The Complex Geometry of Islamic Art & Design: A Short Introduction

When you think of the accomplishments of the Islamic world, what comes to mind? For most of this century so far, at least in the West, the very notion has had associations in many minds with not creation but destruction. In 2002, mathematician Keith Devlin lamented how "the word Islam conjures up images of fanatical terrorists flying jet airplanes full of people into buildings full of even more people" and "the word Baghdad brings to mind the unscrupulous and decidedly evil dictator Saddam Hussein." Ironically, writes Devlin, "the culture that these fanatics claim to represent when they set about trying to destroy the modern world of science and technology was in fact the cradle in which that tradition was nurtured. As mathematicians, we are all children of Islam."

You don't have to dig deep into history to discover the connection between Islam and mathematics; you can simply see it. "In Islamic culture, geometry is everywhere," says the narrator of the brief TED-Ed lesson above. "You can find it in mosques, madrasas, palaces, and private homes."




Scripted by writer and consultant on Islamic design Eric Broug, the video breaks down the complex, abstract geometric patterns found everywhere in Islamic art and design, from its "intricate floral motifs adorning carpets and textiles to patterns of tilework that seem to repeat infinitely, inspiring wonder and contemplation of eternal order."

And the tools used to render these visions of eternity? Nothing more advanced than a compass and a ruler, Broug explains, used to first draw a circle, divide that circle up, draw lines to construct repeating shapes like petals or stars, and keep intact the grid underlying the whole pattern. The process of repeating a geometric pattern on a grid, called tessellation, may seen familiar indeed to fans of the mathematically minded artist M.C. Escher, who used the very same process to demonstrate what wondrous artistic results can emerge from the use of simple basic patterns. In fact, Escher's Dutch countryman Broug once wrote an essay on the connections between his art and that of the Islamic world for the exhibit Escher Meets Islamic Art at Amsterdam's Tropen­mu­seum.

Escher first encountered tessellations on a trip to the Islamic world himself, in the "colorful abstract decorations in the 14th century Alhambra, the well-known palace and fortress complex in Southern Spain," writes Al.Arte's Aya Johanna Daniëlle Dürst Britt. "Although he visited the Alhambra in 1922 after his graduation as a graphic artist, he was already interested in geometry, symmetry and tessellations for some years." His fascinations included "the effect of color on the visual perspective, causing some motifs to seem infinite — an effect partly caused by symmetry." His second visit to Alhambra, in 1936, solidified his understanding of the principles of tessellation, and he would go on to base about a hundred of his own pieces on the patterns he saw there. Those who seek the door to infinity understand that any tradition may hold the keys.

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Based in Seoul, Colin Marshall writes and broadcasts on cities and culture. He’s at work on the book The Stateless City: a Walk through 21st-Century Los Angeles, the video series The City in Cinema, the crowdfunded journalism project Where Is the City of the Future?, and the Los Angeles Review of Books’ Korea Blog. Follow him on Twitter at @colinmarshall or on Facebook.

Trigonometry Discovered on a 3700-Year-Old Ancient Babylonian Tablet

One presumption of television shows like Ancient Aliens and books like Chariots of the Gods is that ancient people—particularly non-western people—couldn’t possibly have constructed the elaborate infrastructure and monumental architecture and statuary they did without the help of extra-terrestrials. The idea is intriguing, giving us the hugely ambitious sci-fi fantasies woven into Ridley Scott’s revived Alien franchise. It is also insulting in its level of disbelief about the capabilities of ancient Egyptians, Mesopotamians, South Americans, South Sea Islanders, etc.

We assume the Greeks perfected geometry, for example, and refer to the Pythagorean theorem, although this principle was probably well-known to ancient Indians. Since at least the 1940s, mathematicians have also known that the “Pythagorean triples”—integer solutions to the theorem—appeared 1000 years before Pythagoras on a Babylonian tablet called Plimpton 322. Dating back to sometime between 1822 and 1762 B.C. and discovered in southern Iraq in the early 1900s, the tablet has recently been re-examined by mathematicians Daniel Mansfield and Norman Wildberger of Australia’s University of New South Wales and found to contain even more ancient mathematical wisdom, “a trigonometric table, which is 3,000 years ahead of its time.”




