The Map of Mathematics: An Animated Video Shows How All the Different Fields in Math Fit Together

If you’re a regular Open Culture reader, you have hopefully thoroughly immersed yourself in The Map of Physics, an animated video–a visual aid for the modern age–that mapped out the field of physics, explaining all the connections between classical physics, quantum physics, and relativity.

You can’t do physics without math. Hence we now have The Map of Mathematics. Created by physicist Dominic Walliman, this new video explains “how pure mathematics and applied mathematics relate to each other and all of the sub-topics they are made from.” Watch the new video above. You can buy a poster of the map here. And you can download a version for educational use here. Enjoy.

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A Retired Math Teacher Helps Students Learn Geometry Through Quilting

Some real talk from retired geometry teacher Wendy Lichtman, above, the author of several math-themed YA novels:

Not many 15-year-olds care that two parallel lines are crossed by a transversal.

“But right here are two parallel lines,” she continues, pointing to a pink and orange quilt. “and these are transversals, and they are at a 90º angle and it feels real. You’ve gotta get it to look right.”

The teenaged participants in the Oakland, California program she founded to demystify geometry through hands-on quiltmaking get it to look right by plotting their designs on graph paper, carefully measuring and cutting shapes from bright calico of their own choosing. (Licthman has committed to buttoning her lip if their favored print is not to her taste.)

Lichtman came up with this creative approach to help a bright student who was in danger of not graduating, having flunked geometry three times.

She details their journey in How to Make a Geometric Quilt, an essay formatted as step-by-step instructions…not for quiltmaking but rather how those in the teaching profession can lead with humility and determination, while maintaining good boundaries.

Some highlights:

6. Sometime after the sewing has begun, and the math notebook is ignored for weeks, begin to worry that your student is not really learning geometry.  She’s learning sewing and she’s learning to fix a broken bobbin, but really, geometry?

7. Remind yourself that this kid needs a quilt as much as she needs geometry.

8. Remember, also, the very, very old woman who taught you hat-making one night long ago.  She had gone to school only through 5th grade because, she said, she was a Black child in the deep south and that’s how it was back then.  Think about how she explained to the hat-making class that to figure out the length of the hat’s brim, you needed to measure from the center to the edge with a string and then do “three of those and a little bit more,” and remember how you sat in awe, because three radii and a little bit more is the definition of pi, and this hat-maker had evidently discovered for herself the formula for circumference.

As the two become better acquainted, the student let her guard down, revealing more about her situation while they swapped stories of their mothers.

But this was no easy A.

In addition to expecting regular, punctual attendance, Lictman stipulated that in order to pass, the student could not give the fruits of her labor away.

(Solid advice for creators of any craft project this ambitious. As Debbie Stoller, author of Stitch ‘n Bitch: The Knitter’s Handbook counsels:

…those who have never knit something have no idea how much time it took. If you give someone a sweater, they may think that you made that in an evening when you were watching a half-hour sitcom. It’s only when people actually attempt to knit that they finally get this realization, this light bulb goes on over their heads, and they realize that, “Wow, this actually takes some skill and some time. I’ve got newfound respect for my grandma.”)

Ultimately, Lichtman concludes that the five credits she awarded her student could not be reduced to something as simple as geometry or quilt-making;

You are giving her credit for something less tangible.  Something like pride.  Five credit hours for feeling she can accomplish something hard that, okay, is slightly related to geometry.

Examples of the current cohort’s work can be seen on Rock Paper Scissors Collective‘s Instagram.

Once completed, these quilts will be donated to Bay Area foster children and pediatric patients at the local Children’s Hospital.

via BoingBoing

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Via Boing Boing

– Ayun Halliday is the Chief Primatologist of the East Village Inky zine and author, most recently, of Creative, Not Famous: The Small Potato Manifesto and Creative, Not Famous Activity Book. Follow her @AyunHalliday.

Why Algorithms Are Called Algorithms, and How It All Goes Back to the Medieval Persian Mathematician Muhammad al-Khwarizmi

In recent decades, a medieval Persian word has come to prominence in English and other major world languages. Many of use it on a daily basis, often while regarding the concept to which it refers as essentially mysterious. The word is algorithm, whose roots go back to the ninth century in modern-day Greater Iran. There lived a polymath by the name of Muhammad ibn Musa al-Khwarizmi, whom we now remember for his achievements in geography, astronomy, and mathematics. In that last field, he was the first to define the principles of “reducing” and “balancing” equations, a subject all of us came to know in school as algebra (a name itself descended from the Arabic al-jabr, or “completion”).

