Zilch. Nada. Bupkis. Yes, I’m taking about Zero (0), a number that seems so essential to our system of numbers, and yet it hasn’t always enjoyed such a privileged place. Far from it.
In this short animation, Britain’s venerable Royal Institution traces the history of zero, a number that emerged in seventh century India, before making its way to China and Islamic countries, and finally penetrating Western cultures in the 13th century. Only later did it become the cornerstone of calculus and the language of computing.
India, we owe you thanks.
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Almost all the biggest math enthusiasts I’ve known have also loved classical music, especially the work of Bach, Mozart, and Beethoven. Of course, as San Francisco Symphony music director Michael Tilson Thomas once put it, you can’t have those three as your favorite composers, because “they simply define what music is.” But don’t tell that to the mathematically minded, on whom all of them, especially Bach and Beethoven, have always exerted a strong pull.
But why? Do their musical compositions have some underlying quantitative appeal? And by the way, “how is it that Beethoven, who is celebrated as one of the most significant composers of all time, wrote many of his most beloved songs while going deaf?” The question comes from a TED-Ed segment and its accompanying blog post by Natalya St. Clair which explains, using the example of the “Moonlight Sonata,” what the formidable composer did it using math. (You might also want to see St. Clair’s other vides: The Unexpected Math Behind Van Gogh’s “Starry Night.”)
“The standard piano octave consists of 13 keys, each separated by a half step,” St. Clair writes. “A standard major or minor scale uses 8 of these keys with 5 whole step intervals and 2 half step ones.” So far, so good. “The first half of measure 50 of ‘Moonlight Sonata’ consists of three notes in D major, separated by intervals called thirds that skip over the next note in the scale. By stacking the first, third, and fifth notes — D, F sharp, and A — we get a harmonic pattern known as a triad.” These three frequencies together create “ ‘consonance,’ which sounds naturally pleasant to our ears. Examining Beethoven’s use of both consonance and dissonance can help us begin to understand how he added the unquantifiable elements of emotion and creativity to the certainty of mathematics.”
Explained in words, Beethoven’s use of mathematics in his music may or may not seem easy to understand. But it all gets clearer and much more vivid when you watch the TED-Ed video about it, which brings together visuals of the piano keyboard, the musical score, and even the relevant geometric diagrams and sine waves. Nor does it miss the opportunity to use music itself, breaking it down into its constituent sounds and building it back up again into the “Moonlight Sonata” we know and love — and can now, having learned a little more about what mathematician James Sylvester called the “music of the reason” underlying the “mathematics of the sense,” appreciate a little more deeply.
It has long been thought that the so-called “Golden Ratio” described in Euclid’s Elements has “implications for numerous natural phenomena… from the leaf and seed arrangements of plants” and “from the arts to the stock market.” So writes astrophysicist Mario Livio, head of the science division for the institute that oversees the Hubble Telescope. And yet, though this mathematical proportion has been found in paintings by Leonardo da Vinci to Salvador Dali—two examples that are only “the tip of the iceberg in terms of the appearances of the Golden Ratio in the arts”—Livio concludes that it does not describe “some sort of universal standard for ‘beauty.’” Most art of “lasting value,” he argues, departs “from any formal canon for aesthetics.” We can consider Livio a Golden Ratio skeptic.
Far on the other end of a spectrum of belief in mathematical art lies Le Corbusier, Swiss architect and painter in whose modernist design some see an almost totalitarian mania for order. Using the Golden Ratio, Corbusier designed a system of aesthetic proportions called Modulor, its ambition, writes William Wiles at Icon, “to reconcile maths, the human form, architecture and beauty into a single system.”
Praised by Einstein and adopted by a few of Corbusier’s contemporaries, Modulor failed to catch on in part because “Corbusier wanted to patent the system and earn royalties from buildings using it.” In place of Leonardo’s Vitruvian Man, Corbusier proposed “Modulor Man” (below) the “mascot of [his] system for reordering the universe.”
