Mathematics Made Visible: The Extraordinary Mathematical Art of M.C. Escher

The eye and the intel­lect play off one anoth­er in sur­pris­ing and beau­ti­ful ways in the art of M.C. Esch­er. Where the Renais­sance mas­ters used shad­ing and per­spec­tive to cre­ate the illu­sion of three-dimen­sion­al depth on two dimen­sion­al sur­faces, Esch­er turned those tricks in on them­selves to cre­ate puz­zles and para­dox­es. He manip­u­lat­ed our fac­ul­ties of per­cep­tion not sim­ply to please the sens­es, but to stim­u­late the mind.

His cool, ana­lyt­ic ten­den­cy was appar­ent from the start. “Mau­rits Esch­er is a good graph­ic artist,” wrote the head­mas­ter of the Haar­lem School of Archi­tec­ture and Dec­o­ra­tive Arts in 1922, the year of Escher’s grad­u­a­tion, “but he lacks the right artis­tic tem­pera­ment.

His work is to too cerebral–neither emo­tion­al nor lyri­cal enough.” Escher’s work became even more cere­bral over time, as it grew in geo­met­ric sophis­ti­ca­tion. In describ­ing what went into the cre­ation of his wood­cuts and engrav­ings, Esch­er wrote:

The ideas that are basic to them often bear wit­ness to my amaze­ment and won­der at the laws of nature which oper­ate in the world around us. He who won­ders dis­cov­ers that this is in itself a won­der. By keen­ly con­fronting the enig­mas that sur­round us, and by con­sid­er­ing and ana­lyz­ing the obser­va­tions that I had made, I end­ed up in the domain of math­e­mat­ics. Although I am absolute­ly inno­cent of train­ing or knowl­edge in the exact sci­ences, I often seem to have more in com­mon with math­e­mati­cians than with my fel­low artists.

The affin­i­ty between Esch­er and math­e­mati­cians is described in the scene above from the the BBC doc­u­men­tary, The Math­e­mat­i­cal Art of M.C. Esch­er. “Math­e­mati­cians know their sub­ject is beau­ti­ful,” says Ian Stew­art of the Uni­ver­si­ty of War­wick. “Esch­er shows us that it’s beau­ti­ful.”

If the BBC clip whets your appetite, be sure to watch Meta­mor­phose: M.C. Esch­er, 1898–1972, a 2002 doc­u­men­tary by Jan Bro­driesz. The one-hour film gives an excel­lent overview of the Dutch artist’s life and work, and fea­tures a rare inter­view with Esch­er, along with scenes of him cre­at­ing his art. If you’re a fan of Esch­er, this film is a must-see.

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!

Relat­ed Con­tent:

Free Online Math Cours­es

M.C. Escher’s Per­pet­u­al Motion Water­fall Brought to Life: Real or Sleight of Hand?

Inspi­ra­tions: A Short Film Cel­e­brat­ing the Math­e­mat­i­cal Art of M.C. Esch­er

Udacity to Launch 5 New Courses, from Statistics to Physics. Shooting for Largest Online Class Ever.

This week­end, the Wall Street Jour­nal pub­lished a gush­ing lit­tle pro­file on Sebas­t­ian Thrun. By now, you prob­a­bly know his bio. Thrun helped invent the self-dri­ving car at Google and taught arti­fi­cial intel­li­gence at Stan­ford, before ditch­ing his tenured teach­ing posi­tion and launch­ing Udac­i­ty, a new ven­ture that offers MOOCs (Mas­sive Open Online Cours­es) to stu­dents every­where.

