# How many times can you fold a sheet of paper?

## FACT: If you fold paper in half 50 times, it will end up about 100 million kilometres thick!

- How many times can you fold a piece of paper?
- Are there relationships between the number of folds, the perimeter and the thickness of the paper?

## You can't fold paper of any thickness more than 7 times

You may have come across this activity that requires you to fold a sheet of paper as many times as possible. Most aspirants end up with about 5 or 6 folds before the thickness of the combined folds of paper makes further folding impossible.

The myth is that it is impossible to fold the paper more than 7 times. However, this was busted by the Mythbusters TV show, where they used parachute-thin material the size of a football field and a rolling pin bulldozer to establish 11 folds!

## One way of folding

Let’s look at one way a square sheet of paper can be folded.

Take the side length of the sheet of paper to be S* *and its thickness to be T. Initially, the perimeter P* *(outside length) is S + S + S + S = 4S* *and the area is A = S × S = S^{2}

After fold 1 (F = 1) the thickness is doubled, the area is halved and the perimeter is reduced.

So we have the thickness as 2T, area is A/2 and the perimeter is P = S + S + S/2 + S/2 = 2S + S

The values for the first 5 folds are as follows.

The last row gives the thickness, area and perimeter for F folds.

We can show the results as a graph.

As the number of folds increases, the perimeter, P, approaches 2S = 200 cm.

Since the area is halved with each fold, its value will approach 0, and since the thickness is doubled each time, its value keeps increasing without bounds.

## The formula

It turns out that some bright mathematicians have determined the minimum size of paper needed to make it possible to fold the paper a given number of times.

The formula is

- S is the minimum length of paper required.
- T is the thickness of the paper before folding begins
- F is the number of folds

For example, suppose we wish to make 5 folds using a square sheet of paper of thickness 0.1 cm.

The side length must be

The graph of S against F for T = 0.1 cm is shown below.

Notice the exponential rate of increase of the length of paper needed as the number of folds increases. You may now appreciate why it is accepted that folds up to about 7 is possible but anything above that number poses a real challenge.

In Mythbusters , 11 folds were made using a football-size ‘paper’ with the assistance of a bulldozer.

Each time, the folds were alternate.

The formula for alternate folding is

Let’s assume the ‘paper’ thickness used in Mythbusters was 0.2 cm. What length is needed to be able to fold it 11 times without mechanical aid?

Using our formula we have

This is about 200 m, which was approximately the length of the football-size area used.

## Your challenge

You might like to investigate the 'corner to corner' method of folding the paper to see how the perimeter, thickness and area change.

Then beat the mathematicians at their own game by coming up with the formula!

## Comments

**John Welford** from Barlestone, Leicestershire on May 11, 2019:

Isn't it amazing the lengths some people will go to in order to prove (or disprove) something of limited importance!