The math and magic of origami | Robert Lang

Prevoditelj: Stjepan Mateljan Recezent: Ivan Stamenković

My talk is "Flapping Birds and Space Telescopes."

Moj je govor "Lepršave ptice i Svemirski teleskopi."

And you would think that should have nothing to do with one another,

Iako bi mislili da to ne nema nikakve veze jedno s drugim,

but I hope by the end of these 18 minutes,

ali nadam se da ćete do isteka ovih 18 minuta,

you'll see a little bit of a relation.

vidjeti ipak barem neku malenu poveznicu.

It ties to origami. So let me start.

Veže se na origami. Krenimo

What is origami?

Što je origami?

Most people think they know what origami is. It's this:

Većina ljudi misli da zna što je origami. To je ovo:

flapping birds, toys, cootie catchers, that sort of thing.

Lepršave ptice, igračke, gatalice, ta vrsta stvari.

And that is what origami used to be.

A to je ono što je origami bio.

But it's become something else.

Ali postao je nešto drugo.

It's become an art form, a form of sculpture.

Postao je oblik umjetnosti, oblik skulpture.

The common theme -- what makes it origami --

Uobičajena tema -- ono što ga čini origamijem --

is folding is how we create the form.

je savijanje kao način stvaranja oblika.

You know, it's very old. This is a plate from 1797.

Vrlo je star. Ovo je panel iz 1797.

It shows these women playing with these toys.

Pokazuje ove žene kako se igraju s igračkama.

If you look close, it's this shape, called a crane.

Ako pobliže osmotrite, to je ovaj oblik, zvan ždral.

Every Japanese kid

Svaki japanski klinac

learns how to fold that crane.

uči kako saviti ždrala.

So this art has been around for hundreds of years,

Ova je vještina bila prisutna stotine godina,

and you would think something

pa bi pomislili da nešto

that's been around that long -- so restrictive, folding only --

što je toliko dugo prisutno -- tako ograničavajuće, samo savijanje --

everything that could be done has been done a long time ago.

sve što je moglo biti učinjeno je učinjeno prije puno vremena.

And that might have been the case.

I to je možda i bio slučaj.

But in the twentieth century,

Ali u dvadesetom stoljeću,

a Japanese folder named Yoshizawa came along,

japanski savijač imenom Yoshizawa je našao

and he created tens of thousands of new designs.

i stvorio desetke tisuća novih dizajnova.

But even more importantly, he created a language,

Ali čak još i važnije, stvorio je jezik,

a way we could communicate,

način na koji možemo komunicirati,

a code of dots, dashes and arrows.

kod točkica, crtica i strelica.

Harkening back to Susan Blackmore's talk,

Oslanjajući se na govor Susan Blackmore,

we now have a means of transmitting information

sad imamo način odašiljanja informacije

with heredity and selection,

sa naslijeđivanjem i izabiranjem,

and we know where that leads.

a znamo gdje to vodi.

And where it has led in origami

A gdje je odvelo u origamiju

is to things like this.

su stvari poput ovih.

This is an origami figure --

Ovo je origami figura --

one sheet, no cuts, folding only, hundreds of folds.

jedna ploha, bez rezova, samo savijanje, stotine preklopa

This, too, is origami,

To je, isto, origami,

and this shows where we've gone in the modern world.

a ovo pokazuje gdje smo otišli u modernom svijetu.

Naturalism. Detail.

Naturalizam. Pojedinosti.

You can get horns, antlers --

Možete dobti rogove, roščiće --

even, if you look close, cloven hooves.

čak, ako gledate pomno, razdijeljena kopita

And it raises a question: what changed?

To potiče pitanje: što se promijenilo?

And what changed is something

A to što se promijenilo je nešto

you might not have expected in an art,

što možda ne bi očekivali u umjetosti,

which is math.

a to je matematika.

That is, people applied mathematical principles

To jest, ljudi su primijenili matematičke principe

to the art,

na umjetnost,

to discover the underlying laws.

da otkriju ishodišne zakone.

And that leads to a very powerful tool.

A to vodi do jako moćnog alata.

The secret to productivity in so many fields --

Tajna proizvodnosti u tako puno polja --

and in origami --

i u origamiju --

is letting dead people do your work for you.

je prepuštanje mrtvim ljudima da za vas odrade vaš posao.

