The math and magic of origami | Robert Lang

Translator: Tadej Pesjak Reviewer: Tilen Pigac - EFZG

My talk is "Flapping Birds and Space Telescopes."

Moj govor se imenuje "Prhutajoče ptice in vesoljski teleskopi".

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

Mislili bi, da to dvoje nima nič skupnega,

but I hope by the end of these 18 minutes,

a upam, da boste po teh 18 minutah

you'll see a little bit of a relation.

opazili vsaj majhno povezavo.

It ties to origami. So let me start.

Tiče se origamija.

What is origami?

Naj začnem. Kaj je origami?

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

Večina misli, da to ve. Namreč to:

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

prhutajoče ptice, razne igrače, itd.

And that is what origami used to be.

To je origami bil včasih.

But it's become something else.

A zdaj je postal nekaj drugega.

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

Postal je umetnost, vrsta kiparjenja.

The common theme -- what makes it origami --

Rdeča nit origamija

is folding is how we create the form.

je zgibanje in način ustvarjanja oblike.

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

Zelo star je. To je plošča iz l. 1797.

It shows these women playing with these toys.

Prikazuje ženske z igračami.

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

Opazite lahko, da imajo obliko žerjava.

Every Japanese kid

Vsak japonski otrok

learns how to fold that crane.

se nauči zgibanja tega žerjava.

So this art has been around for hundreds of years,

Ta umetnost obstaja že več sto let

and you would think something

in mislili bi, da je pri nečem,

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

kar obstaja tako dolgo in je omejeno le na zlaganje,

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

bilo ustvarjeno že vse, kar je možno.

And that might have been the case.

Morda bi res bilo tako.

But in the twentieth century,

A v 20. stoletju

a Japanese folder named Yoshizawa came along,

je japonski zgibalec Yoshizawa

and he created tens of thousands of new designs.

ustvaril več deset tisoč novih oblik.

But even more importantly, he created a language,

A še bolj pomembno, ustvaril je jezik --

a way we could communicate,

način sporazumevanja,

a code of dots, dashes and arrows.

šifro iz pik, črtic in puščic.

Harkening back to Susan Blackmore's talk,

Če se spomnimo govora Susan Blackmore,

we now have a means of transmitting information

smo sedaj zmožni

with heredity and selection,

prenašati podatke preko dednosti in selekcije

and we know where that leads.

in vemo, kam to vodi.

And where it has led in origami

Pri origamiju pa je vodilo

is to things like this.

v takšne stvari.

This is an origami figure --

To je origami figura:

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

en list, brez rezanja, na stotine zgibov.

This, too, is origami,

Tudi to je origami in kaže,

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

kam smo se usmerili v sodobnem svetu.

Naturalism. Detail.

Naturalizem. Podrobnosti.

You can get horns, antlers --

Ustvarimo lahko rogovje

even, if you look close, cloven hooves.

in celo razklana kopita.

And it raises a question: what changed?

Pojavi se vprašanje: kaj se je spremenilo?

And what changed is something

Spremenilo se je nekaj,

you might not have expected in an art,

česar pri umetnosti ne bi pričakovali,

which is math.

namreč matematika.

That is, people applied mathematical principles

Ljudje so pri tej umetnosti

to the art,

uporabili matematične principe,

to discover the underlying laws.

da bi odkrili njene zakonitosti.

And that leads to a very powerful tool.

Nastalo je zelo zmogljivo orodje.

The secret to productivity in so many fields --

Skrivnost učinkovitosti na tolikih področjih,

and in origami --

tudi pri origamiju, je,

is letting dead people do your work for you.

da tvoje delo opravijo mrtveci.

(Laughter)

(Smeh)

Because what you can do is

Svoj problem

take your problem,

lahko spremeniš

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

v problem, ki ga je nekdo že rešil

and use their solutions.

in uporabiš njihove rešitve.

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

Pri origamiju smo to storili tako:

Origami revolves around crease patterns.

Pomembni so vzorci zgibanja.

The crease pattern shown here is the underlying blueprint

Tole tukaj je načrt

for an origami figure.

za origami figuro.

And you can't just draw them arbitrarily.

Ne moremo jih risati po svoje.

They have to obey four simple laws.

Upoštevati morajo štiri zakonitosti.

And they're very simple, easy to understand.

So enostavne in razumljive.

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

Prva je dvobarvnost. Vsak vzorec zgibanja

with just two colors without ever having

lahko pobarvamo z dvema barvama,

the same color meeting.

brez da bi se ista barva stikala.

