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 Computer Science, Creativity, Math, MIT, Music, Neuroscience, Philosophy, Psychology | August 2nd, 2021

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.

Relat­ed Con­tent: 

How a Bach Canon Works. Bril­liant.

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

The Mir­ror­ing Mind: An Espres­so-Fueled Inter­pre­ta­tion of Dou­glas Hofstadter’s Ground­break­ing Ideas

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

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

in Animation, Math, TED Talks | July 13th, 2021

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.

Relat­ed Con­tent:

Beau­ti­ful Equa­tions: Doc­u­men­tary Explores the Beau­ty of Ein­stein & Newton’s Great Equa­tions

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

The Unex­pect­ed Math Behind Van Gogh’s Star­ry Night

Can You Solve These Ani­mat­ed Brain Teasers from TED-Ed?

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

Why the World’s Best Math­e­mati­cians Are Hoard­ing Japan­ese Chalk

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

in Math, Music | May 13th, 2021

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.

 

Relat­ed Con­tent:

David Gilmour, David Cros­by & Gra­ham Nash Per­form the Pink Floyd Clas­sic, “Shine on You Crazy Dia­mond” (2006)

Watch David Gilmour Play the Songs of Syd Bar­rett, with the Help of David Bowie & Richard Wright

Arthur C. Clarke Pre­dicts the Future in 1964 … And Kind of Nails It

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.

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

in Art, Math, Technology, TED Talks | March 23rd, 2021

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.

Relat­ed Con­tent:

An Origa­mi Samu­rai Made from a Sin­gle Sheet of Rice Paper, With­out Any Cut­ting

A Data­base of Paper Air­plane Designs: Hours of Fun for Kids & Adults Alike

MIT Cre­ates Amaz­ing Self-Fold­ing Origa­mi Robots & Leap­ing Chee­tah Robots

Design­er Cre­ates Origa­mi Card­board Tents to Shel­ter the Home­less from the Win­ter Cold

The Art of Let­ter­lock­ing: The Elab­o­rate Fold­ing Tech­niques That Ensured the Pri­va­cy of Hand­writ­ten Let­ters Cen­turies Ago

The Mak­ing of Japan­ese Hand­made Paper: A Short Film Doc­u­ments an 800-Year-Old Tra­di­tion

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

in History, Math | December 18th, 2020

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.

Relat­ed Con­tent:

Arti­fi­cial Intel­li­gence May Have Cracked the Code of the Voyn­ich Man­u­script: Has Mod­ern Tech­nol­o­gy Final­ly Solved a Medieval Mys­tery?

The Enig­ma Machine: How Alan Tur­ing Helped Break the Unbreak­able Nazi Code

How British Code­break­ers Built the First Elec­tron­ic Com­put­er

The Ser­i­al Killer Who Loved Jazz: The Infa­mous Sto­ry of the Axe­man of New Orleans (1919)

The Grue­some Doll­house Death Scenes That Rein­vent­ed Mur­der Inves­ti­ga­tions

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.

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

in Education, Math, MIT | May 27th, 2020

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.

Relat­ed Con­tent:

Free Online Math Cours­es

Albert Ein­stein’s Grades: A Fas­ci­nat­ing Look at His Report Cards

Teacher Calls Jacques Der­ri­da’s Col­lege Admis­sion Essay on Shake­speare “Quite Incom­pre­hen­si­ble” (1951)

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

in Math | February 27th, 2020

shortest math paper

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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Relat­ed Con­tent:

60 Free Online Math Cours­es

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Does Math Objec­tive­ly Exist, or Is It a Human Cre­ation? A New PBS Video Explores a Time­less Ques­tion

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

in Art, Math, Religion | May 20th, 2019

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.

Relat­ed Con­tent:

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

New Iran­ian Video Game, Engare, Explores the Ele­gant Geom­e­try of Islam­ic Art

The Com­plex Geom­e­try of Islam­ic Art & Design: A Short Intro­duc­tion

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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