(Laughter)

(Smijeh)

Because what you can do is

Jer ono što možete napraviti je

take your problem,

uzeti vlastiti problem,

and turn it into a problem that someone else has solved,

i pretvoriti ga u problem koji je netko drugi već riješio,

and use their solutions.

te iskoristiti njihova rješenje.

And I want to tell you how we did that in origami.

A ja vam želim reći kako smo to učinili u origamiju.

Origami revolves around crease patterns.

Origami se vrti oko uzoraka savijanja.

The crease pattern shown here is the underlying blueprint

Uzorak savijanja prikazan ovdje je ishodišni nacrt

for an origami figure.

za origami figuru.

And you can't just draw them arbitrarily.

I ne možete ih samo crtati proizvoljno.

They have to obey four simple laws.

Moraju ispunjavati četiri jednostavna zakona.

And they're very simple, easy to understand.

A ono su vrlo jednostavni, lagani za razumjeti.

The first law is two-colorability. You can color any crease pattern

Prvi je zakon dvobojnost. Možete pobojati bilo koji uzorak savijanja

with just two colors without ever having

sa samo dvije boje bez da se ikada

the same color meeting.

ista boja dodiruje.

The directions of the folds at any vertex --

Smjer preklopa na bilo kojem vrhu --

the number of mountain folds, the number of valley folds --

broj izbočenih nabora, broj udubljenih nabora --

always differs by two. Two more or two less.

uvijek se razlikuje za dva. Dva više ili dva manje.

Nothing else.

Ništa drugo.

If you look at the angles around the fold,

Ako gledate kuteve oko preklopa,

you find that if you number the angles in a circle,

nalazite da ako numerirate kuteve u krug,

all the even-numbered angles add up to a straight line,

svi parni kutevi zbrojeni daju ravnu liniju,

all the odd-numbered angles add up to a straight line.

svi neparni kutevi zbrojeni daju ravnu liniju.

And if you look at how the layers stack,

A ako gledate kako se slojevi slažu,

you'll find that no matter how you stack folds and sheets,

otkrit ćete da bez obzira kako slažete preklope i listove,

a sheet can never

list nikad ne može

penetrate a fold.

prodrijeti u preklop.

So that's four simple laws. That's all you need in origami.

To su četiri jednostavna zakona. To je sve što trebate u origamiju.

All of origami comes from that.

Sav origami dolazi iz toga.

And you'd think, "Can four simple laws

I pomislili bi: "Mogu li četiri jednostavna zakona

give rise to that kind of complexity?"

dopustiti nastanak ovolike složenosti?"

But indeed, the laws of quantum mechanics

Ali zbilja, zakoni kvantne mehanike

can be written down on a napkin,

mogu biti zapisani na ubrus,

and yet they govern all of chemistry,

pa ipak upravljaju svom kemijom,

all of life, all of history.

svim životom, svom poviješću.

If we obey these laws,

Ako se držimo tih zakona,

we can do amazing things.

Možemo činiti zapanjujuće stvari.

So in origami, to obey these laws,

U origamiju, da bismo se držali tih zakona,

we can take simple patterns --

možemo uzeti jednostavne uzorke --

like this repeating pattern of folds, called textures --

poput ovih ponavljajućih uzoraka savijanja, zvanih teksture --

and by itself it's nothing.

sam po sebi nije ništa.

But if we follow the laws of origami,

Ali ako slijedimo zakone origamija,

we can put these patterns into another fold

možemo staviti ove uzorke u druge

that itself might be something very, very simple,

koji po sebi mogu biti nešto jako jako jednostavno,

but when we put it together,

ali kad ih stavimo zajedno,

we get something a little different.

dobijemo nešto malo različito.

This fish, 400 scales --

ova riba, 400 krljušti --

again, it is one uncut square, only folding.

ponavljam, to je jedan nerazrezani kvadrat, samo savijanje.

And if you don't want to fold 400 scales,

A ako ne želite savijati 400 krljušti,

you can back off and just do a few things,

možete odstupiti i samo učiniti nekoliko stvari,

and add plates to the back of a turtle, or toes.

te dodati ploče na leđa kornjače, ili prste.

Or you can ramp up and go up to 50 stars

Ili možete pojačati i otići do 50 zvijezda

on a flag, with 13 stripes.

na zastavi, sa 13 pruga

And if you want to go really crazy,

A ako želite posve poludjeti,

1,000 scales on a rattlesnake.