The directions of the folds at any vertex --

Smeri zgibov pri vsakem vozlišču --

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

število vbočenih in izbočenih zgibov --

always differs by two. Two more or two less.

se vedno razlikuje za dva.

Nothing else.

Dva več ali manj.

If you look at the angles around the fold,

Če kote v zgibu

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

oštevilčimo v krogu, opazimo,

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

da koti z sodimi števili tvorijo ravno črto

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

in isto je s koti z lihimi števili.

And if you look at how the layers stack,

Če pogledamo zlaganje plasti,

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

opazimo, da pri zlaganju zgibov in ploskev,

a sheet can never

ploskev nikoli

penetrate a fold.

ne prebode zgiba.

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

Pri origamiju rabite le te štiri zakonitosti.

All of origami comes from that.

Vsak origami nastane iz njih.

And you'd think, "Can four simple laws

Vprašate se, ali res lahko le z njimi

give rise to that kind of complexity?"

ustvarimo takšno kompleksnost?

But indeed, the laws of quantum mechanics

A tudi zakonitosti kvantne mehanike

can be written down on a napkin,

lahko napišemo na prtiček,

and yet they govern all of chemistry,

pa vseeno vladajo vsej kemiji,

all of life, all of history.

življenju, zgodovini.

If we obey these laws,

Če upoštevamo te zakonitosti,

we can do amazing things.

lahko ustvarimo neverjetne stvari.

So in origami, to obey these laws,

Pri origamiju lahko tako

we can take simple patterns --

uporabimo preproste vzorce --

like this repeating pattern of folds, called textures --

kot ta ponavljajoči vzorec oz. tekstura --

and by itself it's nothing.

ki so sami po sebi nepomembni.

But if we follow the laws of origami,

Če pa sledimo zakonitostim,

we can put these patterns into another fold

jih lahko vstavimo v nek drug zgib,

that itself might be something very, very simple,

ki je sam po sebi zelo preprost,

but when we put it together,

vendar skupaj

we get something a little different.

postaneta nekaj malce drugačnega.

This fish, 400 scales --

Ta riba ima 400 lusk --

again, it is one uncut square, only folding.

ponavljam, en list, samo zgibanje.

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

Če nočete zgibati 400 lusk,

you can back off and just do a few things,

lahko naredite le par stvari:

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

plošče na želvjem oklepu ali prste.

Or you can ramp up and go up to 50 stars

Lahko pa se potrudite in naredite

on a flag, with 13 stripes.

50 zvezd in 13 črt na zastavi.

And if you want to go really crazy,

Če pa si hočete dati duška,

1,000 scales on a rattlesnake.

tisoč lusk na klopotači.

And this guy's on display downstairs,

Razstavljena je spodaj,

so take a look if you get a chance.

oglejte si jo, če utegnete.

The most powerful tools in origami

Najmočnejša orodja pri origamiju

have related to how we get parts of creatures.

nam pomagajo ustvariti dele bitij.

And I can put it in this simple equation.

To lahko opišem s preprosto enačbo.

We take an idea,

Vzamemo zamisel,

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

jo povežemo s kvadratom in dobimo origami figuro.

(Laughter)

(Smeh)

What matters is what we mean by those symbols.

Pomembno je, kaj simboli predstavljajo.

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

Smo res lahko tako natančni?

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

Rogač ima čeljusti, tipalke.

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

Imamo lahko toliko podrobnosti?

And yeah, you really can.

Ja.

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

Kako nam to uspe? Postopek razdelimo

into a few smaller steps.

na več manjših korakov.

So let me stretch out that equation.

Naj razširim to enačbo.

I start with my idea. I abstract it.

Začnem z zamislijo. Jo naredim abstraktno.

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

Najbolj abstraktna je figura iz črtic.

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

Iz nje moram priti do zgibane oblike,

that has a part for every bit of the subject,

ki ima del za vsak del predmeta.

a flap for every leg.

Zavihek za vsako nogo.

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

Ko dobim to obliko, ti. osnovo,

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

lahko noge zožimo, upogibamo,

you can turn it into the finished shape.

spremenimo v končno obliko.

Now the first step, pretty easy.

Prvi korak je enostaven.

Take an idea, draw a stick figure.

Zamisel in figura iz črtic.

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

Tudi zadnji ni težak, ampak srednji --

going from the abstract description to the folded shape --

prehod iz abstraktne v zgibano obliko --

that's hard.

ta je težak.

But that's the place where the mathematical ideas

A tukaj nam lahko matematične zamisli

can get us over the hump.

pomagajo premagati oviro.

