Folding Paper

Until I read this article, I thought it was impossible to fold a piece of paper more than 7 times. Read for yourself...

I knew that this would happen, because it was "Accepted Wisdom" that it was impossible to fold a piece of paper in half 10 times (or seven, or nine, for that matter.). I told him that it couldn't be done, even if he used paper the size of a football field. But I now know that I was wrong.

Suppose that you start with an standard A4 sheet of paper - about 300 mm long, and about 0.05 mm thick.

The first time you fold it in half, it becomes 150 mm long and 0.1 mm thick. The second fold takes it to 75 mm long and 0.2 mm thick. By the 8th fold (if you can get there), you have a blob of paper 1.25 mm long, but 12.8 mm thick. It's now thicker than it is long, and, if you're trying to bend it, seems to have the structural integrity of steel.

A typical claim on the Internet might run, "No matter its size or thickness, no piece of paper can be folded in half more than 7 times", and as you stare sadly at your block of folded paper, you tend to agree.

In fact, if you had a sheet of paper, and folded it in half 50 times, how thick would it be?

The answer is about 100 million kilometres, which is about two thirds of the distance between the Sun and the Earth.

And so Accepted Wisdom on Paper-Folding ruled, until 2001.

That was when a high school student, Britney Gallivan (of Pomona, California) was given a maths problem. She would get an extra maths credit, if she took up the option of solving the problem of folding a sheet in half 12 times. She tried and failed with reasonably-sized sheets of paper.

So she got smart, and used something incredibly thin - gold foil, only 0.28 of millionth of a metre thick. She started with a square sheet, 10 cm by 10 cm. It took lots of determination and practice, as well as rulers, soft paint brushes and tweezers, but she finally succeeded in folding her gold foil in half 12 times. She ended up with a microscopic square sheet of gold foil.

But her maths teacher said that ultra-thin gold foil was too easy - she had to fold paper 12 times.

She studied the problem, and came with two mathematical solutions.

The first solution was for the classical fold-it-this-way, fold-it-that-way method of folding the paper. Here you fold the paper in alternate directions. She derived a formula relating the number of folds possible (n) to the width (w, of the square sheet you start with) and the material's thickness (t):

The second solution was for folding the paper in a single direction. This is the case when you try to fold a long narrow sheet of paper. She derived another formula relating the number of folds possible in one direction (n) to the minimum possible length of material (l) and the material's thickness (t):

When she looked closely, she found that if you are trying to fold the sheet as many times as possible, there are advantages in using a long narrow sheet of paper.

Her formula told her that to successfully fold paper 12 times, she would need about 1.2 km of paper.

After some searching she found a roll of special toilet paper that would suit her needs - and that cost US $85. In January 2002, she went to the local shopping mall in Pomona. With her parents, she rolled out the jumbo toilet paper, marked the halfway point, and folded it the first time. It took a while, because it was a long way to the end of the paper. Then she folded the paper the second time, and then again and again.

After seven hours, she folded her paper for the 11th time into a skinny slab, about 80 cm wide and 40 cm high, and posed for photos. She then folded it another time (to get that 12th fold essential for her extra maths credit), and wrote up her achievement for the Historical Society of Pomona in her 40 page pamphlet, "How to Fold Paper in Half Twelve Times: An "Impossible Challenge" Solved and Explained". She wrote in her pamphlet, "The world was a great place when I made the twelfth fold."

Britney Gallivan succeeded because she was as contrary and determined as Juan Ramon Jiminez, the Spanish poet and winner of the 1956 Nobel Prize for Literature. He wrote, in a metaphor for the questioning and resilient human spirit, "If they give you ruled paper, write the other way."