.' Click for details.

Tammet, above all, is enchanted by the mesmerism of the unknown, which lies at the heart of science and the heart of poetry:

The fact that we have never read an endless book, or counted to infinity (and beyond!) or made contact with an extraterrestrial civilization (all subjects of essays in the book) should not prevent us from wondering: what if? … Literature adds a further dimension to the exploration of those pure possibilities. As Nemirovsky and Ferrara suggest, there are numerous similarities in the patterns of thinking and creating shared by writers and mathematicians (two vocations often considered incomparable.)

In fact, this very link between mathematics and fiction, between numbers and storytelling, underpins much of Tammet’s exploration. Growing up as one of nine siblings, he recounts how the oppressive nature of existing as a small number in a large set spurred a profound appreciation of numbers as sensemaking mechanisms for life:

Effaced as individuals, my brothers, sisters, and I existed only in number. The quality of our quantity became something we could not escape. It preceded us everywhere: even in French, whose adjectives almost always follow the noun (but not when it comes to une grande famille). … From my family I learned that numbers belong to life. The majority of my math acumen came not from books but from regular observations and day-to-day interactions. Numerical patterns, I realized, were the matter of our world.

This awareness was the beginning of Tammet’s synesthetic sensibility:

Like colors, the commonest numbers give character, form, and dimension to our world. Of the most frequent — zero and one — we might say that they are like black and white, with the other primary colors — red, blue, and yellow — akin to two, three, and four. Nine, then, might be a sort of cobalt or indigo: in a painting it would contribute shading, rather than shape. We expect to come across samples of nine as we might samples of a color like indigo—only occasionally, and in small and subtle ways. Thus a family of nine children surprises as much as a man or woman with cobalt-colored hair.

Daniel Tammet. Portrait by Jerome Tabet.

Sampling from Jorge Luis Borges’s humorous fictional taxonomy of animals, inspired by the work of nineteenth-century German mathematician Georg Cantor, Tammet points to the deeper insight beneath our efforts to itemize and organize the universe — something Umberto Eco knew when he proclaimed that “the list is the origin of culture” and Susan Sontag intuited when she reflected on why lists appeal to us. Tammet writes:

Borges here also makes several thought-provoking points. First, though a set as familiar to our understanding as that of “animals” implies containment and comprehension, the sheer number of its possible subsets actually swells toward infinity. With their handful of generic labels (“mammal,” “reptile,” “amphibious,” etc.), standard taxonomies conceal this fact. To say, for example, that a flea is tiny, parasitic, and a champion jumper is only to begin to scratch the surface of all its various aspects.

Second, defining a set owes more to art than it does to science. Faced with the problem of a near endless number of potential categories, we are inclined to choose from a few — those most tried and tested within our particular culture. Western descriptions of the set of all elephants privilege subsets like “those that are very large,” and “those possessing tusks,” and even “those possessing an excellent memory,” while excluding other equally legitimate possibilities such as Borges’s “those that at a distance resemble flies,” or the Hindu “those that are considered lucky.”

[…]

Reading Borges invites me to consider the wealth of possible subsets into which my family “set” could be classified, far beyond those that simply point to multiplicity.

Tammet circles back to the shared gifts of literature and mathematics, which both help cultivate our capacity for compassion:

Like works of literature, mathematical ideas help expand our circle of empathy, liberating us from the tyranny of a single, parochial point of view. Numbers, properly considered, make us better people.

The rest of the essays in Thinking In Numbers, ranging from fascinating biographical anecdotes to speculative fiction imagining young Shakespeare’s first arithmetic lessons in zero, are equal parts mind-bending and soul-stirring, and altogether delightful in innumerable ways. Complement it with Paul Lockhart’s multisensory exploration of the whimsy of math, then revisit the extraordinary feats of other autistic savants, from Gregory Blackstock’s astonishing visual taxonomies to Gilles Trehin’s remarkable imaginary city.

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