Perhaps now, we need an artist to render a “Fractal Man”—or Fractal Gender Non-Specific Person—to represent the latest enthusiastic findings of math in the arts. This time, scientists have quantified beauty in language, a medium sometimes characterized as so imprecise, opaque, and unscientific that the Royal Society was founded with the motto “take no one’s word for it” and Ludwig Wittgenstein deflated philosophy with his conclusion in the Tractatus, “Whereof one cannot speak, thereof one must be silent.” (Speaking, in this sense, meant using language in a highly mathematical way.) Words—many scientists and philosophers have long believed—lie, and lead us away from the cold, hard truths of pure mathematics.
To determine whether the books had fractal structures, the academics looked at the variation of sentence lengths, finding that each sentence, or fragment, had a structure that resembled the whole of the book.
And it isn’t only Joyce. Through a statistical analysis of 113 works of literature, the researchers found that many texts written by the likes of Dickens, Shakespeare, Thomas Mann, Umberto Eco, and Samuel Beckett had multifractal structures. The most mathematically complex works were stream-of-consciousness narratives, hence the ultimate complexity of Finnegans Wake, which Professor Stanisław Drożdż, co-author of the paper published at Information Sciences, describes as “the absolute record in terms of multifractality.” (The graph at the top shows the results of the novel’s analysis, which produced a shape identical to pure mathematical multifractals.)
This study produced some inconsistencies, however. In the graph above, you can see how many of the titles surveyed ranked in terms of their “multifractality.” A close second to Joyce’s classic work, surprisingly, is Dave Egger’s post-modern memoir A Heartbreaking Work of Staggering Genius, and much, much further down the scale, Marcel Proust’s Remembrance of Things Past. Proust’s masterwork, writes Phys.org, shows “little correlation to multifractality” as do certain other books like Ayn Rand’s Atlas Shrugged. The measure may tell us little about literary quality, though Professor Drożdż suggests that “it may someday help in a more objective assignment of books to one genre or another.” Irish novelist Eimear McBride finds this “upshot” disappointing. “Surely there are more interesting questions about the how and why of writers’ brains arriving at these complex, but seemingly instinctive, fractals?” she told The Guardian.
Of the finding that stream-of-consciousness works seem to be the most fractal, McBride says, “By its nature, such writing is concerned not only with the usual load-bearing aspects of language—content, meaning, aesthetics, etc—but engages with language as the object in itself, using the re-forming of its rules to give the reader a more prismatic understanding…. Given the long-established connection between beauty and symmetry, finding works of literature fractally quantifiable seems perfectly reasonable.” Maybe so, or perhaps the Polish scientists have fallen victim to a more sophisticated variety of the psychological sharpshooter’s fallacy that affects “Bible Code” enthusiasts? I imagine we’ll see some fractal skeptics emerge soon enough. But the idea that the worlds-within-worlds feeling one gets when reading certain books—the sense that they contain universes in miniature—may be mathematically verifiable sends a little chill up my spine.
Tom Lehrer earned a BA and MA in mathematics from Harvard during the late 1940s, then taught math courses at MIT, Harvard, Wellesley, and UC-Santa Cruz. Math was his vocation. But, all along, Lehrer nurtured an interest in music. And, by the mid 1950s, he became best known for his satirical songs that touched on sometimes political, sometimes academic themes.
Today we’re presenting one of his classics: “The Elements.” Recorded in 1959, the song features Lehrer reciting the names of the 102 chemical elements known at the time (we now have 115), and it’s all sung to the tune of Major-General’s Song from The Pirates of Penzance by Gilbert and Sullivan. You can hear a studio version below, and watch a nice live version taped in Copenhagen, Denmark, in September 1967.
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Many people still have a major fear of mathematics, having suffered through school and not really having been in the right frame of mind to grasp concepts that we’ve been told will come in handy in our future working lives. When Britons get to the age of 16, many can choose to leave school, escaping the terror of math (or, as they say, maths).
