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.

Relat­ed Con­tent:

Carl Sagan Explains How the Ancient Greeks, Using Rea­son and Math, Fig­ured Out the Earth Isn’t Flat, Over 2,000 Years Ago

The Ele­gant Math­e­mat­ics of Vit­ru­vian Man, Leonar­do da Vinci’s Most Famous Draw­ing: An Ani­mat­ed Intro­duc­tion

Leonar­do da Vin­ci Saw the World Dif­fer­ent­ly… Thanks to an Eye Dis­or­der, Says a New Sci­en­tif­ic Study

Leonar­do da Vinci’s Ear­li­est Note­books Now Dig­i­tized and Made Free Online: Explore His Inge­nious Draw­ings, Dia­grams, Mir­ror Writ­ing & More

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.

Relat­ed Con­tent:

Free Online Math Cours­es

The Map of Math­e­mat­ics: Ani­ma­tion Shows How All the Dif­fer­ent Fields in Math Fit Togeth­er

Cit­i­zen Maths: A Free Online Course That Teach­es Adults the Math They Missed in High School

Free Math Text­books 

Math Mag­ic

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.

Relat­ed Con­tent:

Explore an Inter­ac­tive, Online Ver­sion of Werner’s Nomen­cla­ture of Colours, a 200-Year-Old Guide to the Col­ors of the Nat­ur­al World

The Map of Math­e­mat­ics: Ani­ma­tion Shows How All the Dif­fer­ent Fields in Math Fit Togeth­er

Where to Find Free Text­books

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

Relat­ed Con­tent:

Gui­tarist Randy Bach­man Demys­ti­fies the Open­ing Chord of The Bea­t­les’ “A Hard Day’s Night”

Arti­fi­cial Intel­li­gence Pro­gram Tries to Write a Bea­t­les Song: Lis­ten to “Daddy’s Car”

The Bea­t­les “While My Gui­tar Gen­tly Weeps” Gets a Dreamy New Music Video from Cirque du Soleil

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.

Relat­ed Con­tent:

The Short­est-Known Paper Pub­lished in a Seri­ous Math Jour­nal: Two Suc­cinct Sen­tences

The World Record for the Short­est Math Arti­cle: 2 Words

Free Online Math Cours­es

Free Math Text­books

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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.”

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.

Relat­ed Con­tent:

Steve Mar­tin Writes Song for Hymn-Deprived Athe­ists

Watch Steve Mar­tin Make His First TV Appear­ance: The Smoth­ers Broth­ers Com­e­dy Hour (1968)

Lis­ten as Albert Ein­stein Reads ‘The Com­mon Lan­guage of Sci­ence’ (1941)

Ein­stein Explains His Famous For­mu­la, E=mc², in Orig­i­nal Audio

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.

Infographics Show How the Different Fields of Biology, Chemistry, Mathematics, Physics & Computer Science Fit Together

Ask any­one who’s pur­sued a career in the sci­ences what first piqued their inter­est in what would become their field, and they’ll almost cer­tain­ly have a sto­ry. Gaz­ing at the stars on a camp­ing trip, rais­ing a pet frog, fool­ing around with com­put­ers and their com­po­nents: an expe­ri­ence sparks a desire for knowl­edge and under­stand­ing, and the pur­suit of that desire even­tu­al­ly deliv­ers one to their spe­cif­ic area of spe­cial­iza­tion.

Or, as they say in sci­ence, at least it works that way in the­o­ry; the real­i­ty usu­al­ly unrolls less smooth­ly. On such a jour­ney, just like any oth­er, it might help to have a map.

Enter the work of sci­ence writer and physi­cist Dominic Wal­li­man, whose ani­mat­ed work on the Youtube chan­nel Domain of Sci­ence we’ve pre­vi­ous­ly fea­tured here on Open Cul­ture. (See the “Relat­ed Con­tent” sec­tion below for the links.)

Wal­li­man’s videos astute­ly explain how the sub­fields of biol­o­gy, chem­istry, math­e­mat­ics, physics, and com­put­er sci­ence relate to each oth­er, but now he’s turned that same mate­r­i­al into info­graph­ics read­able at a glance: maps, essen­tial­ly, of the intel­lec­tu­al ter­ri­to­ry. He’s made these maps, of biol­o­gy, chem­istry, math­e­mat­ics, physics, and com­put­er sci­ence, freely avail­able on his Flickr account: you can view them all on a sin­gle page here along with a few more of his info­graph­ics..

