 Myth

There is a limit to the number of times paper can be folded (assuming you have a large enough piece of paper).

## How Many Times Can You Fold a Piece of Paper?

There is no limit to the amount of times one can fold a piece of paper in half if the paper is large enough. However, because the thickness of the paper grows exponentially, a lot of paper would be needed to make more than 8 folds.

Given this, we can say: 1. the idea that paper can’t be folded more than 8 times is a myth, but 2. due to logistics “less than 8” folds is a reasonable limit for humans and normal sheets of paper.

Consider, to fold tissue paper 0.0033 inches thick 12 times, it would take 2417 feet of tissue paper.

Likewise, folding an average size sheet of paper about 50 times gets you to the sun and a little over 100 times gets you folded paper with a thickness the size of known universe (exact mathematics depend on the exact thickness of the paper).

With that said, there are some rules worth noting:

1. Because of exponential growth, even a very thin sheet of paper can grow to unimaginable sizes with only a few dozen folds. This is because the thickness of the paper doubles with every fold (“every time we fold the paper in half, we double the number of layers involved. We have to fold f(x) = 2n-1 sheets of paper to the nth fold.” That means we very quickly can, in theory, fold our way into the solar system and even across the known universe).
2. If a folded section’s length is less than π times the height, the next fold cannot be completed. This is because paper gets “wasted” on the “rounded edges” required to fold the paper over the growing number of layers (the radius of the rounded portion is one half of the total thickness of folded paper, and that number also experiences exponential growth).
3. Because one has to factor in “wasted paper,” the length and width of the given sheet decide the number of times we can fold the paper in half (not energy).
4. Therefore, if one has a “large” enough sheet of paper, there is no limit to the amount of folds one can make (the physical limits are based on things like “needing to fold a piece of paper in space to make the fold”).
5. CAVEAT: The universe has physical limits, and these limits could theoretically affect paper-folding. In other words, even if one had enough paper and could fold it, they could still run up against the laws of physics at some point.

The implications of this and all the logic behind this can be expressed by a poem contained in Folding Paper in Half (a paper based on the work of a grade 11 student in California, Britney C. Gallivan, who had mathematically disproved the paper folding myth):

HOW TO REACH THE SUN …ON A PIECE OF PAPER

A poem by Wes Magee

Take a sheet of paper and fold it,

and fold it again,

and again, and again.

By the 6th fold it will be 1-centimeter thick.

By the 11th fold it will be

32-centimeter thick,

and by the 15th fold – 5-meters.

At the 20th fold it measures 160-meters.

At the 24th fold – 2.5-kilometers,

and by fold 30 it is 160-kilometers high.

At the 35th fold it is 5000-kilometers.

At the 43rd fold it will reach the moon.

And by the fold 52

will stretch from here

to the sun!

Take a piece of paper.

Go on.

TRY IT!

How Folding Paper Can Get You to the Moon.

MUSING: What is true for folding is not true for rolling. One can roll paper many more times, only adding the thickness of the paper with one roll. Perhaps this is why people suggest rolling your clothes in a suitcase instead of folding them.

## How to Fold Tissues Paper to the Moon and Beyond in Theory and 12 Times in Practice

Although paper comes in different thicknesses, let’s consider tissue paper that is 1/10th of a millimeter (0.0033 inches thick).

To fold it 12 times, the formula states it would take: 2417 feet of tissue paper to accomplish this.

All we need to do is double 0.0033 inches for every time we fold the paper and we’ll get a number that grows exponentially (this is the simple version where we don’t consider things like the length needed or radius).

This can be expressed as the exponential function f(x) = x2.:

For our purposes however let’s use the table below which imagines a thickness of 0.1 mm (not .0033 in) to keep it simple (see the math here).

n 2^n km (0.1*10^-6 * 2^n) Comment
0 1 0.1 x 10^-6
1 2 0.2 x 10^-6
2 4 0.4 x 10^-6
3 8 0.8 x 10^-6 finger nail thickness
4 16 1.6 x 10^-6
5 32 3.2 x 10^-6
6 64 6.4 x 10^-6
7 128 12.8 x 10^-6 thickness of a notebook
8 256 25.6 x 10^-6
9 512 51.2 x 10^-6
10 1024 0.1 x 10^-3 width of a hand (incl. thumb)
11 2048 0.2 x 10^-3
12 4096 0.4 x 10^-3 0.4m height of a stool
13 8192 0.8 x 10^-3
14 16384 1.6 x 10^-3 1.6m: an average person’s height (yeah, a short guy)
15 32768 3.3 x 10^-3
16 65536 6.6 x 10^-3
17 131072 13.1 x 10^-3 13m height of a two story house
18 262144 26.2 x 10^-3
19 524288 52.4 x 10^-3
20 1048576 104.9 x 10^-3 quarter of the Sears tower (440m)
…. …. ….
25 33554432 3.4 x 10^0 past the Matterhorn
30 1073741824 107.4 x 10^0 outer limits of the atmosphere
35 34359738368 3.4 x 10^3
40 1099511627776 109.9 x 10^3
45 35184372088832 3.5 x 10^6
50 1125899906842624 112.5 x 10^6 ~ distance to the sun (95 million miles)
55 36028797018963968 3.6 x 10^9
60 1152921504606846976 115.3 x 10^9 size of the solar system?
65 36893488147419103232 3.7 x 10^12 one-third of a light year
70 1180591620717411303424 118.1 x 10^12 11 light years
75 37778931862957161709568 3.8 x 10^15 377 light years
80 1208925819614629174706176 120.9 x 10^15 12,000 light years
85 38685626227668133590597632 3.9 x 10^18 4x the diameter of our galaxy
90 1237940039285380274899124224 123.8 x 10^18 12 million light years
95 39614081257132168796771975168 4.0 x 10^21
100 1267650600228229401496703205376 126.8 x 10^21 (12 billion light years) approx. radius of the known universe?

Note:

Conclusion

The limit to the amount of times paper can be folded is based on the size of the paper. With a large enough bit of paper there is theoretically no limit to the amount of times it can be folded. Nikolai on
Supports this as a Fact.

There is a theoretical limit because with enough paper and enough folds the thing would collapse into a black hole, and you wont be able to fold it anymore. Thomas DeMichele on

That is actually a good point (well a fun point at least). The universe has physical limits, so speaking loosely if there was enough mass in one place it would collapse into a black hole. With that said, I’m not sure how folding paper would result in that? Still a fun thought. Uncle Jeff on
Supports this as a Fact.

This is clear, concise, and brilliantly factual. I have suspected as much for ages, but have never been able to quantify the amount of room the fold itself takes up since that’s a particular tricky equation: in fact, I think you have clarified the fact that that folded paper takes up much more room than a stack of paper or a roll. Now I just need to find a really really thin rolling paper about the size of city and try this out! Thomas DeMichele on

Rolling paper is much more efficient than other methods from what I understand, however with folding specifically you need enough paper to fold around the thickness each fold (not true with stacking). That is why folding gets out of hand so quickly. 🙂