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survivorboi · Jul 21, 2009, 02:31 PM

I read how a girl named Britney folded something in half 12 times. And the equation she used was:

L= (pi * t/6)(2^n + 4)(2^n - 1)

(You can check the equation here, just in case I typed it wrong Folding Paper in Half Twelve Times)

How I'm I suppose to use it to help me when I try to hold a piece of paper 12 times? I read the article, but I'm still confused...

Thanks!

ebaines · Jul 21, 2009, 03:10 PM

This is very interesting - I too had heard the adage that it's impossible to fold a single piece of paper moret han 7 times. But then I recall that the "Myth Busters" did an episode on it in which I think they succeeded at 11 folds, but they used an airplane hanger sized piece of paper and a steam roller!

Britney's analysis is that the minimum length of paper to get n folds - all in the same direction - is:

$ L = \frac {\pi t} 6 (2^n + 4)(2^n -1) $

where t = thickness of the paper.

She assumes that a paper can be defined as being successfully folded if there is some distance where all the folded layers are parallel to each other (otherwise you just have a ball, not really a fold). This leaves some amount of paper that is consumed in the turning at the folds. In the limit, if the distance of paralell-ness approaches zero, you have the minimum length of paper required, in whch case it is all consumed in the folds. She calculats that length by modeling the paper as folding in a perfect semicircle, and the length of the semicircle for any fold is equal to $\pi k t$ , where k is the number of the layer of that fold from the center. For example, in the first fold the length of the fold is $\pi*t$ - which is the length of a semi-circle of radius t. If you have two layers and you fold it the inner layer will be of length $\pi*t$ and the second layer will be of length $2 \pi*t$ , and so on. Her formula is a way of adding up the number and lengths of all the folds.

survivorboi · Jul 21, 2009, 09:27 PM

Originally Posted by ebaines

This is very interesting - I too had heard the old adage that it's impossible to fold a single piece of paper moret han 7 times. But then I recall that the "Myth Busters" did an episode on it in which I think they suceeded at 11 folds, but they used an airplane hanger sized piece of paper and a steam roller!

Britney's analysis is that the minimum length of paper to get n folds - all in the same direction - is:

$ L = \frac {\pi t} 6 (2^n + 4)(2^n -1) $

where t = thickness of the paper.

She assumes that a paper can be defined as being successfuly folded if there is some distance where all the folded layers are parallel to each other (otherwise you just have a ball, not really a fold). This leaves some amount of paper that is consumed in the turning at the folds. In the limit, if the distance of paralell-ness approaches zero, you have the minimum length of paper required, in whch case it is all consumed in the folds. She calculats that length by modeling the paper as folding in a perfect semicircle, and the length of the semicircle for any fold is equal to $\pi k t$ , where k is the number of the layer of that fold from the center. For example, in the first fold the length of the fold is $\pi*t$ - which is the length of a semi-circle of radius t. If you have two layers and you fold it the inner layer will be of length $\pi*t$ and the second layer will be of length $2 \pi*t$ , and so on. Her formula is a way of adding up the number and lengths of all the folds.

Thanks for trying so hard to explain it to me! I guess I'm still a little too young minded to understand this. But my overall questions is: How is this equation going to help me? How do I find out the minimum number of folds to substitute in the equation?

By the way, how did you write all that in math? Did you just copy from somewhere?

HelpinHere · Jul 21, 2009, 09:40 PM

I really cannot expand upon what ebaines explained.
Basically, it is just "when the curve of the fold is larger than the length of the previous fold, it is not actually a fold, and therefore you stop."

I CAN tell you, however, that she used one long roll, toilet paper, taking up an entire supermarket to do this. It is not like folding a piece of paper, because the length is so proportionately (sp?) large compared to the width and thickness, whereas a piece of paper they are MUCH closer.

CLICK HERE to learn how to type in $math$ .

ebaines · Jul 22, 2009, 07:04 AM

[QUOTE=survivorboi;1872592]But my overall questions is: How is this equation going to help me? How do I find out the minimum number of folds to substitute in the equation?
QUOTE]

The purpose of the equation is to tell you how long the paper has to be tin order for you to have a shot at achieving n folds. If you want to know how many folds you can get from a given piece of paper that is L inches long, just sub in for L and t, and find the largest value of n such that the formula yields a value that is not greater than the length of paper.

