How Leonardo da Vinci Drew an Accurate Satellite Map of an Italian City (1502)

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.

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

Complex Math Made Simple With Engaging Animations: Fourier Transform, Calculus, Linear Algebra, Neural Networks & More

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.

3Blue1Brown, a project of former Khan Academy fellow Grant Sanderson, has done the same for dozens of STEM concepts, including such subjects as higher dimensions, cryptocurrencies, machine learning, and neural networks and essentials of calculus and linear algebra like the derivative paradox and “Vectors, what even are they?”

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.

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

A Beautifully-Designed Edition of Euclid’s Elements from 1847 Gets Digitized: Explore the New Online, Interactive Reproduction

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 Information not 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.

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

Did Lennon or McCartney Write the Beatles 1965 Song “In My Life”? A Math Professor, Using Statistics, Solves the Decades-Old Mystery


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.

They present their research in a paper titled “Assessing Authorship of Beatles Songs from Musical Content: Bayesian Classification Modeling from Bags-Of-Words Representations.” In the NPR Weekend Edition interview above, you can hear Stanford mathematician Keith Devlin break down the terms of their project, including that odd phrase “bags-of-words representations,” which “actually goes back to the 1950s,” he says. “Bags-of-words”—like the word clouds we now see on websites—take text, “ignore the grammar” and word order and produce a collection of words. The method was used to generate the first spam filters. Rather than use words, however, the mathematicians decontextualized snippets of sound.

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.

via NPR

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

John Nash’s Super Short PhD Thesis: 26 Pages & 2 Citations

nash thesis

When John Nash wrote "Non Cooperative Games," his Ph.D. dissertation at Princeton in 1950, the text of his thesis (read it online) was brief. It ran only 26 pages. And more particularly, it was light on citations. Nash's diss cited two texts: John von Neumann & Oskar Morgenstern's Theory of Games and Economic Behavior (1944), which essentially created game theory and revolutionized the field of economics; the other cited text, "Equilibrium Points in n-Person Games," was an article written by Nash himself. And it laid the foundation for his dissertation, another seminal work in the development of game theory, for which Nash won the Nobel Prize in Economic Sciences in 1994.

The reward of inventing a new field is having a slim bibliography.

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Note: An earlier version of this post appeared on our site in June, 2015.

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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 a couple of years back...

Would you like to support the mission of Open Culture? Please consider making a donation to our site. It's hard to rely 100% on ads, and your contributions will help us continue providing the best free cultural and educational materials to learners everywhere.

Also consider following Open Culture on Facebook and Twitter and sharing intelligent media with your friends. Or sign up for our daily email and get a daily dose of Open Culture in your inbox. 

Note: An earlier version of this post appeared on our site in April, 2015.

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Steve Martin & Robin Williams Riff on Math, Physics, Einstein & Picasso in a Smart Comedy Routine

Back in 2002, Stanford University mathematics professor Robert Osserman chatted with comedian and banjo player extraordinaire Steve Martin in San Francisco’s Herbst Theatre. The event was called “Funny Numbers” and it was intended to deliver an off-kilter discussion on math. Boy did it deliver.

The first half of the discussion was loose and relaxed. Martin talked about his writing, banjos and his childhood interest in math. “In high school, I used to be able to make magic squares," said Martin. "I like anything kind of 'jumbly.' I like anagrams. What else do I like? I like sex."

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Then Robin Williams, that manic ball of energy, showed up. As you can see from the five videos throughout this post, the night quickly spiraled into comic madness.

They riffed on the Osbournes, Henry Kissinger, number theory, and physics. “Schrödinger, pick up your cat,” barks Williams at the end of a particularly inspired tear. “He’s alive. He’s dead. What a pet!”

When Martin and Williams read passages from Martin’s hit play, Picasso at the Lapin Agile Williams read his part at different points as if he were Marlon Brando, Peter Lorre and Elmer Fudd. At another time, Williams and Martin riffed on the number zero. Williams, for once acting as the straight man, asked Osserman, "I have one quick question, up to the Crusades, the number zero didn't exist, right? In Western civilization.” To which Martin bellowed, “That is a lie! How dare you imply that the number zero…oh, I think he’s right.”

The videos are weirdly glitchy, though the audio is just fine. And the comedy is completely hilarious and surprisingly thought provoking.

Note: An earlier version of this post appeared on our site in September, 2015.

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Jonathan Crow is a Los Angeles-based writer and filmmaker whose work has appeared in Yahoo!, The Hollywood Reporter, and other publications. You can follow him at @jonccrow. And check out his blog Veeptopus, featuring lots of pictures of vice presidents with octopuses on their heads.  The Veeptopus store is here.

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