In a paper published in Historia Mathematica the two conclude that Plimpton 322’s Babylonian creators detailed a “novel kind of trigonometry,” 1000 years before Pythagoras and Greek astronomer Hipparchus, who has typically received credit for trigonometry’s discovery. In the video above, Mansfield introduces the unique properties of this “scientific marvel of the ancient world," an enigma that has “puzzled mathematicians,” he writes in his article, “for more than 70 years.” Mansfield is confident that his research will fundamentally change the way we understand scientific history. He may be overly optimistic about the cultural forces that shape historical narratives, and he is not without his scholarly critics either.

Eleanor Robson, an expert on Mesopotamia at University College London has not published a formal critique, but she did take to Twitter to register her dissent, writing, “for any historical document, you need to be able to read the language & know the historical context to make sense of it. Maths is no exception.” The trigonometry hypothesis, she writes in a follow-up tweet, is “tediously wrong.” Mansfield and Wildberger may not be experts in ancient Mesopotamian language and culture, it's true, but Robson is also not a mathematician. “The strongest argument” in the Australian researchers’ favor, writes Kenneth Chang at The New York Times, is that “the table works for trigonomic calculations.” As Mansfield says, “you don’t make a trigonomic table by accident.”

Plimpton 322 uses ratios rather than angles and circles. “But when you arrange it such a way so that you can use any known ratio of a triangle to find the other side of a triangle,” says Mansfield, “then it becomes trigonometry. That’s what we can use this fragment for.” As for what the ancient Babylonians used it for, we can only speculate. Robson and others have proposed that the tablet was a teaching guide. Mansfield believes “Plimpton 322 was a powerful tool that could have been used for surveying fields or making architectural calculations to build palaces, temples or step pyramids.”

Whatever its ancient use, Mansfield thinks the tablet “has great relevance for our modern world… practical applications in surveying, computer graphics and education.” Given the possibilities, Plimpton 322 might serve as “a rare example of the ancient world teaching us something new,” should we choose to learn it. That knowledge probably did not originate in outer space.

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Josh Jones is a writer and musician based in Durham, NC. Follow him at @jdmagness

Arnold Schoenberg Creates a Hand-Drawn, Paper-Cut “Wheel Chart” to Visualize His 12-Tone Technique

"These go up to eleven," Spinal Tap famously said of the amplifiers that, so they claimed, took them to a higher level in rock music. But the work of Austrian composer Arnold Schoenberg, one of the best-known figures in the history of avant-garde music, went up to twelve — twelve tones, that is. His "twelve-tone technique," invented in the early 1920s and for the next few decades used mostly by he and his colleagues in the Second Viennese School such as Alban Berg, Anton Webern, and Hanns Eisler, allowed composers to break free of the traditional Western system of keys that limited the notes available for use in a piece, instead granting each note the same weight and making none of them central.

This doesn't mean that composers using Schoenberg's twelve-tone technique could just use notes at random in complete atonality, but that a new set of considerations would organize them. "He believed that a single tonality could include all twelve notes of the chromatic scale," writes Bradford Bailey at The Hum, "as long as they were properly organized to be subordinate to tonic (the tonic is the pitch upon which all others are referenced, in other words the root or axis around which a piece is built)." The mathematical rigor underlying it all required some explanation, and often mathematical and musical concepts — mathematics and music being in any case intimately connected — become much clearer when rendered visually.

Hence Schoenberg's twelve-tone wheel chart pictured at the top of the post, one of what Arnold Schoenberg's Journey author Allen Shawn describes as "no fewer than twenty-two different kinds of contraptions" — including "charts, cylinders, booklets, slide rules" — "for transposing and deriving twelve-tone rows" needed to compose twelve-tone music. (See the slide ruler above too.) "The distinction between 'play' and 'work' is already hard to draw in the case of artists," writes Shawn, "but in Schoenberg's case it is especially hard to make since he brought discipline, originality, and playfulness to many of his activities." These also included making special playing cards (two of whose sets you can see here and here) and even his own version of chess.