1000 ljusaka na čegrtuši.

And this guy's on display downstairs,

A ovaj je momak izložen tu vani nešto niže,

so take a look if you get a chance.

pa bacite oko ako ulovite priliku.

The most powerful tools in origami

Najmoćniji alati u origamiju

have related to how we get parts of creatures.

su se bavili kako da dobijemo dijelove stvorenja.

And I can put it in this simple equation.

A mogu ih staviti u ovu jednostavnu jednadžbu.

We take an idea,

Uzmemo ideju,

combine it with a square, and you get an origami figure.

kobiniramo je sa kvadratom, i dobijete origami figuru.

(Laughter)

(Smijeh)

What matters is what we mean by those symbols.

Ono što je bitno je što mislimo sa tim simbolima.

And you might say, "Can you really be that specific?

I možete reći: "Može li se zbilja biti toliko određen?

I mean, a stag beetle -- it's got two points for jaws,

mislim, jelenak - ima dva roga kao vilice,

it's got antennae. Can you be that specific in the detail?"

ima antene. Može li se biti toliko određen u detaljima?"

And yeah, you really can.

I da, zbilja se može.

So how do we do that? Well, we break it down

Pa kako to činimo? Dakle, razbijamo izradu

into a few smaller steps.

u par manjih koraka.

So let me stretch out that equation.

Neka proširim malo tu jednadžbu.

I start with my idea. I abstract it.

Krećem sa idejom. Apstrahiram ju.

What's the most abstract form? It's a stick figure.

Koji je najapstraktniji oblik? Štapićasti lik.

And from that stick figure, I somehow have to get to a folded shape

A od tog štapićastog lika, nekako moram doći do savijenog oblika

that has a part for every bit of the subject,

koji ima dio za svaki komad teme,

a flap for every leg.

krilce za svaku nogu.

And then once I have that folded shape that we call the base,

a onda jednom kad imam savijen oblik koji zovemo baza,

you can make the legs narrower, you can bend them,

možete napraviti noge užima, možete ih savijati,

you can turn it into the finished shape.

možete ih pretvoriti u završni oblik.

Now the first step, pretty easy.

Sad prvi korak, jako je jednostavan.

Take an idea, draw a stick figure.

uzmi ideju, nacrtaj štapićast lik.

The last step is not so hard, but that middle step --

Zadnji korak nije tako težak, ali taj srednji korak --

going from the abstract description to the folded shape --

dolazak od apstraktnog opisa do savijenog oblika --

that's hard.

to jest teško.

But that's the place where the mathematical ideas

Ali to je mjesto gdje nas matematičke ideje

can get us over the hump.

mogu prevesti preko grbe.

And I'm going to show you all how to do that

A pokazat ću vam svima kako to napraviti

so you can go out of here and fold something.

tako da možete otići odavde i savinuti štogod.

But we're going to start small.

Ali počet ćemo sa malim.

This base has a lot of flaps in it.

Ova baza ima puno krilaca

We're going to learn how to make one flap.

Naučit ćemo kako napraviti jedno krilce.

How would you make a single flap?

Kako biste napravili jedno krilce?

Take a square. Fold it in half, fold it in half, fold it again,

Uzmite kvadrat. Presavinite ga na pola, ponovno, presavinite ga opet,

until it gets long and narrow,

sve dok ne postane dug i uzak,

and then we'll say at the end of that, that's a flap.

i onda ćemo na kraju toga reći: 'to je krilce'.

I could use that for a leg, an arm, anything like that.

Mogu to koristiti za nogu, ruku, bilo što nalik tome.

What paper went into that flap?

Koji je papir ušao u to krilce?

Well, if I unfold it and go back to the crease pattern,

Pa, ako ga odmotam i vratim se na uzorak savijanja,

you can see that the upper left corner of that shape

možete vidjeti da je gornji lijevi kut tog oblika

is the paper that went into the flap.

onaj papir koji je ušao u krilce.

So that's the flap, and all the rest of the paper's left over.

Dakle to je krilce, a sav je preostali papir višak.

I can use it for something else.

Mogu ga koristiti za nešto drugo.

Well, there are other ways of making a flap.

Ima i drugih načina da se napravi krilce.

There are other dimensions for flaps.

Ima drugih dimenzija krilca.

