Paper Folding (100 folds)

There was once a question on a game show about folding a piece of paper in half 100 times and if it was possible.  Of course, the answer is "no," but just how big of a NO is it?

I will prove why.

Suppose all other complexities are irrelevant and through some miracle, you COULD fold a piece of paper in half 100 times.  Lets take a look at the thickness of that stack of folded paper.

Lets call the  thickness of the paper  `t`.

Folding one time will create a stack that is `2t` in height.

Folding it again will create a stack `4t` in height

Again will be `8t` in height

The function that described the height would be:

`H=t2^F`

Where:

`H` was the height of the stack

`t` was the thickness of the paper

and

`F` was the number of times you folded the paper in half

To get the height, we need to do a few substitutions on some variables.

The thickness of a sheet of paper is about 0.0025 inches thick

So the height becomes:

`H=(0.0025)(2^100)` inches

`H=(0.0025)(1,267,650,600,228,229,401,496,703,205,276)` inches

`H=3,169,126,500,570,573,503,741,758,013.44` inches

`H=(3,169,126,500,570,573,503,741,758,013.44)/(12)` feet

`H=264,093,875,047,547,791,978,479,834.45333` feet

`H=(264,093,875,047,547,791,978,479,834.45333)/(5280)`miles

`H=50,017,779,365,065,869,692,893.908040404` miles

`H=(50,017,779,365,065,869,692,893.908040404)/(186,000)` light-seconds

`H=268,912,792,285,300,374.69297800021723` light-seconds

`H=(268,912,792,285,300,374.69297800021723)/(60)` light-minutes

`H=4,481,879,871,421,672.9225496333369538` light-minutes

`H=(4,481,879,871,421,672.9225496333369538)/(60)` light-hours

`H=74,697,997,857,027.881859160555615896` light-hours

`H=(74,697,997,857,027.881859160555615896)/(24)` light-days

`H=3,112,416,577,376.161744131689817329` light-days

`H=(3,112,416,577,376.161744131689817329)/(365.25)` light-years

`H=1,136,810,154,886,643.0770440997057794` light-years

Basically, the stack would be so tall that it would take light more than 1.1 quadrillion years to reach the top of it.

While this is an accurate calculation, it is a bit overwhelming and may even be a bit hard to believe. I will show you a few intermediate calculations that will give a little perspective on the thickness of folded paper.  

If you want to find how many folds it takes to achieve a certain thickness, you start with the original equation and solve for `F`.

Since:

`H=t2^F`

We can derive that:

`H\t=2^F`

And therefore:

`Log_2(H\t)=F`

`F=Log_2(H\t)`

Now, we can plug in a height for `H` and thickness for `t` and calculate how many folds (`F`) it will take to achieve that thickness.

For instance, if we want to know how many folds it takes to get a stack that is 1 inch thick, we can plug that into the formula as follows:

`F=Log_2(H\t)`

`F=Log_2(1\0.0025)`

`F=Log_2(400)`

`F=8.6438`

Then we round up:

`F=9`

So after 9 folds the stack will be over an inch tall.

After 13 folds, the stack would be 20.48 inches tall

After 14 folds, the stack would be almost 41 inches tall

After 25 folds, the stack would be over a mile tall (a little over 6,990 feet)

After 43 folds, the stack would reach from the Earth to the moon and beyond.