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

Pokazal vam bo, kako to deluje,

so you can go out of here and fold something.

da boste lahko šli nekaj zgibat.

But we're going to start small.

Začeli bomo počasi.

This base has a lot of flaps in it.

Ta osnova ima veliko zavihov.

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

Naučili se bomo narediti enega.

How would you make a single flap?

Kako bi to storili?

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

Vzameš kvadrat, ga trikrat prepogneš,

until it gets long and narrow,

da postane dolg in ozek

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

in tisto na koncu je zavih.

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

Postane lahko noga, roka itd.

What paper went into that flap?

Koliko papirja smo porabili za zavih?

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

Če papir razgrnemo,

you can see that the upper left corner of that shape

vidimo, da je zgornji levi kot

is the paper that went into the flap.

uporabljen za zavih.

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

Preostali papir lahko uporabimo

I can use it for something else.

za kaj drugega.

Well, there are other ways of making a flap.

Zavih lahko naredimo še drugače.

There are other dimensions for flaps.

Lahko je drugačnih dimenzij.

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

Če naredim ožjega, porabim manj papirja.

If I make the flap as skinny as possible,

Če ga naredim najbolj ozkega,

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

pridem do najmanjše porabe papirja

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

in sicer četrtina kroga.

There's other ways of making flaps.

Lahko jih naredim še drugače.

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

Če dam zavih na rob, porabim polkrog.

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

Če ga dam na sredino, pa cel krog.

So, no matter how I make a flap,

Ni važno, kako zavih naredim,

it needs some part

vedno se porabi

of a circular region of paper.

neki del kroga na papirju.

So now we're ready to scale up.

Čas je za nadgradnjo.

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

Ustvariti hočem nekaj z več zavihi.

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

Kaj rabim? Veliko krogov.

And in the 1990s,

V 90-ih letih

origami artists discovered these principles

so odkrili te principe

and realized we could make arbitrarily complicated figures

in spoznali, da lahko dobimo poljubno

just by packing circles.

zapletene oblike z zlaganjem krogov.

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

Tu nam mrtveci začnejo pomagati.

because lots of people have studied

Veliko ljudi je raziskovalo

the problem of packing circles.

probleme z zlaganjem krogov.

I can rely on that vast history of mathematicians and artists

Oprem se lahko na matematike in umetnike,

looking at disc packings and arrangements.

ki so se ukvarjali s tem.

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

Njihovi vzorci mi pomagajo pri origamiju.

So we figured out these rules whereby you pack circles,

Po teh pravilih zlagaš kroge,

you decorate the patterns of circles with lines

jih opremiš s črtami.

according to more rules. That gives you the folds.

Dobiš zgibe,

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

iz katerih ustvariš osnovo. Jo oblikuješ.

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

Dobiš zgibano obliko -- tukaj ščurka.

And it's so simple.

Tako preprosto je.

(Laughter)

(Smeh)

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

Tako preprosto, da lahko to stori računalnik.

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

Vprašate: "Je to res preprosto?"

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

Pri računalnikih je treba stvari opisati

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

preprosto in tukaj smo jih lahko.

So I wrote a computer program a bunch of years ago

Napisal sem program, "TreeMaker",

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

dobite ga na moji spletni strani,

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

brezplačno. Dela celo na Windowsih.

(Laughter)

(Smeh)

And you just draw a stick figure,

Narišete figuro iz črtic

and it calculates the crease pattern.

in dobite shemo zgibov.

It does the circle packing, calculates the crease pattern,

Zloži kroge in ustvari shemo zgibov

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

in če uporabite to obliko,

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

lahko vidite, da je jelen,

you'll get this crease pattern.

dobite tole shemo zgibov.

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

Če zgibate po črticah,

you'll get a base that you can then shape

dobite osnovo, iz katere oblikujete

into a deer,

jelena

with exactly the crease pattern that you wanted.

s hoteno shemo zgibov.

And if you want a different deer,

Če hočete drugačnega,

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

ne belorepega jelena,

you change the packing,

spremenite zlaganje

and you can do an elk.

in lahko ustvarite vapitija.

Or you could do a moose.

Ali losa.

Or, really, any other kind of deer.

Ali kateregakoli jelena.

These techniques revolutionized this art.

Te tehnike so zrevolucionirale to umetnost.

We found we could do insects,

Lahko smo ustvarili žuželke,

spiders, which are close,

pajke,

things with legs, things with legs and wings,

bitja z nogami, z nogami in krili,

things with legs and antennae.

z nogami in tipalkami.

