## Posts Tagged ‘**Beauty**’

## “Mathematics, rightly viewed, possesses not only truth, but supreme beauty”*…

Maryam Mirzakhani did not enjoy mathematics to begin with. She dreamed of being an author or politician, but as a top student at her all-girls school in Tehran she was still disappointed when her first-year maths exam went poorly. Her teacher believed her – wrongly – to have no particular affinity with the subject.

Soon that would all change. “My first memory of mathematics is probably the time [my brother] told me about the problem of adding numbers from 1 to 100,” she recalled later. This was the story of Carl Gauss, the 18th-century genius whose schoolteacher set him this problem as a timewasting exercise – only for his precocious pupil to calculate the answer in a matter of seconds.

The obvious solution is simple but slow: 1+2+3+4. Gauss’s solution is quicker to execute, and far more cunning. It goes like this: divide the numbers into two groups: from 1 to 50, and from 51 to 100. Then, add them together in pairs, starting with the lowest (1) and the highest (100), and working inwards (2+99, 3+98, and so on). There are 50 pairs; the sum of each pair is 101; the answer is 5050. “That was the first time I enjoyed a beautiful solution,” Mirzakhani told the Clay Mathematics Institute in 2008.

Since then, her appreciation for beautiful solutions has taken her a long way from Farzanegan middle school. At 17 she won her first gold medal at the International Mathematics Olympiad. At 27 she earned a doctorate from Harvard University. The Blumenthal Award and Satter Prize followed, and in 2014 she became the first woman to be awarded the Fields Medal, the highest honour a mathematician can obtain.

Before this particular brand of wonder became perceptible to Mirzakhani, she experienced feelings many of us can relate to: to the indifferent, her subject can seem “cold”, even “pointless”. Yet those who persist will be rewarded with glimpses of conceptual glory, as if gifted upon them by a capricious god: “The beauty of mathematics,” she warned, “only shows itself to more patient followers.”

This concept of “beauty” found in maths has been referred to over centuries by many others; though, like beauty itself, it is notoriously difficult to define…

For an experienced mathematician, the greatest equations are beautiful as well as useful. Can the rest of us see what they see? “What makes maths beautiful?”

[From *The New Humanist*, via the ever-illuminating 3 Quarks Daily]

Maryam Mirzakhani died last Friday, a victim of breast cancer; she was 40. As Peter Sarnak (a mathematician at Princeton University and the Institute for Advanced Study) said, her passing is “a big loss and shock to the mathematical community worldwide.” See also here.

* Bertrand Russell, *A History of Western Philosophy*

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**As we accede to awe,** we might spare a thought for Andrey (Andrei) Andreyevich Markov; he died on this date in 1922. A Russian mathematician, he helped to develop the theory of stochastic processes, especially those now called Markov chains: sequences of random variables in which the future variable is determined by the present variable but is independent of the way in which the present state arose from its predecessors. (For example, the probability of winning at the game of *Monopoly* can be determined using Markov chains.) His work on the study of the probability of mutually-dependent events has been developed and widely applied to the biological and social sciences.

## “Beauty is the first test: there is no permanent place in the world for ugly mathematics”*…

Long-time readers will know of your correspondent’s admiration and affection for Martin Gardner (c.f., e.g., here and here). So imagine his delight to learn from @MartyKrasney of this…

Martin wrote about 300 articles for

Scientific Americanbetween 1952 and 1998, most famously in his legendary “Mathematical Games” column starting in Jan 1957. Many of those articles are now viewed as classics, from his seminal piece on hexaflexagons in Dec 1956—which led to the offer to write a regular column for the magazine—to his breakthrough essays on pentomnoes, rep-tiles, the Soma cube, the art of Escher, the fourth dimension, sphere packing, Conway’s game of Life, Newcomb’s paradox, Mandelbrot’s fractals, Penrose tiles, and RSA cryptography, not forgetting the recurring numerological exploits of his alter ego Dr. Matrix, and the tongue-in-cheek April Fool column from 1975.Many of those gems just listed were associated with beautiful graphics and artwork, so it’s no surprise that Martin scored some

Scientific Americancovers over the years, though as we’ll see below, there’s surprisingly little overlap between his “greatest hits” and his “cover stories.”It’s worth noting that, just as the magazine editors selected the titles under which his original articles appeared—he generally ditched those in favor of his own when he republished them in the spin-off books—artwork submitted was often altered by

Scientific Americanstaff artists…

The full dozen, replete with the cover art, at “A Gardner’s Dozen—Martin’s *Scientific American* Cover Stories.”

* G.H. Hardy

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**As we agree with G.K Chesterton that “the difference between the poet and the mathematician is that the poet tries to get his head into the heavens while the mathematician tries to get the heavens into his head,”** we might send carefully calculated birthday greetings to John Charles Fields, he was born on this date in 1863. A mathematician of accomplishment, he is better remembered as a tireless advocate of the field and its importance– and best remembered as the founder of the award posthumously named for him: The Fields Medal, familiarly known as “the Nobel of mathematics.”

