## 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.^{[1]}^{[2]}^{[3]}

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).^{[4]}

With that said, there are some rules worth noting:

- 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*) =*2*^{n-1}sheets of paper to the*n*^{th }fold.” That means we very quickly can, in theory, fold our way into the solar system and even across the known universe). - 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).
- 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).
- 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”).
**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).^{[5]}

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*) = *x*^{2}.:^{[6]}

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).^{[7]}

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:

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.

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.

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!

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. 🙂