As Shawn describes it, Koalitionsscach, or "Coalition Chess," involves "the armies of four countries arrayed on the four sides of the board, for which he designed and constructed the pieces himself." Instead of an eight-by-eight board, Coalition Chess uses a ten-by-ten, and the pieces on it "represent machine guns, artillery, airplanes, submarines, tanks, and other instruments of war." The rules, which "require that the four players form alliances at the outset," add at least a dimension to the age-old standard game of chess — a form that, like traditional Western music, humanity will still be struggling to master decades and even centuries hence. But apparently, for a mind like Schoenberg's, chess and music as he knew them weren't nearly challenging enough.

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John Coltrane Draws a Picture Illustrating the Mathematics of Music

Based in Seoul, Colin Marshall writes and broadcasts on cities and culture. He’s at work on the book The Stateless City: a Walk through 21st-Century Los Angeles, the video series The City in Cinema, the crowdfunded journalism project Where Is the City of the Future?, and the Los Angeles Review of Books’ Korea Blog. Follow him on Twitter at @colinmarshall or on Facebook.

John Coltrane Draws a Picture Illustrating the Mathematics of Music

Physicist and saxophonist Stephon Alexander has argued in his many public lectures and his book The Jazz of Physics that Albert Einstein and John Coltrane had quite a lot in common. Alexander in particular draws our attention to the so-called “Coltrane circle,” which resembles what any musician will recognize as the “Circle of Fifths,” but incorporates Coltrane’s own innovations. Coltrane gave the drawing to saxophonist and professor Yusef Lateef in 1967, who included it in his seminal text, Repository of Scales and Melodic Patterns. Where Lateef, as he writes in his autobiography, sees Coltrane's music as a "spiritual journey" that "embraced the concerns of a rich tradition of autophysiopsychic music," Alexander sees “the same geometric principle that motivated Einstein’s" quantum theory.

Neither description seems out of place. Musician and blogger Roel Hollander notes, “Thelonious Monk once said ‘All musicans are subconsciously mathematicians.’ Musicians like John Coltrane though have been very much aware of the mathematics of music and consciously applied it to his works.”




Coltrane was also very much aware of Einstein’s work and liked to talk about it frequently. Musican David Amram remembers the Giant Steps genius telling him he “was trying to do something like that in music.”

Hollander carefully dissects Coltrane's mathematics in two theory-heavy essays, one generally on Coltrane’s “Music & Geometry” and one specifically on his “Tone Circle.” Coltrane himself had little to say publically about the intensive theoretical work behind his most famous compositions, probably because he’d rather they speak for themselves. He preferred to express himself philosophically and mystically, drawing equally on his fascination with science and with spiritual traditions of all kinds. Coltrane’s poetic way of speaking has left his musical interpreters with a wide variety of ways to look at his Circle, as jazz musician Corey Mwamba discovered when he informally polled several other players on Facebook. Clarinetist Arun Ghosh, for example, saw in Coltrane's "mathematical principles" a "musical system that connected with The Divine." It's a system, he opined, that "feels quite Islamic to me."

James Patterson Teaches You To Writer A Bestseller. Learn More.

Lateef agreed, and there may be few who understood Coltrane’s method better than he did. He studied closely with Coltrane for years, and has been remembered since his death in 2013 as a peer and even a mentor, especially in his ecumenical embrace of theory and music from around the world. Lateef even argued that Coltrane's late-in-life masterpiece A Love Supreme might have been titled "Allah Supreme" were it not for fear of "political backlash." Some may find the claim tendentious, but what we see in the wide range of responses to Coltrane's musical theory, so well encapsulated in the drawing above, is that his recognition, as Lateef writes, of the "structures of music" was as much for him about scientific discovery as it was religious experience. Both for him were intuitive processes that "came into existence," writes Lateef, "in the mind of the musican through abstraction from experience."