As much use as Wal­li­man’s maps might be to sci­ence-mind­ed young­sters look­ing for the best way to direct their fas­ci­na­tions into a prop­er course of study, they also offer a help­ful reminder to those far­ther down the path — espe­cial­ly those who’ve strug­gled with the blind­ers of hyper­spe­cial­iza­tion — of where their work fits in the grand scheme of things. No mat­ter one’s field, sci­en­tif­ic or oth­er­wise, one always labors under the threat of los­ing sight of the for­est for the trees. Or the realm of life for the bioin­for­mat­ics, bio­physics, and bio­math­e­mat­ics; the whole of math­e­mat­ics for the num­ber the­o­ry, the dif­fer­en­tial geom­e­try, and the dif­fer­en­tial equa­tions; the work­ings of com­put­ers for the sched­ul­ing, the opti­miza­tion, and the boolean sat­is­fi­a­bil­i­ty.

Relat­ed Con­tent:

The Map of Biol­o­gy: Ani­ma­tion Shows How All the Dif­fer­ent Fields in Biol­o­gy Fit Togeth­er

The Map of Com­put­er Sci­ence: New Ani­ma­tion Presents a Sur­vey of Com­put­er Sci­ence, from Alan Tur­ing to “Aug­ment­ed Real­i­ty”

The Map of Math­e­mat­ics: Ani­ma­tion Shows How All the Dif­fer­ent Fields in Math Fit Togeth­er

The Map of Physics: Ani­ma­tion Shows How All the Dif­fer­ent Fields in Physics Fit Togeth­er

The Map of Chem­istry: New Ani­ma­tion Sum­ma­rizes the Entire Field of Chem­istry in 12 Min­utes

The Art of Data Visu­al­iza­tion: How to Tell Com­plex Sto­ries Through Smart Design

Based in Seoul, Col­in Mar­shall writes and broad­casts on cities 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 or on Face­book.

Free: The Best Books for Learning Modern Statistics

A quick fyi: Dan Kopf, an eco­nom­ics reporter, has a tip that seemed worth pass­ing along. Over at Quartz, he writes:

As a for­mer data sci­en­tist, there is no ques­tion I get asked more than, “What is the best way to learn sta­tis­tics?” I always give the same answer: Read An Intro­duc­tion to Sta­tis­ti­cal Learn­ing. Then, if you fin­ish that and want more, read The Ele­ments of Sta­tis­ti­cal Learn­ing. These two books, writ­ten by sta­tis­tics pro­fes­sors at Stan­ford Uni­ver­si­ty, the Uni­ver­si­ty of Wash­ing­ton, and the Uni­ver­si­ty South­ern Cal­i­for­nia, are the most intu­itive and rel­e­vant books I’ve found on how to do sta­tis­tics with mod­ern tech­nol­o­gy… You can down­load them for free.

Find An Intro­duc­tion to Sta­tis­ti­cal Learn­ing in PDF for­mat here. And The Ele­ments of Sta­tis­ti­cal Learn­ing here. Physical/hard copies can be pur­chased respec­tive­ly here and here.

We’d also rec­om­mend sup­ple­ment­ing these resources (both now avail­able in our col­lec­tion of Free Math Text­books) with video-based class­es found on our list of Free Math Cours­es, a sub­set of our big col­lec­tion, 1,700 Free Online Cours­es from Top Uni­ver­si­ties.

If you would like to sign up for Open Culture’s free email newslet­ter, please find it here. Or fol­low our posts on Threads, Face­book, BlueSky or Mastodon.

If you would like to sup­port the mis­sion of Open Cul­ture, 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. You can con­tribute through Pay­Pal, Patre­on, and Ven­mo (@openculture). Thanks!

via Quartz

Relat­ed Con­tent:

Against All Odds: A Gen­tle Intro­duc­tion to Sta­tis­tics Host­ed by Har­vard Geneti­cist Par­dis Sabeti (Free Online Course)

Sta­tis­tics Explained Through Mod­ern Dance: A New Way of Teach­ing a Tough Sub­ject

Flo­rence Nightin­gale Saved Lives by Cre­at­ing Rev­o­lu­tion­ary Visu­al­iza­tions of Sta­tis­tics (1855)

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