For example: given paper 11 inches long and 0.005 inches think, if you set n = 5 you find that the formula gives:

$ L = \frac { \pi * (0.005)} 6 (2^5+4)(2^5-1) \\ =\frac {\pi *(0.005)} 6 (36)(31)\ =\ \pi*(.005)*6*31 = 2.92\ inches. $

So in theory a 5 mil thick piece of paper can be folded 5 times and 2.92 inches will be consumed on the folds. That leaves 8.08 inches for the "flat part," and given 2^5 = 32 layers of paper, that means the length of the flat part is 8.08/32 = 0.25 inches. Again, all this is theory - it assumes that all folds are perfectly round, that the paper isn't crushed in the folds, and that there are no gaps between layers at the folds.

Unknown008 · Jul 22, 2009, 10:06 AM

Psst, you did a typo... it's 32 not 36 like you put in the formula.

ebaines · Jul 22, 2009, 10:26 AM

Originally Posted by Unknown008

Psst, you did a typo... it's 32 not 36 like you put in the formula.

Um... 2^5 + 4 = 36, no?

Unknown008 · Jul 22, 2009, 11:35 AM

Oops, sorry :o was reading too fast...

survivorboi · Jul 22, 2009, 03:06 PM

[QUOTE=ebaines;1873159]

Originally Posted by survivorboi

But my overall questions is: How is this equation going to help me? How do I find out the minimum number of folds to substitute in the equation?
QUOTE]

The purpose of the equation is to tell you how long the paper has to be tin order for you to have a shot at achieving n folds. If you want to know how many folds you can get from a given piece of paper that is L inches long, just sub in for L and t, and find the largest value of n such that the formula yields a value that is not greater than the length of paper.

For example: given paper 11 inches long and 0.005 inches think, if you set n = 5 you find that the formula gives:

$ L = \frac { \pi * (0.005)} 6 (2^5+4)(2^5-1) \\ =\frac {\pi *(0.005)} 6 (36)(31)\ =\ \pi*(.005)*6*31 = 2.92\ inches. $

So in theory a 5 mil thick piece of paper can be folded 5 times and 2.92 inches will be consumed on the folds. That leaves 8.08 inches for the "flat part," and given 2^5 = 32 layers of paper, that means the length of the flat part is 8.08/32 = 0.25 inches. Again, all this is theory - it assumes that all folds are perfectly round, that the paper isn't crushed in the folds, and that there are no gaps between layers at the folds.

Hey! I have 3 questions:

1. How did you get 5 for "n"? How can you just make one up?

2. On the bottom equation, why did you multiply 6 instead of dividing it? The equation said to divide.

3. Why didn't you multiply 36? You just did (pi * 0.005)*6*31
I think you skipped 36...

Shouldn't it be:

{(pi * 0.005)/6} (36) (31)?

THANKS!!

Unknown008 · Jul 23, 2009, 03:08 AM

Ok, that one, I got it right.

1. How did you get 5 for "n"? How can you just make one up?

5 is just a number picked randomly. It could be any other number.

2. On the bottom equation, why did you multiply 6 instead of dividing it? The equation said to divide.

What really happened was $\frac{\pi *0.005}{6}(36)(31)$ , then $(\pi *0.005)(6)(31)$ . Did you see that 36 was divided by 6?

3. Why didn't you multiply 36? You just did (pi * 0.005)*6*31
I think you skipped 36...

Question already answered now.

:)

ebaines · Jul 23, 2009, 05:42 AM

As to why I picked n=5: that's the largest value for n that gives an answer for L less than 11 inches. So according to the formula you can't fold an 11 inch piece of paper more than 5 times.

survivorboi · Jul 23, 2009, 02:28 PM

Originally Posted by Unknown008

Ok, that one, I got it right.

5 is just a number picked randomly. It could be any other number.

What really happened was $\frac{\pi *0.005}{6}(36)(31)$ , then $(\pi *0.005)(6)(31)$ . Did you see that 36 was divided by 6?

Question already answered now.

:)

Oh! So it's just another way of doing it, and if I would have divided it by 6 then times 36, the answer would be the same! It's just another route eh?

Unknown008 · Jul 25, 2009, 12:03 AM

Yup, another, shorter route, but still, same answer, because you can also write it as:

$\frac{\pi * 0.005}{6}(36)(31)$ , or $\frac{\pi * 0.005(36)(31)}{6}$ or $(\pi * 0.005) $\frac{36}{6} $(31)$ , etc.