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I first encountered the world of Maurits Cornelis Escher where many others do: in school. A poster of his 22-foot-long Metamorphosis III hung along the walls of my fourth-grade classroom, where I spent many an idle minute or ten staring at its intricate geometry through which squares became birds, birds became lizards, lizards became fish, and it all somehow arrived at the cliff-like edge of a three-dimensional chessboard. It came as the last of a trilogy of woodcuts Escher made between 1937 and 1968, and a journey through its 1940 predecessor Metamorphosis II ends the 1971 documentary above, M.C. Escher: Adventures in Perception.
Escher himself seemingly had no happy classroom memories. “I hated school,” the narrator quotes him as saying. “The only class I liked at all was art. That doesn’t mean I was any good at it.” Though his work has no doubt inspired many youngsters to take up drawing, woodcutting, and printmaking themselves, it’s surely driven even more of them into mathematics.
Obsessed with perspective, geometry, and pattern (Escher described tessellation as “a real mania to which I have become addicted”), his images have, by the count of mathematician and Escher scholar Doris Schattschneider, led so far to eleven separate strands of mathematical and scientific research.
The twenty-minute Adventures in Perception, originally commissioned by the Netherlands’ Ministry of Foreign Affairs, offers in its first half a meditation on the mesmerizing, often impossible world Escher had created with his art to date. Its second half captures Escher in the last years of his life, still at work in his Laren, North Holland studio. It even shows him printing one of the three titular serpents, threaded through a set of elaborately interlocking circles, of his very last print Snakes. He never actually finished Snakes, whose patterns would have continued on to the effect of infinity, and even says here of his officially complete works that none succeed, “because it’s the dream I tried for that can’t be realized.” But those unrealized dreams have kept the rest of us dreaming, and thinking, ever since.
Zombies, alien overlords, sharks, a mad dictator…math is a dangerous proposition in the hands of TED Ed script writer Alex Gendler.
The recreational mathematics puzzles he retrofits for TED’s educational initiative have been around for hundreds, even thousands of years. In the past, storylines tended to rely on biases 21st-century puzzle solvers would find objectionable. As mathematician David Singmaster told Science News:
One must be a little careful with some of these problems, as past cultures were often blatantly sexist or racist. But such problems also show what the culture was like.… The river crossing problem of the jealous husbands is quite sexist and transforms into masters and servants, which is classist, then into missionaries and cannibals, which is racist. With such problems, you can offend everybody!
Gendler’s updates, animated by Artrake studio, derive their narrative urgency from the sort of crowd pleasing sci fi predicaments that fuel summer blockbusters.
And fortunately for those of us whose brains are permanently stuck in beach mode, he never fails to explain how the characters prevail, outwitting or outrunning the aforementioned zombies, aliens, sharks, and mad dictator.
(No worries if you’re determined to find the solution on your own. Gendler gives plenty of fair warning before each reveal.)
Put your brain in gear, pull the skull-embossed lever, and remember, teamwork — and inductive logic — carry the day!
The prisoner hat riddle, above, hinges on a hierarchy of beliefs and the alien overlord’s willingness to give its nine captives a few minutes to come up with a game plan.
Go deeper into this age old puzzle by viewing the full lesson.
Gendler’s spin on the green-eyed logic puzzle, above, contains two brain teasers, one for the hive mind, and one for an individual acting alone, with a strategy culled from philosopher David Lewis’ Common Knowledge playbook. Here’s the full lesson.
Raring for more? You’ll find a playlist of TED-Ed puzzles by Gendler and others here. The full lesson for the bridge problem at the top of the post is here.
Ayun Halliday, author, illustrator, and Chief Primatologist of the East Village Inky zine, will be leading a free collaborative zine workshop at the Gluestick Fest in Indianapolis Saturday, July 9. Follow her @AyunHalliday
We now regard Alan Turing, the troubled and ultimately persecuted cryptanalyst (and, intellectually, much more besides)—who cracked the code of the German Enigma machine in World War II—as one of the great minds of history. His life and work have drawn a good deal of serious examination since his early death in 1954, and recently his legacy has even given rise to popular portrayals such as that by Benedict Cumberbatch in the film The Imitation Game. So what, more and more of us have started to wonder, forms a mind like Turing’s in the first place?