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Josh Jones is a writer and musician based in Durham, NC. Follow him at @jdmagness

Behold the Ingenious “Ambiguous Cylinder Illusion” (and Then Find Out How It Works)

Created by Kokichi Sugihara, a math professor at Meiji University in Tokyo, the “Ambiguous Cylinder Illusion” wowed audiences at "the Best Illusion of the Year Contest" in 2016. Here's the general gist of the illusion:

The direct views of the objects and their mirror images generate quite different interpretations of the 3D shapes. They look like vertical cylinders, but their sections appear to be different; in one view they appear to be rectangles, while in the other view they appear to be circles. We cannot correct our interpretations although we logically know that they come from the same objects. Even if the object is rotated in front of a viewer, it is difficult to understand the true shape of the object, and thus the illusion does not disappear.

So how do those rectangles look like circles, and vice-versa? The video below--if you care to spoil the illusion--will show you. Find more videos from the Illusion Contest here.

via The Kids Should See This

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How Did Beethoven Compose His 9th Symphony After He Went Completely Deaf?

You don’t need to know anything at all about classical music, nor have any liking for it even, to be deeply moved by that most famous of symphonies, Ludwig van Beethoven’s 9th---“perhaps the most iconic work of the Western musical tradition,” writes The Juilliard Journal in an article about its handwritten score. Commissioned in 1817, the sublime work was only completed in 1824. By that time, its composer was completely and totally deaf. At the first performance, Beethoven did not notice that the massive final choral movement had ended, and one of the musicians had to turn him around to acknowledge the audience.

This may seem, says researcher Natalya St. Clair in the TED-Ed video above, like some “cruel joke,” but it’s the truth. Beethoven was so deaf that some of the most interesting artifacts he left behind are the so-called “conversation books,” kept from 1818 onward to communicate with visitors who had to write down their questions and replies. How then might it have been possible for the composer to create such enduringly thrilling, rapturous works of aural art?




Using the delicate, melancholy “Moonlight Sonata” (which the composer wrote in 1801, when he could still hear), St. Clair attempts to show us how Beethoven used mathematical “patterns hidden beneath the beautiful sounds.” (In the short video below from documentary The Genius of Beethoven, see the onset of Beethoven's hearing loss in a dramatic reading of his letters.) According to St. Clair’s theory, Beethoven composed by observing “the mathematical relationship between the pitch frequency of different notes,” though he did not write his symphonies in calculus. It’s left rather unclear how the composer's supposed intuition of mathematics and pitch corresponds with his ability to express such a range of emotions through music.

We can learn more about Beethoven's deafness and its biological relationship to his compositional style in the short video below with research fellow Edoardo Saccenti and his colleague Age Smilde from the Biosystems Data Analysis Group at Amsterdam’s Swammerdam Institute for Life Sciences. By counting the high and low frequencies in Beethoven’s complete string quartets, a task that took Saccenti many weeks, he and his team were able to show how three distinct compositional styles “correspond to stages in the progression of his deafness,” as they write in their paper (which you can download in PDF here).

The progression is unusual. As his condition worsened, Beethoven included fewer and fewer high frequency sounds in his compositions (giving cellists much more to do). By the time we get to 1824-26, “the years of the late string quartets and of complete deafness”---and of the completion of the 9th---the high notes have returned, due in part, Smilde says, to “the balance between an auditory feedback and the inner ear.” Beethoven’s reliance on his “inner ear” made his music “much and much richer.” How? As one violinist in the clip puts it, he was “given more freedom because he was not attached anymore to the physical sound, [he could] just use his imagination.”

For all of the compelling evidence presented here, whether Beethoven’s genius in his painful later years is attributable to his intuition of complex mathematical patterns or to the total free reign of his imaginative inner ear may in fact be undiscoverable. In any case, no amount of rational explanation can explain away our astonishment that the man who wrote the unfailingly powerful, awesomely dynamic “Ode to Joy” finale (conducted above by Leonard Bernstein), couldn’t actually hear any of the music.

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Josh Jones is a writer and musician based in Durham, NC. Follow him at @jdmagness

Sesame Street’s Count Von Count counts Pi to 10,000 Places: A 5 Hour Recording for Pi Day

March 14 is Pi Day. This oddity will keep the celebration going a good part of the day.

Follow Open Culture on Facebook and Twitter and share intelligent media with your friends. Or better yet, sign up for our daily email and get a daily dose of Open Culture in your inbox. 

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