If I make the flaps skinnier, I can use a bit less paper.

ako učinim krilce tanjim, mogu koristiti nešto manje papira.

If I make the flap as skinny as possible,

Ako učinim krilce onoliko tankim koliko je uopće moguće,

I get to the limit of the minimum amount of paper needed.

dolazim do granice minimalne količine potrebnog papira.

And you can see there, it needs a quarter-circle of paper to make a flap.

a možete vidjeti ovjde, potrebno je četvrt kruga papira da se napravi krilce.

There's other ways of making flaps.

Postoje i drugi načini izrade krilaca.

If I put the flap on the edge, it uses a half circle of paper.

Ako stavim krilce na rub, koristiti će pola kruga papira.

And if I make the flap from the middle, it uses a full circle.

a ako ga napravim u sredini, koristiti će puni krug.

So, no matter how I make a flap,

Dakle, bez obzira kako napravim krilce,

it needs some part

traži neki dio

of a circular region of paper.

kružne regije papira.

So now we're ready to scale up.

dakle sad smo spremni na uvećanje.

What if I want to make something that has a lot of flaps?

Što ako želim nešto što ima puno krilaca?

What do I need? I need a lot of circles.

Što trebam? Trebam puno krugova.

And in the 1990s,

A u 90-ima,

origami artists discovered these principles

umjetnici origamija otkrili su te principe

and realized we could make arbitrarily complicated figures

i shvatili da možemo stvarati koliko god složene figure

just by packing circles.

samo slažući krugove.

And here's where the dead people start to help us out,

I to je gdje su nam mrtvi ljudi počeli pomagati,

because lots of people have studied

Jer puno je ljudi proučavalo

the problem of packing circles.

problem slaganja krugova.

I can rely on that vast history of mathematicians and artists

Mogu se osloniti na veliku povijest matematičara i umjetnika

looking at disc packings and arrangements.

koji su se bavili slaganjem diskova i razmještajem.

And I can use those patterns now to create origami shapes.

A ja sad mogu koristiti te oblike kako bih složio oblike u origamiju.

So we figured out these rules whereby you pack circles,

Tako smo otkrili ta pravila pomoću kojih slažete krugove,

you decorate the patterns of circles with lines

ukrašavate uzorke krugova pomoću linija

according to more rules. That gives you the folds.

u skladu sa još pravila. To vam daje presavijanja.

Those folds fold into a base. You shape the base.

Ta presavijanja se savijaju u baze. Vi oblikujete bazu.

You get a folded shape -- in this case, a cockroach.

Dobijate savijeni oblik -- u ovom slučaju, žohara.

And it's so simple.

I to je tako jednostavno.

(Laughter)

(Smijeh)

It's so simple that a computer could do it.

To je tako jednostavno da to može i računalo.

And you say, "Well, you know, how simple is that?"

Te kažete: "Pa, znate, koliko je to jednostavno?"

But computers -- you need to be able to describe things

Ali računala -- morate biti u stanju opisati stvari

in very basic terms, and with this, we could.

u posve jednostavnim pojmovima, a s tim, možemo.

So I wrote a computer program a bunch of years ago

Pa sam napisao računalni program prije niz godina

called TreeMaker, and you can download it from my website.

zvan TreeMaker, a koji možete skinuti s mog sajta.

It's free. It runs on all the major platforms -- even Windows.

Besplatan je. Vrti se na svim glavnim platformama -- čak i Windowsima.

(Laughter)

(Smijeh)

And you just draw a stick figure,

Vi samo nacrtate štapićasti lik,

and it calculates the crease pattern.

a on izračuna uzorak savijanja.

It does the circle packing, calculates the crease pattern,

On slaže krugove, računa uzorak savijanja,

and if you use that stick figure that I just showed --

a ako koristite onaj štapićasti lik koji sam vam upravo pokazao --

which you can kind of tell, it's a deer, it's got antlers --

za koji nekako možete reći da je jelen, jer ima rogove --

you'll get this crease pattern.

dobti ćete ovaj uzorak savijanja.

And if you take this crease pattern, you fold on the dotted lines,

A ako uzmete taj uzorak savijanja i presavijete po istočkanim linijama,

you'll get a base that you can then shape

dobit ćete bazu koju potom možete oblikovati

into a deer,

u jelena,

with exactly the crease pattern that you wanted.

sa točno onim uzorkom savijanja koji ste željeli.