Also this week­end, Thrun kicked off an effort to “break the stu­dent record for the largest online class ever taught” with his new class, Intro­duc­tion to Sta­tis­tics: Mak­ing Deci­sions Based on Data. It starts June 25, and above you can watch Thrun give a short (Hans Rosling-like) intro­duc­tion to the class.  The course is entire­ly free and open to stu­dents every­where. Stu­dents will receive dynam­ic feed­back along the way, and dili­gent stu­dents will get a cer­tifi­cate of com­ple­tion at the end. So what’s stop­ping you? Cer­tain­ly not mon­ey or geog­ra­phy. Oth­er cours­es start­ing on June 25 include:

Intro to Physics: Land­marks in Physics

Algo­rithms: Crunch­ing Social Net­works

Log­ic & Dis­crete Math­e­mat­ics: Foun­da­tions of Com­put­ing

Soft­ware Test­ing: How to Make Soft­ware Fail

Relat­ed Con­tent:

Cours­era Adds Human­i­ties Cours­es, Rais­es $16 Mil­lion, Strikes Deal with 3 Uni­ver­si­ties

Har­vard and MIT Cre­ate EDX to Offer Free Online Cours­es World­wide

Why the Uni­ver­si­ty Sys­tem, as We Know It, Won’t Last …. and What’s Com­ing Next

Free Online Cer­tifi­cate Cours­es from Great Uni­ver­si­ties: A Com­plete List

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Inspirations: A Short Film Celebrating the Mathematical Art of M.C. Escher

Almost two years ago, Span­ish film­mak­er CristĂłbal Vila shot an exquis­ite lit­tle film, Nature by Num­bers, which cap­tured the ways in which math­e­mat­i­cal con­cepts (Fibonac­ci Sequence, Gold­en Num­ber, etc.) reveal them­selves in nature. And the short then clocked a good 2.1 mil­lion views on YouTube alone.

This week, Vila returns with a new film called Inspi­ra­tions. In this case, the inspi­ra­tion is M.C. Esch­er (1898–1972), the Dutch artist who explored a wide range of math­e­mat­i­cal ideas with his wood­cuts, lith­o­graphs, and mez­zot­ints. Although Esch­er had no for­mal train­ing in math­e­mat­ics beyond sec­ondary school, many math­e­mati­cians count­ed them­selves as admir­ers of his work. (Vis­it this online gallery to get bet­ter acquaint­ed with Escher’s art, and be sure to click on the thumb­nails to enlarge the images). As Vila explains, Inspi­ra­tions tries to imag­ine Escher’s work­place, “what things would sur­round an artist like him, so deeply inter­est­ed in sci­ence in gen­er­al and math­e­mat­ics in par­tic­u­lar.” It’s a three min­utes of unbri­dled imag­i­na­tion.

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!

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Watch Breaking the Code, About the Life & Times of Alan Turing (1996)

Updat­ed on Decem­ber 24, 2013: Yes­ter­day the British gov­ern­ment brought a sad chap­ter to a close when it final­ly issued a posthu­mous par­don to Alan Tur­ing, who was con­vict­ed in 1952 of break­ing laws that crim­i­nal­ized homo­sex­u­al­i­ty. The post you see below was orig­i­nal­ly writ­ten in Feb­ru­ary, 2012, when the ques­tion of Tur­ing being par­doned was still up for debate. The film fea­tured above is still very much worth your while.

This week the British gov­ern­ment final­ly par­doned Alan Tur­ing. One of the great­est math­e­mati­cians of the 20th cen­tu­ry, Tur­ing laid the foun­da­tions for com­put­er sci­ence and played a key role in break­ing the Nazi Enig­ma code dur­ing World War II. In 1952 he was con­vict­ed of homo­sex­u­al­i­ty. He killed him­self two years lat­er, after being chem­i­cal­ly cas­trat­ed by the gov­ern­ment.

On Mon­day, Jus­tice Min­is­ter Tom McNal­ly told the House of Lords that the gov­ern­ment of Prime Min­is­ter David Cameron stood by the deci­sion of ear­li­er gov­ern­ments to deny a par­don, not­ing that the pre­vi­ous prime min­is­ter, Gor­don Brown, had already issued an “unequiv­o­cal posthu­mous apol­o­gy” to Tur­ing. McNal­ly was quot­ed  in the Guardian:

A posthu­mous par­don was not con­sid­ered appro­pri­ate as Alan Tur­ing was prop­er­ly con­vict­ed of what at the time was a crim­i­nal offense. He would have known that his offense was against the law and that he would be pros­e­cut­ed. It is trag­ic that Alan Tur­ing was con­vict­ed of an offense which now seems both cru­el and absurd–particularly poignant giv­en his out­stand­ing con­tri­bu­tion to the war effort. How­ev­er, the law at the time required a pros­e­cu­tion and, as such, long-stand­ing pol­i­cy has been to accept that such con­vic­tions took place and, rather than try­ing to alter the his­tor­i­cal con­text and to put right what can­not be put right, ensure instead that we nev­er again return to those times.

