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 back in 2015.
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Have accomplished abstract geometrical artists come out of any demographic in greater numbers than from the women of South Asia? Not when even the most demanding art-school curriculum can’t hope to equal the rigor of the kōlam, a complex kind of line drawing practiced by women everywhere from India to Sri Lanka to Malaysia to Thailand. Using humble materials like chalk and rice flour on the ground in front of their homes, they interweave not just lines, shapes, and patterns but religious, philosophical, and magical motifs as well — and they create their kōlams anew each and every day.
“Taking a clump of rice flour in a bowl (or a coconut shell), the kōlam artist steps onto her freshly washed canvas: the ground at the entrance of her house, or any patch of floor marking an entrypoint,” writes Atlas Obscura’s Rohini Chaki.
“Working swiftly, she takes pinches of rice flour and draws geometric patterns: curved lines, labyrinthine loops around red or white dots, hexagonal fractals, or floral patterns resembling the lotus, a symbol of the goddess of prosperity, Lakshmi, for whom the kōlam is drawn as a prayer in illustration.”
Kōlams are thought to bring prosperity, but they also have other uses, such as feeding ants, birds, and other passing creatures. Chaki quotes University of San Francisco Theology and Religious Studies professor Vijaya Nagarajan as describing their fulfilling the Hindu “karmic obligation” to “feed a thousand souls.” Kōlams have also become an object of genuine interest for mathematicians and computer scientists due to their recursive nature: “They start out small, but can be built out by continuing to enlarge the same subpattern, creating a complex overall design,” Chaki writes. “This has fascinated mathematicians, because the patterns elucidate fundamental mathematical principles.”
Like any traditional art form, the kōlam doesn’t have quite as many practitioners as it used to, much less practitioners who can meet the standard of mastery of completing an entire work without once standing up or even lifting their hand. But even so, the kōlam is hardly on the brink of dying out: you can see a few of their creators in action in the video at the top of the post, and the age of social media has offered kōlam creators of any age — and now even the occasional man — the kind of exposure that even the busiest front door could never match. Some who get into kōlams in the 21st century may want to create ones that show ever more complexity of geometry and depth of reference, but the best among them won’t forget the meaning, according to Chaki, of the form’s very name: beauty.
Based in Seoul, Colin Marshall writes and broadcasts on cities, language, and culture. His projects include the book The Stateless City: a Walk through 21st-Century Los Angeles and the video series The City in Cinema. Follow him on Twitter at @colinmarshall, on Facebook, or on Instagram.
Here’s the latest from Great Big Story: “Once upon a time, not long ago, the math world fell in love … with a chalk. But not just any chalk! This was Hagoromo: a Japanese brand so smooth, so perfect that some wondered if it was made from the tears of angels. Pencils down, please, as we tell the tale of a writing implement so irreplaceable, professors stockpiled it.”
Head over to Amazon and try to buy it, and all you get is: “Currently unavailable. We don’t know when or if this item will be back in stock.” Indeed, they’ve stockpiled it all.
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Human imagination seems seriously limited when faced with the cosmic scope of time and space. We can imagine, through stop-motion animation and CGI, what it might be like to walk the earth with creatures the size of office buildings. But how to wrap our heads around the fact that they lived hundreds of millions of years ago, on a planet some four and a half billion years old? We trust the science, but can’t rely on intuition alone to guide us to such mind-boggling knowledge.
At the other end of the scale, events measured in nanoseconds, or billionths of a second, seem inconceivable, even to someone as smart as Grace Hopper, the Navy mathematician who invented COBOL and helped built the first computer. Or so she says in the 1983 video clip above from one of her many lectures in her role as a guest lecturer at universities, museums, military bodies, and corporations.
When she first heard of “circuits that acted in nanoseconds,” she says, “billionths of a second… Well, I didn’t know what a billion was…. And if you don’t know what a billion is, how on earth do you know what a billionth is? Finally, one morning in total desperation, I called over the engineering building, and I said, ‘Please cut off a nanosecond and send it to me.” What she asked for, she explains, and shows the class, was a piece of wire representing the distance a signal could travel in a nanosecond.
Now of course it wouldn’t really be through wire — it’d be out in space, the velocity of light. So if we start with a velocity of light and use your friendly computer, you’ll discover that a nanosecond is 11.8 inches long, the maximum limiting distance that electricity can travel in a billionth of a second.
Follow the rest of her explanation, with wire props, and see if you can better understand a measure of time beyond the reaches of conscious experience. The explanation was immediately successful when she began using it in the late 1960s “to demonstrate how designing smaller components would produce faster computers,” writes the National Museum of American History. The bundle of wires below, each about 30cm (11.8 inches) long, comes from a lecture Hopper gave museum docents in March 1985.