And if you want a different deer,

A ako želite različitog jelena,

not a white-tailed deer, but you want a mule deer, or an elk,

ne bjelorepog jelena, nego želite drugu vrstu, ili losa,

you change the packing,

promijenite slaganje,

and you can do an elk.

i možete napraviti soba.

Or you could do a moose.

ili možete napravili losa.

Or, really, any other kind of deer.

Ili, zbilja, bilo koju drugu vrstu jelena.

These techniques revolutionized this art.

Ove su tehnike preokrenule ovu umjetnost.

We found we could do insects,

Otkrili smo da možemo raditi kukce,

spiders, which are close,

Pauke, koji su im bliski,

things with legs, things with legs and wings,

stvari s nogama, stvari s nogama i krilima,

things with legs and antennae.

stvari s nogama i antenama.

And if folding a single praying mantis from a single uncut square

A ako savijanje jedne bogomoljke iz jednog nerazrezanog kvadrata

wasn't interesting enough,

nije dovoljno zanimljivo,

then you could do two praying mantises

onda možete napraviti dvije bogomoljke

from a single uncut square.

iz jednog nerazrezanog kvadrata.

She's eating him.

Ona ga jede.

I call it "Snack Time."

Zovem ga "vrijeme za prezalogajiti".

And you can do more than just insects.

A možete činiti i više od samo kukaca.

This -- you can put details,

Ovo -- možete stavljati detalje,

toes and claws. A grizzly bear has claws.

prste i kandže. Grizli ima kandže.

This tree frog has toes.

Ova gatalinka ima prste.

Actually, lots of people in origami now put toes into their models.

Zapravo, puno ljudi u origamiju sad stavlja prste na svoje modele.

Toes have become an origami meme,

Prsti su postali meme u origamiju,

because everyone's doing it.

jer svi ih rade.

You can make multiple subjects.

Možete uzeti višestruke teme.

So these are a couple of instrumentalists.

Tako je ovo par instrumentalaca.

The guitar player from a single square,

Svirač gitare iz jednog kvadrata.

the bass player from a single square.

Svirač basa iz jednog kvadrata.

And if you say, "Well, but the guitar, bass --

A ako kažete: "Dobro, ali gitara, bas --

that's not so hot.

to nije takva špica.

Do a little more complicated instrument."

Napravite malo složeniji instrument."

Well, then you could do an organ.

U redu, onda možete složiti orgulje.

(Laughter)

(Smijeh)

And what this has allowed is the creation

A to je omogućilo stvaranje

of origami-on-demand.

origamija po narudžbi.

So now people can say, "I want exactly this and this and this,"

Tako da sad ljudi mogu reći: "Želim točno to i to i to,"

and you can go out and fold it.

a vi možete otići i saviti im što žele.

And sometimes you create high art,

Ponekad stvarate visoku umjetnost,

and sometimes you pay the bills by doing some commercial work.

a ponekad plačate račune odrađujući nešto komercijalno.

But I want to show you some examples.

Ali želim vam pokazati neke primjere.

Everything you'll see here,

Sve što ćete vidjeti ovdje,

except the car, is origami.

osim auta, je origami.

(Video)

(Video)

(Applause)

(Pljesak)

Just to show you, this really was folded paper.

Samo da vam pokažem, ovo je zbijla bio savijeni papir.

Computers made things move,

Računala su napravila da se stvari kreću,

but these were all real, folded objects that we made.

Ali to su sve bili pravi, savijeni predmeti koje smo napravili.

And we can use this not just for visuals,

I ne moramo ih koristiti samo za vizualne efekte,

but it turns out to be useful even in the real world.

nego ispada da su korisni i u stvarnom svijetu.

Surprisingly, origami

Iznenađujuće, origami

and the structures that we've developed in origami

i strukture koje smo razvili u origamiju

turn out to have applications in medicine, in science,

imaju primjene u medicini, znanosti,

in space, in the body, consumer electronics and more.

u svemiru, tijelu, potrošačkoj elektronici i drugdje.

And I want to show you some of these examples.

I želim vam pokazati neke od tih primjera.

One of the earliest was this pattern,

Jedan od najranijih je bio ovaj uzorak,

this folded pattern,

ovaj savijeni uzorak,

studied by Koryo Miura, a Japanese engineer.

koji je proučavao Koryo Miura, japanski inženjer.