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

Če ena bogomolka iz enega kvadrata

wasn't interesting enough,

ni dovolj zanimiva,

then you could do two praying mantises

lahko iz enega kvadrata

from a single uncut square.

ustvarimo dve.

She's eating him.

Ona jé njega.

I call it "Snack Time."

Imenujem ga "Čas za malico".

And you can do more than just insects.

Ustvarimo lahko ne le žuželke.

This -- you can put details,

Dodamo lahko podrobnosti:

toes and claws. A grizzly bear has claws.

prste in kremplje. Grizli ima kremplje.

This tree frog has toes.

Ta žaba ima prste.

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

Veliko ljudi dodaja prste.

Toes have become an origami meme,

Postali so modna muha,

because everyone's doing it.

ki jo vsi posnemajo.

You can make multiple subjects.

Ustvarimo lahko več predmetov.

So these are a couple of instrumentalists.

Tukaj sta dva glasbenika.

The guitar player from a single square,

Oba, kitarist in basist,

the bass player from a single square.

sta ustvarjena iz enega kvadrata.

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

Če rečete: Kitara in bas

that's not so hot.

nista zanimiva,

Do a little more complicated instrument."

naredite bolj zapleteno glasbilo.

Well, then you could do an organ.

Ustvarimo lahko orgle.

(Laughter)

(Smeh)

And what this has allowed is the creation

S tem lahko ustvarimo

of origami-on-demand.

origami na zahtevo.

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

Kar ljudje zahtevajo,

and you can go out and fold it.

to lahko ustvarimo.

And sometimes you create high art,

Včasih ustvarimo umetnost,

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

včasih pa samo komercialen izdelek.

But I want to show you some examples.

Naj vam pokažem primer.

Everything you'll see here,

Vse, kar boste videli,

except the car, is origami.

razen avta, je origami.

(Video)

(Video)

(Applause)

(Aplavz)

Just to show you, this really was folded paper.

To je res bil le zgiban papir.

Computers made things move,

Premikal jih je računalnik,

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

a vsi so bili zgibani predmeti.

And we can use this not just for visuals,

Niso le vizualni pripomočki,

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

ampak so tudi uporabni v resničnem svetu.

Surprisingly, origami

Presenetljivo so origami

and the structures that we've developed in origami

in strukture, ustvarjene z njim,

turn out to have applications in medicine, in science,

uporabni v medicini, znanosti,

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

vesolju, telesu, elektroniki itd.

And I want to show you some of these examples.

Naj vam pokažem par primerov.

One of the earliest was this pattern,

Eden zgodnejših

this folded pattern,

je ta zgiban vzorec,

studied by Koryo Miura, a Japanese engineer.

ki ga je preučeval inženir, Koryo Miura.

He studied a folding pattern, and realized

Ugotovil je,

this could fold down into an extremely compact package

da se lahko zloži v zelo kompakten paket,

that had a very simple opening and closing structure.

ki se enostavno odpira in zapira.

And he used it to design this solar array.

Z njim je ustvaril ta solarni zbiralec.

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

1995 je poletel

in 1995.

z japonskim teleskopom.

Now, there is actually a little origami

Preprost origami je tudi

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

v teleskopu Jamesa Webba.

The telescope, going up in space,

Ko gre teleskop v vesolje,

it unfolds in two places.

se na dveh delih razgrne.

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

Vzorec je preprost,

you wouldn't even call that origami.

težko bi mu rekli origami.

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

Niso rabili pomoči umetnikov origamija.

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

Če pa želimo nekaj večjega,

then you might need some origami.

bo pomoč origamija potrebna.

Engineers at Lawrence Livermore National Lab

V laboratoriju Lawrenca Livermora

had an idea for a telescope much larger.

so si zamislili mnogo večji teleskop,

They called it the Eyeglass.

imenovan "Okular".

The design called for geosynchronous orbit

Potuje po geosinhroni orbiti,

25,000 miles up,

42.000 km visoko,

100-meter diameter lens.

z lečo premera 100 m,

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

torej velikosti nogometnega igrišča.

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

Za to so se zanimali dve skupini:

planetary scientists, who want to look up,

planetarni znanstveniki, ki bi gledali gor

and then other people, who wanted to look down.

in drugi ljudje, ki bi gledali dol.

Whether you look up or look down,

Ne glede, kam gledamo,

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

kako ga spravimo v vesolje? Z raketo.

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

A te so majhne, zato ga moramo pomanjšati.

How do you make a large sheet of glass smaller?

Kako pomanjšamo veliko ploščo stekla?

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

Nekako jo je treba zložiti.

So you have to do something like this.

Potrebno je nekaj takega --

This was a small model.

to je pomanjšan model.

