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## Exponential Thinking Pt. 2

by | Mar 9, 2018

A prior IFOD explored the “second-half to the chess board” as an example of how humans struggle to think exponentially. That IFOD is here: Exponential Thinking

Here’s another great example of how hard it is to think exponentially. Question: How thick would a piece of paper be if you could fold it in half 50 times (which btw, is impossible)? Answer below – it’s probably way thicker than you think.

One fold – two layers
Two folds – four layers
Three folds – eight layers
Four folds – 16 layers
Five folds – 32 layers
Six folds – 64 layers
Seven folds – 128 layers
Eight folds – 256 layers (this was thought to be the limit of how many times you can fold a piece of paper in half – but this is discussed more below)
Nine folds – 512 layers
10 folds – 1,024 layers

Now it really explodes. Assuming that a piece of paper is about .1 millimeter in thickness, here are the stats for folding a piece of paper:

• 10 folds is 1,024 layers which is 4 Inches thick (makes sense, a 500 ream packet of copy paper is about 2 inches high).
• 20 folds is 1,048,576 layers  which is  344 feet (a little over half the height of the St. Louis Gateway Arch)
• 30 folds is 1,073,741,874 layers which is about 66 miles.
• 40 folds is 1,099,510,000,000 layers which is about 70,000 miles (about a 1/3 of the way to the moon)
• 50 folds is 1,259,000,000,000,000 layers which is about 70 million million miles
• 51 folds is 2,518,000,000,000,000 layers which is 140 million miles (note that the sun is 93 million miles away)

At 60 folds it has the diameter of the solar system. At 103 folds it Is as thick as the observable universe. Repeated doubling is an explosive process, obviously.

How many times can a piece of paper actually be folded in half? Until 2002 it was thought that no matter how big your piece of paper was the limit was about 8 times. The problem is that it gets too thick to fold. Pull out a piece of paper and try it. I can get six with a normal sized piece of paper. Here’s folding a normal sheet of paper with a hydraulic press 7 times (I am so glad to learn there is a “hydraulic press channel” where they just schmoosh stuff with a press for fun):

In 2002, for extra credit in high school math class, Brittany Gallivan proved that you can fold a piece of paper in half 12 times with this mathematical formula:

Where L is the minimum possible length of the material, t is material thickness, and n is the number of folds possible in one direction. After extensive experimentation, she folded a sheet of gold foil 12 times, breaking the record. This was using alternate directions of folding (and a really big piece of paper/foil) In one day Britney was the first person to set the record for folding paper in half 9, 10, 11 or 12 times. In May of 2007 she graduated from UofC Berkeley. Here’s a picture of the 11th fold (one more to go).

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