# Folding Paper in Half - Exponential Decay - Fraction Denominators Resemble Binary Numbers

I had the Mathletes folding a paper plate in half. They were only able to do this four times: 1/2, 1/4, 1/8, 1/16. The children noticed that each time the paper got a half smaller, the denominator of the fraction doubled like the binary numbers.

Then we used a standard piece of 8.5” x 11” paper. The most anyone could do was 7 folds: 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128.

Then we used a piece of 11” x 17” piece of paper which is twice the size of the standard paper. The children hypothesized that they would be able to do 8 folds since one fold will equal the standard piece. However, some of them reasoned that we would only be able to fold it 7 times since the first fold would give us a piece of paper that was twice the thickness. These children were right: maximum of 7 folds 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128.

Then we used a large flip chart piece of paper and again only 7 folds. What is it about the number 7 when we exponentially decay by 50%? I had the walk half the distance to a wall, and then repeat this until the next half was too small to be  recognizable. Again, we were only able to walk to the wall 7 times before we were right up against it.

The myth: You can't fold a paper in half more than 6-8 times.  The current record is 11 times done by Britney Gallivan on Myth Busters at  https://www.youtube.com/watch?v=kRAEBbotuIE

At the end of class, we layed out a roll of toilet paper (unused) as far as it would go and then proceeded to fold it in half and were only able to get 7 true folds. The children were excited to try this at home with alternative media such as wax paper, silver foil, saran wrap, etc.

The reality: Given a paper large enough—and enough energy—you can fold it as many times as you want. The problem: If you fold it 103 times, the thickness of your paper will be larger than the observable Universe: 93 billion light-years. Seriously.

How can a 0.0039-inch-thick paper get to be as thick as the Universe?

The answer is simple: Exponential growth. The average paper thickness in 1/10th of a millimeter (0.0039 inches.) If you perfectly fold the paper in half, you will double its thickness.

Folding the paper in half a third time will get you about the thickness of a nail.

Seven folds will be about the thickness of a notebook of 128 pages.

10 folds and the paper will be about the width of a hand.

23 folds will get you to one kilometer—3,280 feet.

30 folds will get you to space. Your paper will be now 100 kilometers high.

Keep folding it. 42 folds will get you to the Moon. With 51 you will burn in the Sun.

Now fast forward to 81 folds and your paper will be 127,786 light-years, almost as thick as the Andromeda Galaxy, estimated at 141,000 light-years across.

90 folds will make your paper 130.8 million light-years across, bigger than the Virgo Supercluster, estimated at 110 million light-years. The Virgo Supercluster contains the Local Galactic Group—with Andromeda and our own Milky Way—and about 100 other galaxy groups.

And finally, at 103 folds, you will get outside of the observable Universe, which is estimated at 93 billion light-years in diameters.

1. Try folding large objects in half multiple times to see if you can fold it more than 6-8 times. Use paper, newspaper, strips of paper, toilet paper, etc. Write the fraction on each appropriate section after each fold such as 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128, 1/256.
2. Use the paper with 4 and 8 segments to section off a given fraction. So, if you choose 1/2, shade in one half on the top segment. Then in the section below, shade in half of the remaining piece to get 1/4 (1/2 x 1/2 = 1/4), then shade in half of the remaining piece to get 1/8 (1/2 x 1/4 = 1/8), then shade in half of the remaining piece to get 1/16 (1/2 x 1/8 = 1/16), and so on.
3. If you have time, try other fractions such as 2/3, or 3/4, or 4/5.