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How Many Times Can You Fold a Piece of Paper?

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YouTube / astroboy0969

When I was a kid, I learned that there was a limit to the number of times a piece of paper could be folded. It was a lesson in exponential growth, the idea being that each fold doubles the paper's thickness, and even with something as thin as paper, quickly you'll end up with an unmanageable mess, too thick to fold further.

But the big question was always: Okay, so how many times can a given piece of paper be folded? In a brief third-grade science lesson we tried this experiment with various kid-sized pieces of paper, and often got to around six folds—and I just did it now with a large sticky note, and again got to six folds easily. Somebody (I can't recall whether it was our teacher or a fellow student) imparted the sage wisdom: seven folds is the most. This seemed plausible, because it seemed to hold up to all the testing a room full of savvy eight-year-olds could manage. Case closed: The universe only allowed for seven paper-folds on a given sheet. Oh, our minds would be blown in a few decades.

In January 2002, Britney Gallivan, then a junior in high school, folded a 4,000-foot-long roll of toilet paper to prove that 12 folds were possible (note that she used single-direction folding, given the long, narrow nature of her paper; my class had been using multi-directional folding, but still—wow). What's more, she did this after deriving a paper folding theorem (yes, it involves pi) that allows calculation of maximum folds based on paper thickness, length, and/or direction of folding, and accounts for the loss of usable paper at the edges due to the rounding that comes with extreme folding. That is some math magic right there, with empirical proof to boot.

Since Gallivan's proof, people have gotten up to quite a bit of fun with this. In 2007, the MythBusters tried the experiment and got nearly as far—but needed heavy machinery and used multi-directional folding, requiring a truly gigantic piece of paper to start with. Take a look:

Then in 2012, students at St. Mark's School in Southborough, Massachusetts visited MIT to attempt 13 single-direction folds. They didn't actually use Gallivan's single-sheet method, instead choosing to layer the first 64 sheets (equivalent to six folds) on top of each other and then begin the folding, but this is still a lot of fun:

For more on Gallivan's achievement (and the math), read this page from The Historical Society of Pomona Valley.

See also: Folding Space-Time Using a Music Box

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History
When Math Discoveries Led to Banned Numbers
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iStock

The literature world has seen more than its share of controversy. The best stories tend to provoke the strongest reactions—both positive and negative—in readers, which is why so many classic books have been banned at one point or another. But even a more objective field like math isn’t immune to conflict. In its new video, TED-Ed rounds up the numbers that caused such a stir when they were introduced that they were banned in math circles.

One of the earliest examples comes from ancient Greece. A mathematician named Hippasus was having trouble solving certain equations with fractions and whole numbers alone, so he came up with irrational numbers to make these values easier to express. The ruling school of thought at the time dictated that everything in nature could be explained elegantly with the numbers that already existed. Threatened by Hippasus’s new notion, his fellow mathematicians rejected the irrational numbers and had him exiled.

Other numbers have been banned for legal reasons. When Arab traders brought their positional number system, which included zero, to Italy in the Middle Ages, Florence banned it from record-keeping fearing that they would be easier to forge than Roman numerals. The Arabic way of counting also led to the rise of negative numbers, which were regarded with disdain by many experts into the 19th century. For more banned numbers, including some that are prohibited today, check out the full story below.

[h/t TED-Ed]

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Euclid
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Live Smarter
An Ex-Google Engineer Just Reinvented the Measuring Cup
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Euclid

Recognizing a problem most people didn’t even know they had, former Google and Facebook software engineer Joshua Redstone has made a bold claim for his recent Kickstarter venture: He’s developed a better measuring cup.

According to the Boston Business Journal, Redstone spent four years tinkering with a solution to something that had long annoyed him as an amateur chef: Traditional measuring cups, which are stocky and not very well tapered, don’t do a great job of accurately measuring their own contents. Redstone believes the shape of a cup determines its success, particularly when a cook overfills a liquid or solid by a tiny amount. The smaller the volume, the more the problem is magnified.

Euclid

Redstone’s cup, Euclid, resolves the issue. According to the Kickstarter page: “With traditional measuring cups, the smaller the amount, the harder it is to measure accurately. The culprit? The shape. Straight sides magnify errors when measuring lower down in the cup. Some have tried to solve this problem with conical measuring cups, but their results fall short of Euclid’s by up to 60 percent. Euclid is the only measuring cup with a mathematically optimal, tapered design for consistent accuracy across amounts.”

Euclid is just about ready to overshoot its $30,000 Kickstarter goal. Backers can pay $24 for the cup now, or wait until it’s available at retail for a slightly higher price to be determined. The cup is scheduled for release in May 2018.

[h/t Boston Business Journal]

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