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

Lečo razdelimo, dodamo pregibe.

But this pattern's not going to work

A to ne bo delovalo, če hočemo

to get something 100 meters down to a few meters.

100-metrsko lečo zložiti na par metrov.

So the Livermore engineers,

Inženirji iz Livermora

wanting to make use of the work of dead people,

so hoteli izkoristili delo mrtvecev,

or perhaps live origamists, said,

oz. živih origamistov in rekli:

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

"Poglejmo, če že kdo to počne".

So they looked into the origami community,

Stopili so v stik z origamisti

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

in začel sem delati z njimi.

And we developed a pattern together

Skupaj smo razvili vzorec,

that scales to arbitrarily large size,

primeren za poljubne velikosti,

but that allows any flat ring or disc

s katerim lahko katerikoli ploski disk

to fold down into a very neat, compact cylinder.

zložimo v kompakten cilinder.

And they adopted that for their first generation,

To so storili pri prvi generaciji

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

velikosti 5 m.

But this is a five-meter telescope --

A ta 5-metrski teleskop

has about a quarter-mile focal length.

ima goriščno razdaljo 400 m.

And it works perfectly on its test range,

Pri testiranju se je odlično odrezal

and it indeed folds up into a neat little bundle.

in se res zložil v majhen sveženj.

Now, there is other origami in space.

V vesolju so še drugi origamiji.

Japan Aerospace [Exploration] Agency flew a solar sail,

JAXA so razprli solarno jadro

and you can see here that the sail expands out,

in tukaj lahko vidite,

and you can still see the fold lines.

da so črte zgibanja še opazne.

The problem that's being solved here is

Tako so rešili težavo,

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

kjer mora velika ploskev priti do cilja,

but needs to be small for the journey.

med potovanjem pa biti majhna.

And that works whether you're going into space,

To deluje pri odhodu v vesolje

or whether you're just going into a body.

ali vstopu v telo.

And this example is the latter.

To je primer drugega.

This is a heart stent developed by Zhong You

To žilno opornico je razvil Zhong You

at Oxford University.

iz Oxfordske univerze.

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

Odpira zadelane arterije,

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

a mora biti mnogo manjša

through your blood vessels.

za pot skozi krvne žile.

And this stent folds down using an origami pattern,

Zloži se po vzorcu, ki uporablja

based on a model called the water bomb base.

ti. osnovo za vodno bombo.

Airbag designers also have the problem

Tudi oblikovalci zračnih blazin

of getting flat sheets

se soočajo s težavo,

into a small space.

kako ravno ploskev spraviti v majhen prostor.

And they want to do their design by simulation.

Pomagajo si s simulacijo.

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

Na računalniku morajo ugotoviti,

to flatten an airbag.

kako sploščiti zračno blazino.

And the algorithms that we developed

Algoritmi, s katerimi

to do insects

izdelujemo žuželke,

turned out to be the solution for airbags

so se izkazali za rešitev

to do their simulation.

pri njihovih simulacijah.

And so they can do a simulation like this.

Simulacija je taka.

Those are the origami creases forming,

To so pregibi origamija.

and now you can see the airbag inflate

Zdaj vidimo napihovanje blazine

and find out, does it work?

in ugotovimo, če deluje.

And that leads

To vodi

to a really interesting idea.

k zelo zanimivi zamisli.

You know, where did these things come from?

Od kod so prišle te stvari?

Well, the heart stent

Žilna opornica

came from that little blow-up box

od napihljive škatle,

that you might have learned in elementary school.

ki ste se je morda učili v šoli.

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

Osnova je ista, ti. vodna bomba.

The airbag-flattening algorithm

Algoritem pri zračnih blazinah

came from all the developments

je prišel iz razvoja

of circle packing and the mathematical theory

zlaganja krogov in matematične teorije,

that was really developed

uporabljene pri izdelovanju

just to create insects -- things with legs.

žuželk -- bitij z nogami.

The thing is, that this often happens

To se pogosto zgodi

in math and science.

v matematiki in znanosti.

When you get math involved, problems that you solve

Matematiko uporabimo pri zadevah

for aesthetic value only,

estetske narave,

or to create something beautiful,

ali pri ustvarjanju nečesa lepega,

turn around and turn out

potem pa se izkaže,

to have an application in the real world.

da so te zadeve resnično uporabne.

And as weird and surprising as it may sound,

Naj se sliši še tako nenavadno,

origami may someday even save a life.

a origami bo morda nekomu rešil življenje.

Thanks.

Hvala.

(Applause)

(Aplavz)