Boxing The Ring
August 12, 2024 2:40 AM Subscribe
A solution to an ancient geometry problem — how to slice up a circle (into about 10200 pieces) and rearrange it into a square.
I now present the nerdiest math joke that I know:
Q: What's an anagram of Banach-Tarski?posted by Johnny Assay at 4:25 AM on August 12 [26 favorites]
A: Banach-Tarski Banach-Tarski.
Did they use more than a straightedge and compass?
posted by GenjiandProust at 5:09 AM on August 12 [5 favorites]
posted by GenjiandProust at 5:09 AM on August 12 [5 favorites]
Worse, they use the (shudder) axiom of choice!
posted by SaltySalticid at 5:34 AM on August 12 [9 favorites]
posted by SaltySalticid at 5:34 AM on August 12 [9 favorites]
Math like this makes my brain hurt. “We figured out you can break the circle into a bunch of defined pieces! But we can’t actually describe any of the pieces.”
How in the heck can one prove that this is possible if the pieces are indescribable? How are the pieces defined if they can’t be described in any way?
This, friends, is why in my career I stayed with simple pursuits, like “how do brains work”
posted by caution live frogs at 5:38 AM on August 12 [8 favorites]
How in the heck can one prove that this is possible if the pieces are indescribable? How are the pieces defined if they can’t be described in any way?
This, friends, is why in my career I stayed with simple pursuits, like “how do brains work”
posted by caution live frogs at 5:38 AM on August 12 [8 favorites]
How in the heck can one prove that this is possible if the pieces are indescribable? How are the pieces defined if they can’t be described in any way?
Reading this, the behavior of much software becomes more comprehensible (though still not explicable) when you keep in mind that computer science is a branch of mathematics.
posted by Ayn Marx at 5:56 AM on August 12 [1 favorite]
Reading this, the behavior of much software becomes more comprehensible (though still not explicable) when you keep in mind that computer science is a branch of mathematics.
posted by Ayn Marx at 5:56 AM on August 12 [1 favorite]
Even more nuts:
Those well-defined pieces from the circle do not fill the entire square. Additional pieces are still necessary to cover a tiny portion of the square. This portion is so tiny that it has no area, and mathematicians refer to it as a “set of measure zero.”
Scientist: "The area of the parts we divided the circle up into do not fill the area of the square."
Me: "How much area does the missing parts consist of?"
Scientist: "Zero."
Me: "So the circle DOES fill the area of the square."
Scientist: *sighs aggressively and starts rubbing his temples* Now, listen here...
posted by AzraelBrown at 6:07 AM on August 12 [13 favorites]
Those well-defined pieces from the circle do not fill the entire square. Additional pieces are still necessary to cover a tiny portion of the square. This portion is so tiny that it has no area, and mathematicians refer to it as a “set of measure zero.”
Scientist: "The area of the parts we divided the circle up into do not fill the area of the square."
Me: "How much area does the missing parts consist of?"
Scientist: "Zero."
Me: "So the circle DOES fill the area of the square."
Scientist: *sighs aggressively and starts rubbing his temples* Now, listen here...
posted by AzraelBrown at 6:07 AM on August 12 [13 favorites]
How in the heck can one prove that this is possible if the pieces are indescribable?
The pieces aren't necessarily indescribable. The authors just don't know what they are.
It's like a ship in a bottle: we know that the ship got in there, but we don't know how. (Well I am sure I could just Google that -- but imagine that I am the first person to be presented with such a thing).
We know that the ancient Egyptians built the pyramids. We don't know precisely how. We know that bombardier beetles evolved. Etc.
posted by novalis_dt at 6:24 AM on August 12
The pieces aren't necessarily indescribable. The authors just don't know what they are.
It's like a ship in a bottle: we know that the ship got in there, but we don't know how. (Well I am sure I could just Google that -- but imagine that I am the first person to be presented with such a thing).
We know that the ancient Egyptians built the pyramids. We don't know precisely how. We know that bombardier beetles evolved. Etc.
posted by novalis_dt at 6:24 AM on August 12
> Worse, they use the (shudder) axiom of choice!
They don't! On page 20, 3.4. Making the Pieces Borel.
posted by lucidium at 6:28 AM on August 12 [6 favorites]
They don't! On page 20, 3.4. Making the Pieces Borel.
posted by lucidium at 6:28 AM on August 12 [6 favorites]
[i]Me: "So the circle DOES fill the area of the square."[/i]
Well, suppose you had pieces that covered all of the circle except for one point, its center. A single point has zero area. So you've covered 100% of the area of the circle. But you haven't covered the whole circle, you've covered the circle except for one point. That's the distinction that's being made.
posted by escabeche at 6:45 AM on August 12 [7 favorites]
Well, suppose you had pieces that covered all of the circle except for one point, its center. A single point has zero area. So you've covered 100% of the area of the circle. But you haven't covered the whole circle, you've covered the circle except for one point. That's the distinction that's being made.
posted by escabeche at 6:45 AM on August 12 [7 favorites]
Worse, they use the (shudder) axiom of choice!
I told you not to do it!
posted by Navelgazer at 6:58 AM on August 12 [3 favorites]
I told you not to do it!
posted by Navelgazer at 6:58 AM on August 12 [3 favorites]
> How in the heck can one prove that this is possible if the pieces are indescribable? How are the pieces defined if they can’t be described in any way?
Very much in mathematics is the result of understanding that if P is true, Q must also be true, and finding the P that appears to have nothing to do with Q that can easily be shown to be true. In this way you get proof of the existence of Q without any details of what Q is like, other than by recondite mapping via P (which has nothing to do with aspects of Q that are obvious, and can only reveal a tiny subset of what Q describes).
posted by Aardvark Cheeselog at 7:34 AM on August 12 [3 favorites]
Very much in mathematics is the result of understanding that if P is true, Q must also be true, and finding the P that appears to have nothing to do with Q that can easily be shown to be true. In this way you get proof of the existence of Q without any details of what Q is like, other than by recondite mapping via P (which has nothing to do with aspects of Q that are obvious, and can only reveal a tiny subset of what Q describes).
posted by Aardvark Cheeselog at 7:34 AM on August 12 [3 favorites]
Thanks lucidium, I was pretty sure the earlier results must rest on uncountable choice but I had a suspicion part of the latest stuff was that they didn't. Mostly I wanted to make a joke about how freaky the axiom of choice is :)
posted by SaltySalticid at 9:11 AM on August 12 [1 favorite]
posted by SaltySalticid at 9:11 AM on August 12 [1 favorite]
Without wishing to be a geometry snob, 'an ancient geometry problem' hasn't been solved, just a goalpost-changing restatement, a whipper-snapper at barely a century old.
Not that it isn't an achievement, just that it isn't the achievement promised by the breathless headline, and headlines like that need to get off my lawn. buys, shakes walking stick
posted by BCMagee at 9:45 AM on August 12 [3 favorites]
Not that it isn't an achievement, just that it isn't the achievement promised by the breathless headline, and headlines like that need to get off my lawn. buys, shakes walking stick
posted by BCMagee at 9:45 AM on August 12 [3 favorites]
If your pieces of circle covered all the points of the square except those with rational coordinates, that would still only leave out a set of measure zero.
posted by jamjam at 12:01 PM on August 12 [2 favorites]
posted by jamjam at 12:01 PM on August 12 [2 favorites]
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posted by HearHere at 4:10 AM on August 12 [1 favorite]