Like the age of the dinosaurs, the nanosecond may only represent a small fraction of the incomprehensibly small units of time scientists are eventually able to measure—and computer scientists able to access. “Later,” notes the NMAH, “as components shrank and computer speeds increased, Hopper used grains of pepper to represent the distance electricity traveled in a picosecond, one trillionth of a second.”
At this point, the map becomes no more revealing than the unknown territory, invisible to the naked eye, inconceivable but through wild leaps of imagination. But if anyone could explain the increasingly inexplicable in terms most anyone could understand, it was the brilliant but down-to-earth Hopper.
When I look at maps from centuries ago, I wonder how they could have been of any use. Not only were they filled with mythological monsters and mythological places, but the perspectives mostly served an aesthetic design rather than a practical one. Of course, accuracy was hard to come by without the many mapping tools we take for granted—some of them just in their infancy during the Renaissance, and many more that would have seemed like outlandish magic to nearly everyone in 15th century Europe.
Everyone, it sometimes seems, but Leonardo da Vinci, who anticipated and sometimes steered the direction of futuristic public works technology. None of his flying machines worked, and he could hardly have seen images taken from outer space. But he clearly saw the problem with contemporary maps. The necessity of fixing them led to a 1502 aerial image of Imola, Italy, drawn almost as accurately as if he had been peering at the city through a Google satellite camera.
“Leonardo,” says the narrator of the Vox video above, “needed to show Imola as an ichnographic map,” a term coined by ancient Roman engineer Vitruvius to describe ground plan-style cartography. No streets or buildings are obscured, as they are in the maps drawn from the oblique perspective of a hilltop or mountain. Leonardo undertook the project while employed as Cesare Borgia’s military engineer. “He was charged with helping Borgia become more aware of the town’s layout.” For this visual aid turned cartographic marvel, he drew from the same source that inspired the elegant Vitruvian Man.
While the visionary Roman builder could imagine a god’s eye view, it took someone with Leonardo’s extraordinary perspicacity and skill to actually draw one, in a startlingly accurate way. Did he do it with grit and moxie? Did he astral project thousands of miles above the city? Was he in contact with ancient aliens? No, he used geometry, and a compass, the same means and instruments that allowed ancient scientists like Eratosthenes to calculate the circumference of the earth, to within 200 miles, over 2000 years ago.
Leonardo probably also used an instrument called a bussola, a device that measures degrees inside a circle—like the one that surrounds his city map. Painstakingly recording the angles of each turn and intersection in the town and measuring their distance from each other would have given him the data he needed to recreate the city as seen from above, using the bussola to maintain proper scale. Other methods would have been involved, all of them commonly available to surveyors, builders, city planners, and cartographers at the time. Leonardo trusted the math, even though he could never verify it, but like the best mapmakers, he also wanted to make something beautiful.
It may be difficult for historians to determine which inaccuracies are due to miscalculation and which to deliberate distortion for some artistic purpose. But license or mistakes aside, Leonardo’s map remains an astonishing feat, marking a seismic shift from the geography of “myth and perception” to one of “information, drawn plainly.” There’s no telling if the archetypal Renaissance man would have liked where this path led, but if he lived in the 21st century, he’d already have his mind trained on ideas that anticipate technology hundreds of years in our future.
In many an audio engineering course, I’ve come across the Fourier Transform, an idea so fundamental in sound production that it seems essential for everyone to know it. My limited understanding was, you might say, functional. It’s some kind of mathematical reverse engineering machine that turns waveforms into frequencies, right? Yes, but it’s much more than that. The idea can seem overwhelming to the non-mathematically-inclined among us.
The Fourier Transform, named for French mathematician and physicist Jean-Baptiste Joseph Fourier, “decomposes” any wave form into frequencies, and “virtually everything in the world can be described via a waveform,” writes one introduction to the theory. That includes not only sounds but “electromagnetic fields, the elevation of a hill versus location… the price of your favorite stock versus time,” the signals of an MRI scanner.
The concept “extends well beyond sound and frequency into many disparate areas of math and even physics. It is crazy just how ubiquitous this idea is,” notes the 3Blue1Brown video above, one of dozens of animated explorations of mathematical concepts. I know far more than I did yesterday thanks to this comprehensive animated lecture. Even if it all seems old hat to you, “there is something fun and enriching,” the video assures us, “about seeing what all of its components look like.”
Things get complicated rather quickly when we get into the dense equations, but the video illustrates every formula with graphs that transform the numbers into meaningful moving images.
In shorter lessons, you can learn to count to 1000 on two hands, or, just below, learn what it feels like to invent math. (It feels weird at first.)
Sanderson’s short courses “tend to fall into one of two categories,” he writes: topics “people might be seeking out,” like many of those mentioned above, and “problems in math which many people may not have heard of, and which seem really hard at first, but where some shift in perspective makes it both doable and beautiful.” These puzzles with elegantly clever solutions can be found here. Whether you’re a hardcore math-head or not, you’ll find Sanderson’s series of 3Blue1Brown animations illuminating. Find them all here.
