Take an Intellectual Odyssey with a Free MIT Course on Douglas Hofstadter’s Pulitzer Prize-Winning Book Gödel, Escher, Bach: An Eternal Golden Braid

In 1979, mathematician Kurt Gödel, artist M.C. Escher, and composer J.S. Bach walked into a book title, and you may well know the rest. Douglas R. Hofstadter won a Pulitzer Prize for Gödel, Escher, Bach: an Eternal Golden Braid, his first book, thenceforth (and henceforth) known as GEB. The extraordinary work is not a treatise on mathematics, art, or music, but an essay on cognition through an exploration of all three — and of formal systems, recursion, self-reference, artificial intelligence, etc. Its publisher settled on the pithy description, “a metaphorical fugue on minds and machines in the spirit of Lewis Carroll.”

GEB attempted to reveal the mind at work; the minds of extraordinary individuals, for sure, but also all human minds, which behave in similarly unfathomable ways. One might also describe the book as operating in the spirit — and the practice — of Herman Hesse’s Glass Bead Game, a novel Hesse wrote in response to the data-driven machinations of fascism and their threat to an intellectual tradition he held particularly dear. An alternate title (and key phrase in the book) Magister Ludi, puns on both “game” and “school,” and alludes to the importance of play and free association in the life of the mind.




Hesse’s esoteric game, writes his biographer Ralph Freedman, consists of “contemplation, the secrets of the Chinese I Ching and Western mathematics and music” and seems similar enough to Hofstadter’s approach and that of the instructors of MIT’s open course, Gödel, Escher, Bach: A Mental Space Odyssey. Offered through the High School Studies Program as a non-credit enrichment course, it promises “an intellectual vacation” through “Zen Buddhism, Logic, Metamathematics, Computer Science, Artificial Intelligence, Recursion, Complex Systems, Consciousness, Music and Art.”

Students will not study directly the work of Gödel, Escher, and Bach but rather “find their spirits aboard our mental ship,” the course description notes, through contemplations of canons, fugues, strange loops, and tangled hierarchies. How do meaning and form arise in systems like math and music? What is the relationship of figure to ground in art? “Can recursion explain creativity,” as one of the course notes asks. Hofstadter himself has pursued the question beyond the entrenchment of AI research in big data and brute force machine learning. For all his daunting erudition and challenging syntheses, we must remember that he is playing a highly intellectual game, one that replicates his own experience of thinking.

Hofstadter suggests that before we can understand intelligence, we must first understand creativity. It may reveal its secrets in comparative analyses of the highest forms of intellectual play, where we see the clever formal rules that govern the mind’s operations; the blind alleys that explain its failures and limitations; and the possibility of ever actually reproducing workings in a machine. Watch the lectures above, grab a copy of Hofstadter’s book, and find course notes, readings, and other resources for the fascinating course Gödel, Escher, Bach: A Mental Space Odyssey archived here. The course will be added to our list, 1,700 Free Online Courses from Top Universities.

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

An Introduction to Nicolas Bourbaki, One of the Most Influential Mathematicians of All Time … Who Never Actually Lived

In 20th-century mathematics, the renowned name of Nicolas Bourbaki stands alone in its class — the class, that is, of renowned mathematical names that don’t actually belong to real people. Bourbaki refers not to a mathematician, but to mathematicians; a whole secret society of them, in fact, who made their name by collectively composing Elements of Mathematic. Not, mind you, Elements of Mathematics: “Bourbaki’s Elements of Mathematic — a series of textbooks and programmatic writings first appearing in 1939—pointedly omitted the ‘s’ from the end of ‘Mathematics,'” writes JSTOR Daily’s Michael Barany, “as a way of insisting on the fundamental unity and coherence of a dizzyingly variegated field.”

That’s merely the tip of Bourbaki’s iceberg of eccentricities. Formed in 1934 “by alumni of the École normale supérieure, a storied training ground for French academic and political elites,” this group of high-powered mathematical minds set about rectifying their country’s loss of nearly an entire generation of mathematicians in the First World War. (While Germany had kept its brightest students and scientists out of battle, the French commitment to égalité could permit no such favoritism.) It was the pressing need for revised and updated textbooks that spurred the members of Bourbaki to their collaboratively pseudonymous, individually anonymous work.