The deci­sion came as a dis­ap­point­ment to thou­sands of peo­ple around the world who had peti­tioned for a for­mal par­don dur­ing the cen­te­nary year of Tur­ing’s birth. The Guardian also quot­ed an email sent by Amer­i­can math­e­mati­cian Den­nis Hejhal to a British col­league:

i see that the House of Lords reject­ed the par­don Feb 6 on what are for­mal grounds.

if law is X on date D, and you know­ing­ly break law X on date D, then you can­not be par­doned (no mat­ter how wrong or flawed law X is).

the real rea­son is OBVIOUS. they do not want thou­sands of old men say­ing par­don us too.

Efforts to obtain a par­don for Tur­ing are con­tin­u­ing. British cit­i­zens and UK res­i­dents can still sign the peti­tion.

To learn more about Tur­ing’s life, you can watch the 1996 BBC film Break­ing the Code (above, in its entire­ty), fea­tur­ing Derek Jaco­bi as Tur­ing and Nobel Prize-win­ning play­wright Harold Pin­ter as the mys­te­ri­ous “Man from the Min­istry.” Direct­ed by Her­bert Wise, the film is based on a 1986 play by Hugh White­more, which in turn was based on Andrew Hodge’s 1983 book Alan Tur­ing: The Enig­ma.

Break­ing the Code moves back and forth between two time frames and two very dif­fer­ent codes: one mil­i­tary, the oth­er social. The film runs 91 min­utes, and has been added to our col­lec­tion of Free Movies Online.

 

Three Passions of Bertrand Russell (and a Collection of Free Texts)

“Three pas­sions, sim­ple but over­whelm­ing­ly strong, have gov­erned my life,” wrote Bertrand Rus­sell in the pro­logue to his auto­bi­og­ra­phy: “the long­ing for love, the search for knowl­edge, and unbear­able pity for the suf­fer­ing of mankind.”

This five minute video, a pre­view of a three-part series pro­duced in 2005 for Ontario pub­lic tele­vi­sion called “The Three Pas­sions of Bertrand Rus­sell,” fea­tures a record­ing of Rus­sell read­ing pas­sages from the pro­logue, enti­tled “What I Have Lived For.” You can read the orig­i­nal text at the Bertrand Rus­sell Soci­ety, an excel­lent online resource, that also makes avail­able free books by Rus­sell, includ­ing:

You can also down­load the first edi­tion of Rus­sel­l’s land­mark 1910–13 col­lab­o­ra­tion with Alfred North White­head, Prin­cip­ia Math­e­mat­i­ca, as well as many of Rus­sel­l’s essays, includ­ing:

To explore the full list of avail­able resources, and to learn how you can sup­port the soci­ety’s activ­i­ties, vis­it the Bertrand Rus­sell Soci­ety web­site.

Also don’t miss some great Rus­sell mate­r­i­al in our own archives, includ­ing all six of his 1948 BBC Rei­th Lec­tures, a clip from a Cana­di­an tele­vi­sion inter­view fea­tur­ing his views on God, and his elo­quent 1959 mes­sage to the future.

Incredible Mental Math Gymnastics on “Countdown”

Count­down is a British TV game show revolv­ing around words and num­bers. In the num­bers round, con­tes­tants select six of twen­ty-four shuf­fled tiles with num­bers on them. Next, a com­put­er gen­er­ates a ran­dom three-dig­it tar­get num­ber and the con­tes­tants have thir­ty sec­onds to get as close to that num­ber as pos­si­ble by com­bin­ing the six num­bers through addi­tion, sub­trac­tion, mul­ti­pli­ca­tion and divi­sion. This mem­o­rable episode of Count­down aired in March 1997 and starred James Mar­tin and his rather unusu­al way of arriv­ing at the tar­get num­ber of 952.

