Folding Paper 100 Times

What is Intuition? One definition I found when doing a quick search was

The direct perception of truth, fact, etc., independent of any reasoning process; immediate apprehension.

Recently I read an interesting fact that would makes most peoples intuitions give them a very incorrect answer.

But first recounting the old myth that states; you can’t fold a paper more than 7 times on it self, has been debunked famously by Mythbusters in an episode where they folded a football sized paper 11 times.

Then there is THIS Britney girl (no not the singer, she would struggle to fold the paper more than twice) who did her little calculation and then ordered about a 1.2 kilometre long toilet paper roll and folded it 12 times. The picture has her on the 11th fold.

EDIT: Apr-2016

Here is a video of what happens when you get the help of a hydrolic press to try to fold a piece of paper more than 7 times

 

But now lets go back to the question

The Question

If you could theoretically fold a 1mm thick piece of paper how thick would it be on the 100th fold?

Thinking about it now, the prior talk of folding and the included pictures may steer your intuition in the right direction, but I would still venture that you would be way off.

The Answer

*drum-roll*

Thicker than that observable diameter of the universe.

The thickness of the paper would be 133,989,789,471 light years across which is (133.99 billion light-years distance)

Compare this to the diameter of the observable universe which is estimated to be 93 billion, or 9.3 × 1010, light years across in length.

Explanation and Assumptions:

The first assumption made was the thickness of the paper. I chose to use the thickness of 1mm, which is actually rather thick if you think about it, more of a cardboard. The usual A4 paper is around 0.1 mm thick, which is 10 times thinner than what I used. I found some sites that use the A4 thickness and thus their calculations give them a thickness of the paper after 100 folds to around 12-13 billion light-years.

A light-year is a distance that light travels in a vacuum in one Julian calendar year. For example the time it takes for light from the Sun to reach Earth is around 8 minutes and 20 seconds or for light to travel from the closest star (not the Sun) to Earth would be about 4.22 light years. One light year is about 9,460,730,472,580.8 kilometers.

The diameter of the observable universe was taken from HERE

Here’s a site doing the calculations, it also used the 1mm assumption for the paper and is working in meters.

This little problem is an example of exponential growth. The basic idea is to take width of paper x 2^100 which is the amount of folds.

There is a similar illustration of exponential growth with the following story of rice/wheat and a chessboard found HERE

Comparisons

I will now give some comparisons to objects of the same length (thickness) at different stages of folding.

Amount of fold: 4

Thickness of paper: 1.6cms

Comparison: about the size of a button or a coffee bean

 

 

Amount of fold: 5

Thickness of paper: 3.2 cms

Comparison: Length of a quail egg

 

 

Amount of fold: 7

Thickness of paper: 12.8 cms

Comparison: Average height of a cup

 

 

 

Amount of fold: 10

Thickness of paper: 1.024 meters

Comparison: A diameter of a large beach ball or a 300Mhz wavelength that falls in the range of Australian TV channels being broadcast

 

 

Amount of fold: 12

Thickness of paper: 4.096 meters

Comparison: About the length of a VW Beetle

 

Amount of fold: 17

Thickness of paper: 131.07 meters

Comparison: A bit less than the Great Pyramid of Giza inEgypt

 

 

 

Amount of fold: 20

Thickness of paper: 1,048.58 meters (1.04858 kilometres)

Comparison: About the height of the proposed Kingdom Tower in Saudi Arabia. It is also more than the height ofAngel Falls (979m), the world’s highest waterfall.

 

 

Amount of fold: 23

Thickness of paper: 8.38 kilometres

Comparison: Approaching the height of the world’s highest mountain;Mount Everest(8,848 m)

Amount of fold: 30

Thickness of paper: 1073.74 kilometres

Comparison: More than the drive fromSydney to Brisbane(939 kms). It is also approaching the diameter of Charon, the biggest moon of Pluto (1,207 kms)

 

 

Amount of fold: 37

Thickness of paper: 137,438 kilometres

Comparison: Nearly equivalent to the diameter of Jupiter (142,984 km)

Amount of fold: 44

Thickness of paper: 17,592,186 kilometres (17.5 million kilometres)

Comparison: About the distance of a light-minute. The time it takes for light to travel one minute in a vacuum (1.799×107 km)

Amount of fold: 47

Thickness of paper: 140,737,488 kilometres (140.7 million kilometres)

Comparison: The distance from Earth to the Sun is about 149 million kilometres

Amount of fold: 50

Thickness of paper: 1,125,899,906 kilometres (1.1 billion kilometres or 1.1 terameters [Tm])

Comparison: About the distance of a light-hour. The time it takes for light to travel one hour in a vacuum 1.079 Tm

Amount of fold: 63

Thickness of paper: 9,223,372,036,854 kilometres

Comparison: About the distance of a light-year. The time it takes for light to travel one year in a vacuum (9,460,730,472,580.8 km or 9.5 petametres or 10^12 kilometers)

Amount of fold: 70

Thickness of paper: 1,180,591,620,717,410 kilometres (124.8 light-years)

Comparison: Nearing the diameter of Rosette nebula (130 light years or 1.22986869 × 1015 kilometres)

 

Amount of fold: 76

Thickness of paper: 75,557,863,725,914,300 kilometres (7,986.64 light-years)

Comparison: Longer than the diameter of Small Magellanic Cloud dwarf galaxy (7000 light-years across)

 

Amount of fold: 81

Thickness of paper: 1,208,925,819,614,630,000 kilometres (127,786 light-years)

Comparison: Approaching the diameter of the Andromeda Galaxy that is estimated to be 141,000 light years across

Amount of fold: 87

Thickness of paper: 154,742,504,910,673,000,000 kilometres (16.3 million light-years)

Comparison: It’s now larger than the diameter of the Virgo Cluster, which is a prominent cluster of galaxies (15 million light-years in diameter). It’s also more than the distance to Centaurus A galaxy from our Sun (15 million light-years)

Amount of fold: 90

Thickness of paper: 1,237,940,039,285,380,000,000 kilometres (130.8 million light-years across)

Comparison: It’s now longer than the diameter of the Virgo Supercluster (110 million light-years). The Virgo Supercluster has over 100 galaxy clusters (one of them being the Virgo cluster mentioned earlier). The Supercluster contains within it a cluster of about 30 galaxies called ‘Local group’. Within that is the Milky Way galaxy, which contains Earth.

Amount of fold: 97

Thickness of paper: 158,456,325,028,529,000,000,000 kilometres (16.74 billion light-years across)

Comparison: Distance to far-away quasars

Amount of fold: 100

Thickness of paper: 1,267,650,600,228,230,000,000,000 kilometres (133.99 billion light-years across)

Comparison: As stated before diameter of the observable universe is estimated to be 93 billion light-years across.