“Yet instead of writing textbooks,” explains Quanta‘s Kevin Hartnett, “they ended up creating something completely novel: free-standing books that explained advanced mathematics without reference to any outside sources.” The most distinctive feature of this already unusual project “was the writing style: rigorous, formal and stripped to the logical studs. The books spelled out mathematical theorems from the ground up without skipping any steps — exhibiting an unusual degree of thoroughness among mathematicians.”  Not that Bourbaki lacked playfulness: “In fanciful and pun-filled narratives shared among one another and alluded to in outward-facing writing,” adds Barany, “Bourbaki’s collaborators embedded him in an elaborate mathematical-political universe filled with the abstruse terminology and concepts of modern theories.”

You can get an animated introduction to Bourbaki, which survives even today as a still-prestigious and at least nominally secret mathematical society, in the TED-Ed lesson above. In the decades after the group’s founding, writes lesson author Pratik Aghor, “Bourrbaki’s publications became standard references, and the group’s members took their prank as seriously as their work.” Their commitment to the front was total: “they sent telegrams in Bourbaki’s name, announced his daughter’s wedding, and publicly insulted anyone who doubted his existence. In 1968, when they could no longer maintain the ruse, the group ended their joke the only way they could: they printed Bourbaki’s obituary, complete with mathematical puns.” And if you laugh at the mathematical pun with which Aghor ends the lesson, you may carry a bit of Bourbaki’s spirit within yourself as well.

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Based in Seoul, Colin Marshall writes and broadcasts on cities, language, and culture. His projects include the Substack newsletter Books on Cities, 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 or on Facebook.

Pink Floyd’s David Gilmour Composes a Soundtrack to Arthur C. Clarke’s Documentary Fractals: The Colors of Infinity

An observer once called the Mandelbrot Set “The Thumbprint of God,” the simple equation that led to the discovery of fractal geography, chaos theory, and why games like No Man’s Sky even exist. In 1994, Arthur C. Clarke, writer of both science fiction and science fact, narrated a one-hour documentary on the new mathematics, called Fractals: The Colors of Infinity. If that sounds familiar, dear reader, it’s because we’ve told you about it long ago. But it’s worth revisiting, and it’s worth mentioning that the soundtrack was created by Pink Floyd’s David Gilmour.

To be honest, at first I wasn’t really hearing that Floyd vibe, just some pleasant synth-strings you could find on any number of documentaries. But then Clarke explains the implication of the Mandelbrot equation, ending it with “This really is infinity.” And then Boom, the acid hit.




Or rather, the rainbow computer graphics of the endless zoom hit, and it was unmistakably Gilmour—cue up 5:19 and be careful with that fractal, Eugene. This happens again at 14:30, 25:12, 31:07, 35:46, 38:22, 43:22, 44:51, and 50:06 for those with an itchy scrubbing finger. But stick around for the whole doc, as the history of how we got to the equation, its precedents in nature and art, and the implications only hinted at in the program, all make for interesting viewing.

The music will remind you in places of “Shine On Your Crazy Diamond”, “Obscured by Clouds,” and “On the Run.” When a DVD was released years later, a special feature isolated just Gilmour’s music and the fractal animation.

Gilmour has contributed soundtrack work to other programs. He has an uncredited performance on Guy Pratt’s soundtrack from 1995’s Hackers; incidental music for 1992’s Ruby Takes a Trip with Ruby Wax; and a 1993 documentary on the arts and drug use called The Art of Tripping.

There are no official releases of this soundtrack work, but one user has put up 16 minutes of the Colours of Infinity music over at SoundCloud.

 

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Ted Mills is a freelance writer on the arts who currently hosts the Notes from the Shed podcast and is the producer of KCRW’s Curious Coast. You can also follow him on Twitter at @tedmills, and/or watch his films here.

