You can grab a paper right now and start folding it in halves. After 4 or 5 folds, it will be pretty difficult to continue folding. The world record for folding a piece of paper is 12 times. All of this is okay and you could read this just about anywhere. But, it is what follows that I strongly disagree with and I call upon any challenger to change my mind.
Consider the statement:
You can't fold anything more than a certain number of times.
Lets investigate if it is indeed true for everything. Liquids and gases are out of the question because you can't fold them and when it comes to solids; things like paper, cloth and just about anything that can fold, seems to obey the rule. But there are always exceptions, Have you ever rolled a dough flat and then folded it in half? If you have then you know you can fold and flatten the dough an infinite number of times. Hence the statement is incorrect i.e. it is physically possible to fold an object as many times required with proportional forces as long as required.
Now if it is physically feasible then how high can the folded object become. In the case of the dough, when you flatten the dough, the two layers of dough will combine with the one below and undo the growth in height unless you put flour between the two layers.
Now you could keep creating layers of dough and find the maximum height the dough reaches, but take it from me, even if you fold it a hundred times, the dough won't be higher than your waist.
But lets continue with more facts from the internet.
Consider the statement:
26 folds would make the paper thicker than the height of Mount Everest.
So if someone has studied pure mathematics all their life, they would explain it as follows:
Assume paper thickness is 1cm and height of Mount Everest is 9000m.
Each time you fold the paper, height becomes double. Therefore:
Let h = paper height. f = number of folds. t = paper initial thickness. Then:
h = t * 2 ^ f
Now lets investigate what the height of the paper would be if we folded 26 times:
h = 0.001 * 2 ^ 26 👉2 ^ 26 = 67,108,864 👉 h = 67,109m
(after folding 24 times we first surpass height of mount Everest)
That is obviously more than the height of Mt. Everest but is it physically possible for geometric growth to reach such high numbers at all in reality.
Lets think in terms of individual atoms. Consider each circle below as an atom so a total of 10 atoms:
OOOOOOOOOO
Now if we fold in half we get:
OOOOO
OOOOO
If we continue folding, we get:
OO
OO
OOO
OOO
and eventually, we will get a huge tower of atoms:
O
O
O
O
O
O
O
O
O
O
Now if were to fold one more time, what do you think will happen? Will it continue growing or start shrinking? It has to fold in half now but in the other direction.
OO
OO
OO
OO
OO
The first statement about 26 folds would lead you to believe that there will be this huge exponential growth each time you fold the paper but in reality you hit a point where folding further will simply halve the height and eventually bring it back to where you started from.
Suppose we can endlessly fold a paper would it really reach as great a height as Mount Everest. For that we need to know the size of an atom and how many atoms there would be in a piece of paper. From there we can find out the maximum size the piece of paper could reach by folding.
Before calculating, it is quite obvious no structure can stay in place if it is a single atom wide but compared to the stuff other physicists assume, this is nothing.
So how many atoms are there in an paper of size 10cm by 10cm with a thickness of 1 mm?
Atoms have an average radius of 0.1nm and for simplicity suppose they are square shaped then an atom has an area of 1e-18 m and we have to fill an area of 0.1m by 0.1m = 0.01 m. Therefore the number of atoms is equal to 0.01/1e-18 = 1e16 atoms in a single layer.
Since one atom has a height of 0.2 nm and the thickness is 0.001 m. There will be 5 million layers of atoms. With total number of atoms equal to
1e16*5e6 = 5e22
If we stack them all on top of each other which will happen after the last fold we will get a height of:
5e22*2e-10 = 1e10 kilo meters or 10 billion kilo meters.
And that would be the maximum height the paper can reach. It is certainly much higher than Mount Everest. How high will this stack of paper be? Distance from Earth to Pluto is about 5 Billion Kilometers. Which means our stack of paper will go as far as twice the distance between earth and Pluto but not more than that.
So what is the maximum number of times you can fold a 10cm by 10cm paper before it folds in on itself? The answer is about 54 folds.
After 54 folds this paper will begin transposing like a matrix halving in size until it becomes a single layer of atoms in the other direction. Of course a larger paper will fold for much longer and reach larger distances but there will always be a limit. It will not follow an exponential growth as depicted by many but a sinusoidal growth just like everything else in the universe.
This could be the proof that every exponential is actually a sinusoid in the making until we hit the limit. That is probably how I would have arrived at Euler's identity if he had not done it already.
Euler's Identity:
eix = cos x + isin x
Which means every exponential growth is made up of a real cosine and an imaginary sine. Two sinusoids.
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