Pythagoras Punches Proverb

January 30, 2020 8:58 AM

Math proves that Round Peg can Fit into a Square Hole. "If that still leaves you scratching your head, be sure to watch how [Stanford mathematician] Tokieda folds the paper. He does so in a specific manner that transforms the sheet from two to three dimensions. In doing that, he brings two sides of a square together and forms a larger opening for which the coaster can pass through without a problem."
posted by storybored (16 comments total)

As soon as I read the description, I knew exactly what he did without seeing the video. I love this sort of spatial thinking!
posted by Big Al 8000 at 9:05 AM on January 30, 2020

When I was a kid, there was a similar trick where you could pass a nickel through a dime-sized hole. (For non USians, the nickel is a little larger than a dime coin.)

1. Start by tracing a circle around a dime onto a piece of paper.

2. Then fold the paper once, so the crease runs right across the hole, and the hole is facing down. The folded paper now looks like a little semicircle was cut out of the bottom of it.

3. Pick up the paper and place the nickel inside the folded paper, sitting on top of the hole. You will see the larger coin protruding a bit out of the gap.

4. Now pull the top corners of the folded paper toward each other. As you do so, the cut-out “mouth” will widen, just enough for the nickel to pass through.

This was widely shared in my social group of elementary school kids back in the ‘70s.
posted by darkstar at 9:08 AM on January 30, 2020 [4 favorites]

Yeah, I remember something similar from kid's TV when I was much younger, along these lines.
posted by biffa at 9:09 AM on January 30, 2020 [1 favorite]

I recommend skipping the linked article, which adds nothing, and going direct to the video.
posted by timdiggerm at 9:11 AM on January 30, 2020 [4 favorites]

biffa, that’s the one!
posted by darkstar at 9:12 AM on January 30, 2020

That reminds me of the childhood trick to step through a hole in a postcard.
posted by Big Al 8000 at 9:12 AM on January 30, 2020 [10 favorites]

Always easy for Topologists.
posted by sammyo at 9:36 AM on January 30, 2020

There is a bit more about Tokieda's curiosities in this Quanta article from a couple years ago. He had a pretty interesting path into mathematics, starting as a philology professor and then transitioning via an accidentally discovered biography of physicist Lev Davidovich Landau:

Landau says to his son [Igor] “Igor, you’re here. What’s the indefinite integral of dx over sin x?”
Well, Igor takes out a scratch sheet of paper and starts doing calculations, but somehow he can’t get it. Landau says, “Igor, you regard yourself as an educated adult, yet you’re incapable of performing such a simple task.”

When I read this, I took it as a personal criticism. I regarded myself, rather arrogantly, as a very educated person, but I had never heard of calculus in my life. I hadn’t the slightest idea what this sequence of symbols meant.

I decided, as a personal revenge on Landau, to study the subject up to the point where I could solve this exercise.
posted by Think_Long at 10:11 AM on January 30, 2020 [5 favorites]

Er, just noticed an omission in what I wrote for first step in doing this trick. After you trace the circle, you obviously have to cut out the hole. :)
posted by darkstar at 10:11 AM on January 30, 2020 [2 favorites]

I consider myself reasonably well-read and versed in math/geometry tricks, but that...literally made me gasp.

Cool!
posted by notsnot at 10:25 AM on January 30, 2020

Thanks for that article Think_Long! It is a worthy read, his outlook reminds me a bit of Vonnegut (crossed with Feynman):

One very common question that comes up at the end of a lecture is, “Does all this have any practical applications?” It’s really intriguing because this question is asked in almost exactly the same words wherever I go. It’s like listening to a prerecorded message.

I ask them, what do you think constitutes a practical application? It’s very surprising....people converge within five to 10 minutes onto two categories of practical applications. One is, if you manage to make several million dollars instantly. The other is, if you manage to kill millions of people instantly. Many people are actually kind of shocked by their own answers.

Then I tell them that, well, I don’t know about other people, but I have a practical application for my toys. When I show my toys to some children, they seem to be happy. If that’s not a practical application, what is?
posted by storybored at 10:30 AM on January 30, 2020 [9 favorites]

Igor, you regard yourself as an educated adult, yet you’re incapable of performing such a simple task

I just type it into Wolfram Alpha nowadays, what does that make me?
posted by polymodus at 11:47 AM on January 30, 2020

I, too, can perform seemingly impossible tasks by redefining the parameters.

It's a cute trick, but the circle didn't go through a square hole. A rectangular cross-section with length equal to the diameter and width equal to the coaster's thickness went through the hole. Which was rendered to no longer be a square, either.

Normally I try not to be too nit-picky, but if we're allowed to fold the square into a slot, why not just fold the coaster instead so that it fits through the square?
posted by explosion at 1:06 PM on January 30, 2020 [7 favorites]

This is the real world example of the old punchline that goes, "The mathematician says, 'If we define a square hole as...' "
posted by Therapeutic Amputations at 1:15 PM on January 30, 2020

OK, but isn't the proverb "a square peg into a round hole"?

Have I been saying it wrong all these years?
posted by madajb at 1:46 PM on January 30, 2020 [1 favorite]

This looks like the basis of a very profitable bar bet!
posted by sensate at 1:48 PM on January 30, 2020 [2 favorites]