The Mathematics Behind Origami, the Ancient Japanese Art of Paper Folding

The two characters at the core of origami (折り紙), one of the best-known Japanese words around the world, mean “folding” and “paper.” You might well have guessed that, but given the variety and elaborateness of the constructions produced by origami masters over the past few centuries, the simplicity of the practice’s basic nature bears repeating. Those masters must develop no slight degree of manual dexterity, it goes without saying, but also a formidable mathematical understanding of their medium. In many cases that understanding is intuitive; in the TED-Ed lesson above, origami artist Evan Zodl makes it explicit.

Zodl’s lesson explains that “though most origami models are three-dimensional, their crease patterns are usually designed to fold flat, without introducing any new creases or cutting the paper.”(Incidentally, the Japanese word for paper art involving cuts is kirigami, or 切り紙.)




An “abstract, 2D design” thus becomes, in the origami master’s hands, “a 3D form,” but only in accordance with a set of four simple rules Zodl explains. He does so clearly and understandably — and in a way that for many of us may exhume buried geometry-class memories — but like actual works of origami, they’re better shown than described: hence the vivid accompanying animations of Charlotte Arene.

Origami’s principles and products may be fascinating to contemplate, but “the ability to fold a large surface into a compact shape” has also proven to have serious real-world applications. Zodl points to an origami-based re-imagination of “the traditional stent graft, a tube used to open and support damaged blood vessels.” This in addition to “airbags, solar arrays, self-folding robots, and even DNA nanostructures” — as well as a massive “star shade” for space telescopes that blocks the glare of nearby stars. If you’d like to get started on your own tactile understanding of all this, do have a look at Zodl’s own Youtube channel, as well as others like Origami Instructions. Don’t let the elaborately folded flowers, boats, or animals you’ve seen intimidate you; start with a simple box and work your way up from there. If origami shows us anything, after all, it’s that complexity begins with simplicity.

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Based in Seoul, Colin Marshall writes and broadcasts on cities, language, and culture. His projects include the Substack newsletter Books on Cities, 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 or on Facebook.

Three Amateur Cryptographers Finally Decrypted the Zodiac Killer’s Letters: A Look Inside How They Solved a Half Century-Old Mystery

If we envision serial killers as figures who taunt law enforcement with cryptic messages sent to the media, we do so in large part because of the Zodiac Killer, who terrorized northern California in the late 1960s and early 70s. Though he seems to have stopped killing more than half a century ago, he remains an object of great fascination (and even became the subject of David Fincher’s acclaimed film Zodiac in 2007). As thoroughly as the case has been investigated, much remains unknown — not least what he actually said in some of his coded letters. But just this month, a team of three cryptography enthusiasts managed to break one of the Zodiac’s ciphers, finally revealing the contents of a 51-year old letter.

The Zodiac wrote this particular communiqué in a transposition cipher, which, as Ars Technica’s Dan Goodin writes, uses “rules to rearrange the characters or groups of characters in the message.” In the case of the 340, named for the number of symbols, the content “was probably rearranged by manipulating triangular sections cut from messages written into rectangles.” For the past half-century, nobody could successfully return the text to its original arrangement, but in 2020, there’s an app for that. Or rather, a software engineer named David Oranchak, a mathematician named Sam Blake, and a programmer named Jarl Van Eycke made an app for that. Goodin quotes Oranchak as saying the three had been “working on and off on solving the 340 since 2006.”




You can see Oranchak explain how he and his collaborators finally cracked the 340’s cipher in the video at the top of the post, the final episode of his five-part series Let’s Crack the Zodiac. This wasn’t a matter of simply whipping up the right piece of artificial intelligence and letting it rip: they had to generate hundreds of thousands of permutations of the message as well as attempts at decryptions of those messages. And even when recognizable words and phrases began to emerge in the results — “TRYING TO CATCH ME,” “THE GAS CHAMBER” — quite a bit of trial, error, and thought, remained to be done. It helped that Oranchak knew his Zodiac history, such as that someone claiming to be the killer mentioned not wanting to be sent to the gas chamber when he called in to a local television show on October 20, 1969, two weeks before the 340 was received.

