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.

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.