Exponential growth: How folding paper can get you to the Moon

Kako možemo doći do Mjeseca savijajući papir

Prevoditelj: Ivan Stamenković Recezent: Tilen Pigac - EFZG

(Glazba)

Koliko puta možete presaviti komad papira?

How many times can you fold a piece of paper?

Pretpostavimo da imamo jako fini papir,

Assume that one had a piece of paper that was very fine,

kao onaj na kojem se tiska Biblija.

like the kind they typically use to print the Bible.

U stvarnosti, čini se kao komad svile.

In reality, it seems like a piece of silk.

Da bi kvalificirali ove ideje,

To qualify these ideas,

let's say you have a paper

recimo da imate papir koji je debeo tisućinu centimetra.

that's one-thousandth of a centimeter in thickness.

To je 10 na minus treću centimetara,

That is 10 to the power of minus three centimeters,

which equals .001 centimeters.

što je jednako .001 centimetara.

Pretpostavimo da imate veliki komad papira,

Let's also assume that you have a big piece of paper,

kao što je stranica iz novina.

like a page out of the newspaper.

Now we begin to fold it in half.

Sad ćemo ga presavinuti na pola.

Što mislite koliko ga puta tako možemo presavinuti?

How many times do you think it could be folded like that?

And another question:

I još jedno pitanje:

If you could fold the paper over and over, as many times as you wish,

ako biste mogli presaviti papir uvijek iznova, koliko god puta želite,

say 30 times,

recimo 30 puta, što mislite koja bi debljina takvog papira bila?

what would you imagine the thickness of the paper would be then?

Prije nego nastavite,

Before you move on,

I encourage you to actually think about a possible answer to this question.

ohrabrujem vas da razmislite o mogućem odgovoru na ovo pitanje.

OK. Kad smo savili papir jednom,

OK.

After we have folded the paper once,

sada je debeo dvije tisućine centimetra.

it is now two thousandths of a centimeter in thickness.

Ako ga još jednom presavijemo po pola,

If we fold it in half once again,

the paper will become four thousandths of a centimeter.

postat će četiri tisućine centimetra.

With every fold we make, the paper doubles in thickness.

Sa svakim savijanjem, papir postaje dvostruko deblji.

I ako ga nastavimo savijati uvijek iznova,

And if we continue to fold it again and again,

uvijek na pola, mogli bi se suočiti sa sljedećom situacijom

always in half, we would confront the following situation

after 10 folds.

nakon 10 presavijanja.

Two to the power of 10,

Dva na desetu potenciju,

meaning that you multiply two by itself 10 times,

što znači da pomnožite 2 sa samim sobom 10 puta,

je tisuću i 24 tisućine centimetra,

is one thousand and 24 thousandths of a centimeter,

which is a little bit over one centimeter.

što je nešto malo preko jednog centimetra.

Pretpostavimo da nastavljamo presavijati papir na pola.

Assume we continue folding the paper in half.

Što će se onda dogoditi?

What will happen then?

If we fold it 17 times,

Ako ga presavijemo 17 puta,

we'll get a thickness of two to the power of 17,

dobit ćemo debljinu od 2 na sedamnaestu,

which is 131 centimeters,

što je 131 centimetar,

and that equals just over four feet.

i to je nešto preko četiri stope.

If we were able to fold it 25 times,

Ako ga uspijemo presaviti 25 puta,

then we would get two to the power of 25,

dobili bismo dva na dvadesetpetu,

which is 33,554 centimeters,

što je 33,554 centimetra,

nešto preko 1.100 stopa.

just over 1,100 feet.

Tako bi postao visok otprilike kao Empire State Building.

That would make it almost as tall as the Empire State Building.

Vrijedi ovdje zastati i razmisliti na trenutak.

It's worthwhile to stop here and reflect for a moment.

Folding a paper in half, even a paper as fine as that of the Bible,

Presaviti papir na pola, makar i papir tako fin kao onaj iz Bibilije,

25 times would give us a paper almost a quarter of a mile.

25 puta bi nam dao četvrt milje papira.

What do we learn?

Što smo naučili?

This type of growth is called exponential growth,

Ovakav rast zove se eksponencijalni rast,

and as you see, just by folding a paper

i kao što vidite, slagajući papir

we can go very far, but very fast too.

možemo otići vrlo daleko, ali i vrlo brzo.

Da sažmemo, ako savijemo papir

Summarizing, if we fold a paper 25 times,

25 puta, debljina je četvrt milje.

the thickness is almost a quarter of a mile.

Trideset puta, debljina je 6,5 milja,

30 times, the thickness reaches 6.5 miles,

što je otprilike visina na kojoj lete avioni.

which is about the average height that planes fly.

40 times, the thickness is nearly 7,000 miles,

40 puta, gotovo 7.000 milja,

or the average GPS satellite's orbit.

ili prosječna orbita GPS satelita.

48 times, the thickness is way over one million miles.

Četrdesetosam puta, debljina je daleko preko milijun milja.

Ako zamislite udaljenost Zemlje i Mjeseca

Now, if you think that the distance between the Earth and the Moon

koja je manja od 250.000 milja,

is less than 250,000 miles,

ako počnemo sa komadom papira iz Biblije

then starting with a piece of Bible paper

and folding it 45 times, we get to the Moon.

i savijemo ga 45 puta, možemo doći do Mjeseca.

I ako ga presavijemo još jednom,

And if we double it one more time,

vratimo se natrag na Zemlju.

we get back to Earth.

Lekcija: Adrian Paenza Naracija: Adrian Paenza Animacija: TED-ED Tim