For two millennia, Euclid’s Elements, the foundational ancient work on geometry by the famed Greek mathematician, was required reading for educated people. (The “classically educated” read them in the original Greek.) The influence of the Elements in philosophy and mathematics cannot be overstated; so inspiring are Euclid’s proofs and axioms that Edna St. Vincent Millay wrote a sonnet in his honor. But over time, Euclid’s principles were streamlined into textbooks, and the Elements was read less and less.
In 1847, maybe sensing that the popularity of Euclid’s text was fading, Irish professor of mathematics Oliver Byrne worked with London publisher William Pickering to produce his own edition of the Elements, or half of it, with original illustrations that carefully explain the text.
“Byrne’s edition was one of the first multicolor printed books,” writes designer Nicholas Rougeux. “The precise use of colors and diagrams meant that the book was very challenging and expensive to reproduce.” It met with little notice at the time.
Byrne’s edition—The First Six Books of The Elements of Euclid in which Coloured Diagrams and Symbols are Used Instead of Letters for the Greater Ease of Learners—might have passed into obscurity had a reference to it in Edward Tufte’s Envisioning Informationnot sparked renewed interest. From there followed a beautiful new edition by TASCHEN and an article on Byrne’s diagrams in mathematics journal Convergence. Rougeux picked up the thread and decided to create an online version. “Like others,” he writes, “I was drawn to its beautiful diagrams and typography.” He has done both of those features ample justice.
As in another of Rougeux’s online reproductions—his Werner’s Nomenclature of Colours—the designer has taken a great deal of care to preserve the original intentions while adapting the book to the web. In this case, that means the spelling (including the use of the long s), typeface (Caslon), stylized initial capitals, and Byrne’s alternate designs for mathematical symbols have all been retained. But Rougeux has also made the diagrams interactive, “with clickable shapes to aid in understanding the shapes being referenced.”
He has also turned all of those lovely diagrams into an attractive poster you can hang on the wall for quick reference or as a conversation piece, though this semaphore-like arrangement of illustrations—like the simplified Euclid in modern textbooks—cannot replace or supplant the original text. You can read Euclid in ancient Greek (see a primer here), in Latin and Arabic, in English translations here, here, here, and many other places and languages as well.
For an experience that combines, however, the best of ancient wisdom and modern information technology—from both the 19th and the 21st centuries—Rougeux’s free, online edition of Byrne’s Euclid can’t be beat. Learn more about the meticulous process of recreating Byrne’s text and diagrams (illustrated above) here.
In 2009, guitarist Randy Bachman of the Guess Who and Bachman-Turner Overdrive had the rare opportunity to hear the individual tracks that make up that mythic opening chord in the Beatles’ “A Hard Day’s Night,” an enigma that has baffled musicians for decades. Bachman found that it’s actually made up of a combination of different chords played all at once by George, John, and Paul. The discovery made for a great story, and Bachman told it the following year on his CBC radio show. Unbeknownst to him, it seems, another Canadian Beatles lover, Dalhousie University math professor Jason Brown, claimed he had cracked the code the previous year, without setting foot in Abbey Road.
Instead, Brown used what is called a Fourier Analysis, based on work done in the 1820s by French scientist Joseph Fourier, which reduces sounds into their “constituent sine or cosine waves.” The problem with Bachman’s explanation, as Eliot Van Buskirk notes at Wired, is that the chord “contains a note that would be impossible for the Beatles’ two guitarists and bassist to play in one take.” Since there was no overdubbing involved, something else must have been happening. Through his mathematical analysis, Brown determined that something else to have been five notes played on the piano, apparently by George Martin, “who is known to have doubled on piano George Harrison’s solo on the track.”
After ten years of work, Brown has returned with the solution to another longtime Beatles mystery, this time with a little help from his colleagues, Harvard mathematicians Mark Glickman and Ryan Song. The problem: who wrote the melody for “In My Life,” Rubber Soul’s nostalgic ballad? The song is credited to the crack team of Lennon-McCartney, but while the two agreed that Lennon penned the lyrics, both separately claimed in interviews to have written the music. Brown and his collaborators used statistical methods to determine that it was, in fact, Lennon who wrote the whole song.
In an analysis of “about 70 songs from Lennon and McCartney… they found there were 149 very distinct transitions between notes and chords.” These are unique to one or the other songwriters. “When you do the math,” Devlin says, it turns out “the probability that McCartney wrote it was .o18—that’s essentially zero.” Why might Paul have misremembered this—even saying specifically in a 1984 Playboy interview that he recalled “going off for half an hour and sitting with a Mellotron… writing the tune”? Who knows. Mashable has reached out to McCartney’s publicist for comment. But in the final analysis, says Devlin, “I would go with mathematics” over faulty human memory.
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