He studied a folding pattern, and realized

Proučavao je ovaj uzorak presavijanja, i shvatio

this could fold down into an extremely compact package

da bi se dalo ispresavijati u krajnje kompaktno pakiranje

that had a very simple opening and closing structure.

koje bi imalo vrlo jednostavnu strukturu otvarajna i zatvaranja.

And he used it to design this solar array.

I iskoristio ju je da osmisli ovu solarnu ploču.

It's an artist's rendition, but it flew in a Japanese telescope

Ovo je umjetnički prikaz, ali poletjela je na japanskom teleskopu

in 1995.

1995.

Now, there is actually a little origami

Sad, ima zapravo nešto malo origamija

in the James Webb Space Telescope, but it's very simple.

i u svemirskom teleskopu James Webb, ali je vrlo jednostavan.

The telescope, going up in space,

Teleskop, odlazeći u svemir,

it unfolds in two places.

odmotava se na dva mjesta,

It folds in thirds. It's a very simple pattern --

Zamata se u trećinama. To je vrlo jednostavan uzorak --

you wouldn't even call that origami.

ne biste ga čak niti nazvali origami.

They certainly didn't need to talk to origami artists.

Sigurno da nisu trebali pričati sa umjetnicima origamija.

But if you want to go higher and go larger than this,

Ali ako želite ići više i graditi veće od tog,

then you might need some origami.

onda bi vam moglo zatrebati nešto origamija.

Engineers at Lawrence Livermore National Lab

Inženjeri u nacionalnom laboratoriju Lawrence Livermore

had an idea for a telescope much larger.

su imali ideju puno većem teleskopu.

They called it the Eyeglass.

Zvali su ga Okular.

The design called for geosynchronous orbit

Dizajn je bio predviđen za geosinkronu orbitu

25,000 miles up,

na 25.000 milja,

100-meter diameter lens.

s lećom promjera 100 metara.

So, imagine a lens the size of a football field.

Dakle zamislite leću veličine nogometnog igrališta.

There were two groups of people who were interested in this:

Bile su dvije grupe ljudi koje su bile zainteresirane za ovo:

planetary scientists, who want to look up,

planetarni znanstvenici, koji su htjeli gledati gore,

and then other people, who wanted to look down.

i onda neki drugi ljudi, koji su htjeli gledati dolje.

Whether you look up or look down,

Gledali vi gore ili dolje,

how do you get it up in space? You've got to get it up there in a rocket.

kako ćete to podići u svemir? Morate doći tamo sa raketom.

And rockets are small. So you have to make it smaller.

A rakete su malene. Tako da to trebate učiniti još manjim.

How do you make a large sheet of glass smaller?

Kako učiniti veliku plohu stakla manjom?

Well, about the only way is to fold it up somehow.

Pa, jedini je način da se nekako smota.

So you have to do something like this.

Tako da morate napraviti nešto ovakvo.

This was a small model.

Ovo je bio maleni model.

Folded lens, you divide up the panels, you add flexures.

Savijena leća, podijelite ploče, dodate krivine

But this pattern's not going to work

Ali ovaj uzorak neće dostajati

to get something 100 meters down to a few meters.

da smanjite nešto od 100 metara na samo nekoliko metara.

So the Livermore engineers,

Pa su inženjeri iz Livermora,

wanting to make use of the work of dead people,

želeći iskoristiti rad mrtvih ljudi,

or perhaps live origamists, said,

ili možda živih origamista, rekli:

"Let's see if someone else is doing this sort of thing."

"Ajde da vidimo radi li netko drugi ovu vrstu stvari."

So they looked into the origami community,

Pa su pogledali u origami zajednicu,

we got in touch with them, and I started working with them.

stupili smo u kontakt, te sam počeo raditi s njima.

And we developed a pattern together

Pa smo zajedno razvili uzorak

that scales to arbitrarily large size,

koji se smanjuje na proizvoljnu veličinu,

but that allows any flat ring or disc

ali koji dopušta bilo koji plosnati prsten ili disk

to fold down into a very neat, compact cylinder.

da se savine u vrlo uredan, kompaktni cilindar.

And they adopted that for their first generation,

Usvojili su to za svoju prvu generaciju,

which was not 100 meters -- it was a five-meter.

koja nije bila 100 metara -- bila je 5 metara.

