Paper folding fun

Dear MatAl:

Long time listener, first time caller. Me and my cohorts were messing around with the age old mystery of how many times sheet of paper can be folded. We got as high as eight folds. What is the magical number? Does the paper density and size matter? And finally, why is there a limited amount of folding?

-- Rob, the net

The elves decide this would be a perfect begin-the-year question, since everybody's got a ton of foldable stuff around, everything from wrapping paper to old tamale corn husks. A few days ago, the elves started folding stuff in half to see if they couldn't beat eight. Most of the time, they couldn�t beat seven. But once they were through, the trash looked very tidy, what with all the neatly creased and stacked failed experiments. Grandma plans to make this an Alice family tradition.

It has been an old assumption that for whatever reason, eight was the maximum number of folds that could be made in a piece of paper, folding it in half repeatedly. This seemed to hold true whether the paper was folded each time along its length or the folds were alternated along the length and the width. No one demonstrated that the rule of folds was not true until a few years ago when a high school math teacher gave the assignment to his class: Fold a piece of paper 12 times.

An annoyingly bright girl with the unpromising name of Britney-- Britney Gallivan-- presented Teach with a four-inch-by-four-inch piece of microscopically thin gold leaf painstakingly folded 12 times. Clever, but gold leaf isn't paper, he observed. But Britney was on the right track. She considered the thickness of a particular piece of paper relative to the width of the sheet, chewed a while on her pencil eraser, and came up with formula that would predict how wide a piece of paper would have to be, given its thickness, to be folded N times along its length. There's more to it than this, but in part, it's based on the idea that on fold number one, you end up with double the original paper thickness and only half its original width. On fold number two, you have four times the original thickness and one fourth of the original width. If you continue to double one and halve the other, very quickly a sheet of ordinary notebook paper or a newspaper sheet is in an unfoldable lump.

Britney managed to find an industrial-sized roll of toilet paper 4000 feet long. According to her formula, this was the length of the sheet she would need if she was to fold the tissue lengthwise 12 times. She and her parents unrolled the paper along a hall in a shopping mall and started measuring and folding. And danged if it didn't work. She even wrote a booklet detailing the math logic in her experiment and came up with a second equation for predicting the number of alternating length-width folds for a given sheet. We can only assume Britney got an A that semester. Grandma's still trying to track down her source for the TP, and the elves sure want to know what it would be used for besides cracking math myths.