Today, a good few of us have come to resent algorithms even more than algebra. This is perhaps because algorithms are most popularly associated with the deep, unseen workings of the internet, a system with ever increasing influence over the things we do, the information we receive, and even the people with whom we associate.

Provided sufficient data about us and the lives we lead, so we’re given to understand, these algorithms can make better decisions for us than we can make for ourselves. But what exactly are they? You can get one answer from “Why Algorithms Are Called Algorithms,” the BBC Ideas video at the top of the post.

For Western civilization, al-Khwarizmi’s most important book was Concerning the Hindu Art of Reckoning, which was translated into Latin three centuries after its composition. Al-Khwarizmi’s Latinized name “Algoritmi” gave rise to the word algorismus, which at first referred to the decimal number system and much later came to mean “a set of step-by-step rules for solving a problem.” It was Enigma codebreaker Alan Turing who “worked out how, in theory, a machine could follow algorithmic instructions and solve complex mathematics. This was the birth of the computer age.” Now, much further into the computer age, algorithms “are helping us to get from A to B, driving internet searches, making recommendations of things for us to buy, watch, or share.”

The algorithm giveth, but the algorithm also taketh away — or so it sometimes feels as we make our way deeper into the twenty-first century. In the other BBC Ideas video just above, Jon Stroud makes an investigation into both the nature and the current uses of this mathematical concept. The essential job of an algorithm, as the experts explain to him, is that of processing data, these days often in large quantities and of various kinds, and increasingly with the aid of sophisticated machine-learning processes. In making or influencing choices humans would once have handled themselves, algorithms do present a risk of “de-skilling” as we come to rely on their services. We all occasionally feel gratitude for the blessings those services send our way, just as we all occasionally blame them for our dissatisfactions — making the algorithm, in other words, into a thoroughly modern deity.

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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.

Bach on a Möbius Strip: Marcus du Sautoy Visualizes How Bach Used Math to Compose His Music

“A mathematician’s favorite composer? Top of the list probably comes Bach.” Thus speaks a reliable source on the matter: Oxford mathematician Marcus du Sautoy in the Numberphile video above. “Bach uses a lot of mathematical tricks as a way of generating music, so his music is highly complex,” but at its heart is “the use of mathematics as a kind of shortcut to generate extraordinarily complex music.” As a first example du Sautoy takes up the “Musical Offering,” and in particular its “crab canon,” the genius of which has previously been featured here on Open Culture.

Written out, Bach’s crab canon “looks like just one line of music.” But “what’s curious is that when you get to the end of the music, there’s the little symbol you usually begin a piece of music with.” This means that Bach wants the player of the piece to “play this forwards and backwards; he’s asking you to start at the end and play it backwards at the same time.” His composition thus becomes a two-voice piece made out of just one line of music going in both directions. It’s the underlying mathematics that make this, when played, more than just a trick but “something beautifully harmonic and complex.”

To understand the crab canon or Bach’s other mathematically shaped pieces, it helps to visualize them in unconventional ways such as on a twisting Möbius strip, whose ends connect directly to one another. “You can make a Möbius strip out of any piece of music,” says du Sautoy as he does so in the video. “The stunning thing is that when you then look at this piece of music” — that is the fifth canon from Bach’s Goldberg Variations — “the notes that are on one side are exactly the same notes as if this thing were see-through.” (Naturally, he’s also prepared a see-through Bach Möbius strip for his viewing audience.)

In 2017 du Sautoy gave an Oxford Mathematics Public Lecture on “the Sound of Symmetry and the Symmetry of Sound.” In it he discusses symmetry as present in not just the Goldberg Variations but the twelve-tone rows composed in the 20th century by Arnold Schoenberg and even the very sound waves made by musical instruments themselves. Just this year, he collaborated with the Oxford Philharmonic Orchestra to deliver “Music & Maths: Baroque & Beyond,” a presentation that draws mathematical connections between the music, art, architecture, and science going on in the 17th and 18th centuries. Bach has been dead for more than a quarter of a millennium, but the connections embodied in his music still hold revelations for listeners willing to hear them — or see them.

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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.

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.

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