A few years ago, mathematics writer Alex Bellos received, from “an old friend who teaches at Sherborne, the school Turing attended between 1928 and 1930,” some “new information about the computer pioneer and codebreaker’s school years” in the form of “the list of books Turing took out from the school library while he was a pupil.” Bellos lists them as follows:
Illusions
Journal of the Chemical Society, vols. 95, 96, 97
“As you can see, and as you might expect,” writes Bellos, “heavy on the sciences. The AJ Evans, a memoir about the author’s escape from imprisonment in the First World War, is the only non-scientific book.” He also notes that “the physics books he took out all look very serious, but the maths ones are lighthearted: the Lewis Carroll and the Rouse Ball, which for decades was the classic text in recreational maths problems.” Sherborne archivist Rachel Hassall, who provided Bellos with the list, also told him that “the book chosen by Turing for his school prize was a copy of the Rouse Ball. Even teenage geniuses like to have fun.”
If you, too, would like to do a bit of the reading of a genius — or, depending on how quantitatively your own mind works, just have some fun — you can download for free most of these books the young Turing checked out of the school library. Programmer and writer John Graham-Cumming originally found and organized all the links to the texts on his blog; you can follow them there or from the list in this post. And if you know any youngsters in whom you see the potential to achieve history’s next Turing-level accomplishment, send a few e‑books their way. Why read Harry Potter, after all, when you can read A Selection of Photographs of Stars, Star-Clusters & Nebulae, together with information concerning the instruments & the methods employed in the pursuit of celestial photography?
Of course you can! All it takes is a device with a built-in spelling app, an innovation of which no eighth grader in the far western reaches of bluegrass area Kentucky could have conceived back in 1912.
They were, however, expected to be able to name the waters though which an English vessel would pass en route to Manila via the Suez Canal.
Can you?
While we’re at it, how much do you really know about the human liver? Enough to locate it, identify its secretions, and discourse on its size relative to other bodily glands?
If you answered yes, congratulations. There’s a good chance you’d be promoted to high school back in 1912. Not bad for a kid attending a one-room school in rural Bullit County.
And now for some extra credit, name the last battles of the Civil War, the War of 1812, and the French and Indian War. Commanding officers, too…
That’s the sort of multipart question that awaited the eighth graders converging on the Bullit County courthouse for 1912’s common exam, above. The very same courthouse in which the modern day Bullitt County History Museum is located. A civic-minded individual donated a copy of the test to this institution, and the staff put it online, thinking it might be fun for latter-day specimens like you and me to see how we measure up.
So—just for fun—try typing the phrase “commanding officer last battle french & indian war” into your search engine of choice. Forget instant gratification. Embrace the anxiety!
Thank god the Internet was there to define “kalsomining” for me. Even with the aid of a calculator, math is not my strong suit. That said, I’m usually good enough with words to get the narrative gist of any story problem.
Usually.
I confess, I was so demoralized by my ignorance, I couldn’t have dreamed of attempting to figure out how much it would cost to “kalsomine” a 20 x 16 x 9 foot room, especially with a door and window involved.
Fortunately, the Bullit County Genealogical Society has seen fit to provide an online answer sheet, a digital luxury that would have gobsmacked their forebears.
SPOILER: $8.01. That’s the amount it would’ve cost to kalsomine your room at 1912 prices. (A steal, considering that a quart of White Wash Pickling Water Based Stain will run you $12.37 a quart at a nationally known hardware superstore today.)
Go ahead, take that test.
If you quail at the prospect of faring poorly against a rural 1912 eighth grader, just imagine how well he or she would do, teleported to 2016, and forced to contend with such mysteries as cyber bullying, gender politics, and offensive eggplant emojis…
Ayun Halliday is an author, illustrator, and Chief Primatologist of the East Village Inky zine. She lives in fear that her youngest child will pen a memoir titled I Was a Homeschooled 8th Grader and Other Chillling True Life Tales. Follow her @AyunHalliday
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.
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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.
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