Here is another one to bamboozle your brain. Can you pass a circular disk through a slightly smaller square hole?

Instinct tells you no. But you haven’t seen this latest video from Numberphile, in which Standford University’s Tadashi Tokieda demonstrates that, by folding a sheet of paper in just the right way, a round peg really can go in a square hole.

“I made a square hole in this sheet of paper, and a coaster, a circular coaster,” explains Tokieda in the video. “I fold the paper in a mysterious fashion, and I can pass the coaster through the hole.”

But, as Tokieda stresses, he is not cheating at all. “I didn’t stretch, let alone tear, and yet when I fold the sheet back in a judicious way, the coaster does go through the square that is bigger than the hole. How is this possible?”

Well thankfully for the likes of you and me, Tokieda doesn’t leave us baffled for too long, and gets down to explaining exactly how he achieved the seemingly impossible.

“I’m willing to give away the secret for free on this occasion,” Tokieda says. “It has to do with the intrinsic, or inner dimension, of this piece of paper, which is two dimensions, and the fact that this sheet evolves, or flourishes, in the ambient three dimensional space. There is some elbow room, there is some ambient space.”

Ahhh, well I’m glad that’s cleared up!

Nope? Still as confused as us? Well it has to do with the fact that, while in two dimensions the hole is indeed too small for the coaster to fit through, by taking the paper into three dimensions, you are able to bring two sides of the square together, which forms a wider slit than the disk and allows it to pass through.

“This is all possible because when we do this maneuver, you’re allowing the whole thing to come out into 3D and then come back down [into 2D],” Tokieda continues. “This fact that you can escape into the ambient third dimension and come back in... gives you this.”

If you’re still a little fuzzy, then watch Tokieda in action. He’s much better at explaining it all, and there is even some Pythagoras thrown in. And you thought you’d never need to use that math again, didn’t you?

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