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Limits to Paper Folding

Date: 05/09/2002 at 23:16:06
From: E Neville-Lamb
Subject: Limits

It is said that a piece of paper can be folded not more than nine 
times.  I understand that the thickness increases as a 
geometrical progression, but what is the determining limit and 
how is it established?  Why nine times and not eight or ten?

Date: 05/10/2002 at 09:16:10
From: Doctor Peterson
Subject: Re: Limits

Hi, E.

It's just a rule of thumb, not an absolute limit; I thought I'd 
heard something more like 7, but it depends on the size of the 
paper and how hard you work to fold it.

Consider both the thickness and the width of the folded paper 
after N folds, if it starts with thickness t and width w; I'll 
first assume we are always folding in the same direction, making 
a narrow strip with a constant thickness, rather than always 
folding across the narrow dimension as you would in reality. Then 
after N folds, the thickness is 2^N t and the width is 2^-N w, 
since the former doubles each time and the latter is halved. The 
ratio of thickness to width is then

    (2^N t)/(2^-N w) = 2^(2N) t/w

So, what is a reasonable ratio to start with? Suppose we use 8.5 
by 11 paper and treat 11 inches as the width; and suppose that 
500 sheets are 1 inch thick, so t is 0.002 inch. Then t/w is 
0.002/11 = 0.0002. Let's suppose for very thin paper it is 
0.0001. After 7 folds the ratio becomes

    2^14 * 0.0001 = 1.6

That means the thickness is more than the width! And that is what 
limits the folding; if you manage to fold it at all, it's hard to 
think of it as folding when it's all bend and no flat paper.

Now suppose we took the same paper and started with, say, a 
50-foot width; that's about 50 times as wide as our single sheet, 
so the initial t/w is 1/50 as big. Notice that 50 is less than 
2^6, so just three more folds will make up for the difference; 
after 10 folds the ratio becomes

    2^20 * 0.000002 = 2

So depending on exactly when you consider it impossible to go on, 
and just how big the paper is to start with, you might get up to 
10 folds, but that's pretty extreme.

If you take into account the fact that you will really be folding 
in alternating directions to use up both width and height, you 
can refine this a little. We can say that the width will be cut 
in half every two folds rather than every fold, while the 
thickness continues to be doubled every fold; so the thickness 
ratio will really be

    (2^N t) / (2^(N/2) w) = 2^(3N/2) t/w

For our last example, assuming a 50 by 50 foot sheet (!), after 
10 folds the ratio will be

    2^15 * 0.000002 = 0.066

but after 12 folds it will be 8 times as great, or 0.5. This is a 
significant improvement, and we can still get another fold or two 
out of it. But of course, I chose a ridiculously large sheet!

The important point, of course, is that to get just one more fold 
out of a sheet, you will have to multiply the width by 2^(3/2) = 
2.8, so the width, within reasonable bounds, has little effect.

- Doctor Peterson, The Math Forum 

Date: 03/25/2003 at 05:19:33
From: Britney Gallivan
Subject: Folding Paper in Half 12 Times

I originally did work on the paper-folding problem, solved for limits
for several cases, checked the Internet, etc. and wrote a paper on the
limits of folding paper. I had the paper and work I wrote checked by
Professor Benjamin of Harvey Mudd College in February of 2002. I then
published the booklet and have sent it to quite a few mathematicians.
I asked for permission from Dr. Ken to quote the only Dr. Math work
then available (31 Dec 1994, PaperFolding). Dr. Ken's work did not 
describe actual limits.

I was checking the Web and noticed Dr. Peterson's article "Limits to
Paper Folding" done in May of 2002.  This was after I had derived
equations and folded a piece of paper in half 12 times per calculated
limits and written my booklet on the subject. If it had been written
before my publication I certainly would have included it as a reference. 
It does address the correct approach.

For your interest, the booklet I made is described at:

   How to Fold Paper in Half Twelve Times - Historical Society of 
     Pomona Valley

Thank you. It is a quality Web site that you operate.
Britney Gallivan
Associated Topics:
High School Exponents
Middle School Exponents
Middle School Measurement
Middle School Ratio and Proportion

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