Wednesday, March 30, 2011

The Exponential Folding

How many times do you need to fold a piece of paper (A4 size) in half, to make it reach the moon? Assume that the paper is 0.1 mm thick and that the distance to the moon is 384,403 kilometers (238,857 miles) — the accepted average.

When you calculate this, you should find that there are only 42 folds necessary. Welcome to the world of exponentials!

The number might be small, but it is easy to explain why. When you fold a 0.1 mm-thick paper in half, the result is 0.2 mm. And when you fold that in half again, it will become 0.4 mm. Once more, 0.8 mm. The thickness of the paper doubles each fold: it's an exponential growth.

Consider 4 folds. That's 0.1 × 2 × 2 × 2 × 2. A quicker way to write this is 0.1 × 24. Now we can easily turn this into a formula:
0.0001 × 2n = 384403000
Those numbers are in meters. With the distance to the moon plugged in, we want to find ‘n’: the number of times you need to fold.
n = log2(384403000 ÷ 0.0001) = log(3844030000000) ÷ log(2) ≈ 42

Using the above, we can also find that to reach the edge of the observable universe (46 billion light-years; 441 septillion meters) there would be 102 folds needed (← click to see). Less than you thought? I encourage you to try and fold it so many times!

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