## “It is amazing how complete is the delusion that beauty is goodness”*…

Conversely, it’s amazing how how complete the delusion that all mistakes are ugly…

Corey Johnson runs a Tumblr called “Art of the Glitch,” where he posts images that he’s captured of erratic irregularities in analog technology, but only those that meet the requirements of his personal interest in glitch art…

“There was a precision and a refinement to that particular glitch style that I’ve been chasing after in my own work,” Johnson says. He’s not interested in the total destruction that some glitch artists practice; he sees the glitch as “more a storytelling tool than an aesthetic unto itself.” More resolutely, he says he’s looking for that “weird balance of destruction and tangibility.”

No more is this obvious in his latest series of images that have been created from obstreperous VCR errors. These often skew a single subject—the centerpiece of his story—especially faceless people: juddering skulls wrapped in pallid skin with sudden bands of discordant color ripping across them like the scratch of a claw. Add to this the inescapable repetition of the GIF, and these images almost seem depraved, resembling hell’s endless torture of its sinners…

More at “The Creepy Beauty of VCR Errors,” and “Art of the Glitch.”

* Leo Tolstoy, *The Kreutzer Sonata*

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**As we glory in glitches,** we might send delightfully-drawn birthday greetings to Paul Gustave Doré; he was born on this date in 1832. An engraver, illustrator, and sculptor, Dore is probably best-remembered as the man who showed us Heaven and Hell: the canonical illustrator of works by Rabelais, Balzac, Milton, Cervantes, and Dante.

## “I know numbers are beautiful. If they aren’t beautiful, nothing is”*…

Euler’s identity: Math geeks extol its beauty, even finding in it hints of a mysterious connectedness in the universe. It’s on tank tops and coffee mugs [and tattoos]. Aliens, apparently, carve it into crop circles (in 8-bit binary code). It’s appeared on

The Simpsons. Twice.What’s the deal with Euler’s identity? Basically, it’s an equation about numbers—specifically, those elusive constants π and

e. Both are “transcendental” quantities; in decimal form, their digits unspool into infinity. And both are ubiquitous in scientific laws. But they seem to come from different realms: π (3.14159 …) governs the perfect symmetry and closure of the circle; it’s in planetary orbits, the endless up and down of light waves.e(2.71828 …) is the foundation of exponential growth, that accelerating trajectory of escape inherent to compound interest, nuclear fission, Moore’s law. It’s used to model everything that grows…Now, maybe you’ve never thought of math equations as “beautiful,” but look at that result: It combines the five most fundamental numbers in math—0, 1,

e, i, and π—in a relation of irreducible simplicity. (Even more astonishing if you slog through the proof, which involves infinite sums, factorials, and fractions nested within fractions within fractions like matryoshka dolls.) And remember,eand π are infinitely long decimals with seemingly nothing in common; they’re the ultimate jigsaw puzzle pieces. Yet they fit together perfectly—not to a few places, or a hundred, or a million, but all the way to forever…But the weirdest thing about Euler’s formula—given that it relies on imaginary numbers—is that it’s so immensely useful in the real world. By translating one type of motion into another, it lets engineers convert messy trig problems (you know, sines, secants, and so on) into more tractable algebra—like a wormhole between separate branches of math. It’s the secret sauce in Fourier transforms used to digitize music, and it tames all manner of wavy things in quantum mechanics, electronics, and signal processing; without it, computers might not exist…

More marvelous math at “The Baffling and Beautiful Wormhole Between Branches of Math.”

[TotH to @haarsager]

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**As we wonder if Descartes wasn’t right when he wrote that “everything turns into mathematics,”** we might spare a thought for Persian polymath Omar Khayyam; the mathematician, philosopher, astronomer, epigrammatist, and poet died on this date in 1131. While he’s probably best known to English-speakers as a poet, via Edward FitzGerald’s famous translation of the quatrains that comprise the *Rubaiyat of Omar Khayyam*, Omar was one of the major mathematicians and astronomers of the medieval period. He is the author of one of the most important works on algebra written before modern times, the *Treatise on Demonstration of Problems of Algebra,* which includes a geometric method for solving cubic equations by intersecting a hyperbola with a circle. His astronomical observations contributed to the reform of the Persian calendar. And he made important contributions to mechanics, geography, mineralogy, music, climatology, and Islamic theology.

## 1 picture, 1000 words, and all that…

Readers **will recall** David McCandless and his site, **Information is Beautiful**, on which he reminds us, day in day out, that good design is “beauty that works”– in his case more specifically, that gorgeous graphics can also be powerful communicators about issues that matter. Consider **his recent depiction of relative personnel commitments in Afghanistan**:

click **here** for accompanying graphics

**As we cherish perspective**, we might recall that it was on this date in 1791 that Mozart’s blissful *Die Zauberflote* (*The Magic Flute*) premiered in Vienna.

Librettist Emanuel Schikaneder performing in the role of Papageno