But we shouldn’t live in fear, so along comes Citizen Maths, a UK-based free online course that purports to help adults catch up with Level 2 math (aka what a 16-year-old should know) without getting hit with a ruler or a spit wad. The course is funded by the UFI Charitable Trust, which focuses on providing free education for adults.
The Citizen Maths course currently consists of three units—Proportion, Uncertainty, and Representation. Additional sections on Pattern and Measurements will soon follow. All units come with videos and tests that take about an hour of the viewer’s time. As the narrator says, you can “learn in safety, without fear of being told off or exposed.” The full course takes, on average, about 20 hours.
And the tutorials bring in the real world, not just the abstract. Ratios and odds are experienced through roulette, horse racing, and playing dice. Understanding insurance comes into the tutorial on making decisions. Modeling is explained by trying to understand weather patterns. And proportion is explained through baking recipes and making cocktails.
As of this post, three of the five sections are available, with the complete course due up by next year. You can find more advanced Math courses in our collection of Free Online Math Courses.
Ted Mills is a freelance writer on the arts who currently hosts the FunkZone Podcast. You can also follow him on Twitter at @tedmills, read his other arts writing at tedmills.com and/or watch his films here.
I found it difficult to wrap my head around the sheer quantities of information Savage shoehorns into the seven minute video, giving similarly voluble and omnivorous mathmusician Vi Hart a run for her money. Clearly, he understands exactly what he’s talking about, whereas I had to take the review quiz in an attempt to retain just a bit of this new-to-me material.
I’m glad he glossed over Feynman’s childhood fascination with inertia in order to spend more time on the lesser known of his three subjects. Little Feynman’s observation of his toy wagon is charming, but the Nobel Prize winner’s life became an open book to me with Jim Ottaviani and Leland Myrick’s excellent graphic biography. What’s left to discover?
How about Eratosthenes? I’d never before heard of the Alexandrian librarian who calculated the Earth’s circumference with astonishing accuracy around 200 BC. (It helped that he was good at math and geography, the latter of which he invented.) Inspiration fuels the arts, much as it does science, and I’d like to learn more about him.
Ditto Fizeau, whom Savage describes as a less sexy scientific swashbuckler than methodical fact checker, which is what he was doing when he wound up cracking the speed of light in 1849. Two centuries earlier Galileo used lanterns to determine that light travels at least ten times faster than sound. Fizeau put Galileo’s number to the test, experimenting with his notched wheel, a candle, and mirrors and ultimately setting the speed of light at a much more accurate 313,300 Km/s. Today’s measurement of 299792.458 km/s was arrived at using technology unthinkable even a few decades ago.
Personally, I would never think to measure the speed of light with something that sounds like a zoetrope, but I might write a play about someone who did.
Ayun Halliday is an author, illustrator, and Chief Primatologist of the East Village Inky zine. Her play, Fawnbook, opens in New York City later this fall. Follow her @AyunHalliday
Euler’s conjecture, a theory proposed by Leonhard Euler in 1769, hung in there for 200 years. Then L.J. Lander and T.R. Parkin came along in 1966, and debunked the conjecture in two swift sentences. Their article — which is now open access and can be downloaded here — appeared in the Bulletin of the American Mathematical Society. If you’re wondering what the conjecture and its refutation are all about, you might want to ask Cliff Pickover, the author of 45 books on math and science. He brought this curious document to the web last week.
If you came of age during the 1960s and 1970s, you knew the voice of Jane Barbe, aka “The Telephone Lady.” Her voice appeared on telephone company recordings and voicemail systems across the US. Just listen to this clip, and you will immediately know who I’m talking about.
Anyway, Ms. Barbe apparently had more in her repertoire than “If you’d like to make a call, please hang up.” A lot more.
Above, we have her reciting the first 1,000 digits of Pi. The backstory remains unknown to us. We did some research and came up bone dry. If you know something about the circumstances behind the recording, please let us know in the comments section below. To read along with a transcript of the recording, just click here.
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