One YouTube user sug­gest­ed a dif­fer­ent way: 6 x 75 = 450; 450 Ă· 50 = 9; 100 + 3 = 103; 9 x 103 = 927; 927 + 25 = 952

I found yet anoth­er way: 100 + 3 = 103; 103 x 6 = 618; 618 x 75 = 46,350; 46,350 Ă· 50 = 927; 927 + 25 = 952

What about you? Any more sug­ges­tions?

By pro­fes­sion, Matthias Rasch­er teach­es Eng­lish and His­to­ry at a High School in north­ern Bavaria, Ger­many. In his free time he scours the web for good links and posts the best finds on Twit­ter.

Relat­ed Con­tent:

Free Math Cours­es

Math­e­mat­ics in Movies: Har­vard Prof Curates 150+ Scenes

Mul­ti­pli­ca­tion: The Vedic Way

To Infinity and Beyond: A Mind-Bending Documentary from the BBC

Infin­i­ty. It’s a puz­zling con­cept. Is it real, or a math­e­mat­i­cal fic­tion?

Aris­to­tle believed infin­i­ty could only be poten­tial, nev­er actu­al. To speak of an actu­al infin­i­ty, he argued, is to fall into log­i­cal con­tra­dic­tion: “The infi­nite turns out to be the con­trary of what it is said to be,” Aris­to­tle wrote in the Physics. “It is not what has noth­ing out­side it that is infi­nite, but what always has some­thing out­side it.”

Aris­totle’s log­ic rest­ed on com­mon sense: the belief that the whole is always greater than the part. But in the late 19th Cen­tu­ry, Georg Can­tor and Richard Dedekind turned com­mon sense upside down by demon­strat­ing that the part can be equal to the whole. Can­tor went on to show that there are many orders of infinity–indeed, an infin­i­ty of infini­ties.

But what rela­tion does the Pla­ton­ic realm of pure math­e­mat­ics have to the phys­i­cal world? Physics is an empir­i­cal sci­ence, but that has­n’t stopped the­o­rists from imag­in­ing the mind-bog­gling con­se­quences of an infi­nite uni­verse. To Infin­i­ty and Beyond, a one-hour BBC Hori­zon spe­cial fea­tur­ing inter­views with lead­ing math­e­mati­cians and physi­cists, is an enter­tain­ing explo­ration of a sub­ject which, by def­i­n­i­tion, you won’t be able to wrap your mind around.

Relat­ed Con­tent:

Dan­ger­ous Knowl­edge: 4 Bril­liant Math­e­mati­cians & Their Drift to Insan­i­ty

Futur­ist Arthur C. Clarke on Mandelbrot’s Frac­tals

Math­e­mat­ics in Movies: Har­vard Prof Curates 150+ Scenes

Mathematics in Movies: Harvard Prof Curates 150+ Scenes

Oliv­er Knill teach­es cal­cu­lus, lin­ear alge­bra and dif­fer­en­tial equa­tions at Har­vard, and, sev­er­al years back, he pulled togeth­er a fair­ly nifty col­lec­tion of Math­e­mat­ics Scenes in Movies. Over 150 films are rep­re­sent­ed here, every­thing from Good Will Hunt­ing, A Beau­ti­ful Mind, Juras­sic Park (above) to Alice in Won­der­land (1951), The Mal­tese Fal­con and Apoc­a­lypse Now. You can watch each scene in flash for­mat on Knil­l’s site, or down­load them as a quick­time file. And, math buffs, don’t miss our col­lec­tion of Free Online Math Cours­es, a sub­set of our meta list, 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!

Relat­ed Con­tent:

Miss USA 2011: Should Schools Teach Evo­lu­tion? … or Math?

Mul­ti­pli­ca­tion: The Vedic Way

The Math Guy Radio Archive

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