But this is a five-meter telescope --

ali to je 5-metarski teleskop --

has about a quarter-mile focal length.

s fokalnom duljinom od oko četvrt milje.

And it works perfectly on its test range,

I radi savršeno na svojem probnom poligonu,

and it indeed folds up into a neat little bundle.

i zaista se sklapa u zgodan mali paket.

Now, there is other origami in space.

Sad, postoji i drugi origami u svemiru.

Japan Aerospace [Exploration] Agency flew a solar sail,

Japanska aeronautička i svemirska istraživačka agencija je podigla solarno jedro,

and you can see here that the sail expands out,

i ovdje možete vidjeti kako se jedro širi,

and you can still see the fold lines.

i ovdje još vidite linije preklopa.

The problem that's being solved here is

Problem koji je riješen ovdje je

something that needs to be big and sheet-like at its destination,

nešto što treba biti veliko i nalik plohi na odredištu,

but needs to be small for the journey.

ali mora biti maleno tijekom puta.

And that works whether you're going into space,

a to radi išli vi u svemir,

or whether you're just going into a body.

ili u tijelo.

And this example is the latter.

A ovo je primjer potonjeg.

This is a heart stent developed by Zhong You

Ovo je srčani stent koji je razvio Zhong You

at Oxford University.

sa sveučilišta u Oxfordu.

It holds open a blocked artery when it gets to its destination,

On drži otvorenom blokiranu arteriju jednom kad dođe na svoje odredište,

but it needs to be much smaller for the trip there,

ali mora biti puno manji za put do tamo.

through your blood vessels.

kroz vaše krvne žile.

And this stent folds down using an origami pattern,

A ovaj se stent previja koristeći uzorak iz origamija,

based on a model called the water bomb base.

temeljen na modelu koj se zove baza vodene bombe.

Airbag designers also have the problem

Dizajneri zračnih jastuka također imaju problem

of getting flat sheets

stiskanja ravnih ploha

into a small space.

u mali prostor.

And they want to do their design by simulation.

Oni žele napraviti svoj dizajn simulacijom.

So they need to figure out how, in a computer,

Pa moraju shvatiti kako, na računalu,

to flatten an airbag.

spljoštiti zračni jastuk.

And the algorithms that we developed

A algoritmi koje smo razvili

to do insects

da radimo kukce

turned out to be the solution for airbags

je ispao riješenje za zračne jastuke

to do their simulation.

da izvrše njihovu simulaciju.

And so they can do a simulation like this.

Tako da mogu raditi simulacije poput ove.

Those are the origami creases forming,

Ovo se oblikuju origami nabori,

and now you can see the airbag inflate

i sad možete vidjeti kako se zračni jastuk napuhuje

and find out, does it work?

i promisliti, radi li?

And that leads

A to vodi

to a really interesting idea.

do zbilja zanimljive ideje.

You know, where did these things come from?

Znate, otkud su ove stvari došle?

Well, the heart stent

Pa, srčani stent

came from that little blow-up box

dolazi iz one male kutije na napuhavanje

that you might have learned in elementary school.

koju ste mogli naučiti u osnovnoj školi.

It's the same pattern, called the water bomb base.

To je isti uzorak, zvan baza vodene bombe.

The airbag-flattening algorithm

Algoritam za spljoštavanje zračnog jastuka

came from all the developments

dolazi iz svog razvoja

of circle packing and the mathematical theory

slaganja krugova i matematičke teorije

that was really developed

koja je zbilja bila razvijena

just to create insects -- things with legs.

smao da stvori kukce -- stvari s nogama.

The thing is, that this often happens

Stvar je, da se ovo često dešava

in math and science.

u matematici i znanosti.

When you get math involved, problems that you solve

Kada uključite matematiku, problem koji ste riješili

for aesthetic value only,

samo radi estetske vrijednosti,

or to create something beautiful,

ili da stvorite nešto prekrasno,

turn around and turn out

se vrti uokolo i ispada

to have an application in the real world.

da ima primjenu u stvarnom svijetu.

And as weird and surprising as it may sound,

I koliko god čudno i iznenađujuće može zvučati,

origami may someday even save a life.

origami može jednog dana čak i spasiti život.

Thanks.

Hvala.

(Applause)

(Pljesak)