Was it really him? The 340, when finally decoded — a process complicated by the mistakes the Zodiac made, not just in spelling but in executing his laborious, fully analog encryption process — seems to provide the answer:

I HOPE YOU ARE HAVING LOTS OF FUN IN TRYING TO CATCH ME
THAT WASNT ME ON THE TV SHOW
WHICH BRINGS UP A POINT ABOUT ME
I AM NOT AFRAID OF THE GAS CHAMBER
BECAUSE IT WILL SEND ME TO PARADICE ALL THE SOONER
BECAUSE I NOW HAVE ENOUGH SLAVES TO WORK FOR ME
WHERE EVERYONE ELSE HAS NOTHING WHEN THEY REACH PARADICE
SO THEY ARE AFRAID OF DEATH
I AM NOT AFRAID BECAUSE I KNOW THAT MY NEW LIFE IS
LIFE WILL BE AN EASY ONE IN PARADICE DEATH

“The message doesn’t really say a whole lot,” admits Oranchak. “It’s more of the same attention-seeking junk from Zodiac. We were disappointed that he didn’t put any personally identifying information in the message, but we didn’t expect him to.” The Zodiac Killer remains unidentified, and indeed remains one of recent history’s more compelling villains, not just to those with an interest in true crime, but to those with an interest in cryptography as well. For two more messages still remain to be decoded, and in one of them he offers a short cipher that, he writes, contains his name — but then, if there’s any correspondent we shouldn’t rush to take at his word, it’s this one.

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Based in Seoul, Colin Marshall writes and broadcasts on cities, language, and culture. His projects include the Substack newsletter Books on Cities, 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.

This Is What an 1869 MIT Entrance Exam Looks Like: Could You Have Passed the Test?

The late 19th Century was the time of Charles Darwin and James Clerk Maxwell, of Thomas Edison and Alexander Graham Bell. It was a golden age of science and technology. So you might wonder how hard it was to get into one of the top technical universities in that era.

The answer, according to this video? Not very hard.

At least that was the case in 1869 at the Massachusetts Institute of Technology, or MIT,  as the young Australian science and math teacher Toby Hendy explains on her excellent YouTube channel, Tibees. MIT was brand new and desperate for tuition revenue in 1869, so the object of the test wasn’t to whittle a massive field of applicants down to a manageable size. It was simply to make sure that incoming students could handle the work.




MIT opened in 1865, just after the end of the Civil War. The idea was to create a European-style polytechnic university to meet the demands of an increasingly industrial economy. The original campus was in Boston, across the Charles River from its current location in Cambridge. Only 15 students signed up in 1865. Tuition was $100 for the whole year. There was no formal entrance test. According to an article from the school’s Archives and Special Collections,

The “conditions for admission” section of MIT’s catalogue for 1865-66 indicates that candidates for admission as first year students must be at least sixteen years old and must give satisfactory evidence “by examination or otherwise” of a competent training in arithmetic, geometry, English grammar, geography, and the “rudiments of French.” Rapid and legible handwriting was also stressed as being “particularly important.” By 1869 the handwriting requirement and French had been dropped, but algebra had been added and students needed to pass a qualifying exam in the required subject areas. An ancillary effect was to protect unqualified students from disappointment and professors from wasting their time.

A couple of years earlier, in 1867, the MIT Executive Committee reported that faculty members had felt it necessary to ask parents of “some incompetent and inattentive students to withdraw them from the school, wishing to spare them the mortification of an examination which it was certain they could not pass.”

Nowadays, the students who make it into MIT have average SAT and ACT scores in the 99th percentile. Of 21,312 first-year applicants hoping to join the Class of 2023, only 1,427 made it. That’s an admission rate of 6.7 percent. What a difference 150 years can make!

To take the 1869 entrance examination in English, Algebra, Geometry and Arithmetic, and to see the correct answers, visit this cached article from the MIT website.

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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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Discover Kōlams, the Traditional Indian Patterns That Combine Art, Mathematics & Magic

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.

“Feeding A Thousand Souls: Kōlam” by Thacher Gallery at the University of San Francisco is licensed under CC BY-SA 2.0

“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.”

Colorful Kolam – Sivasankaran – Own work

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.”

“Kolam” by resakse is licensed under CC BY-ND 2.0

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.

Read more about kōlams at Atlas Obscura.

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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.

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