### Math blogging: Banach-Tarski

Is it possible to slice a golf ball into pieces and reassemble the pieces to form the Statue of Liberty? Not a golf-ball-sized reproduction of the Statue of Liberty, mind you, but a life-sized one, solid and with no holes or gaps? According to a really bizarre mathematical paradox, the answer is "yes," at least theoretically.

As a physicist by training, I had more advanced courses in mathematics than is probably healthy. I first encountered the Banach-Tarski paradox in one of these classes about 20 years ago, and it has been one of my white whales ever since.

The paradox states that it is possible to decompose any arbitrary shape and size into a finite number of pieces, and, through a series of translations and rotations, reassemble those pieces to form another object *of arbitrary shape or size.*

I'll let that sink in for a moment. One popular formulation is that it's possible to deconstruct a pea and reassemble the pieces to form a sphere the size of the sun -- again, with no gaps or holes. The most classic formulation (and a rather boring one if you ask me) is that you can disassemble a sphere and reassemble the pieces to form *two* spheres equal in size to the first.

This is mathematically provable, but I had a *real* problem with this. As a physicist, it offended every notion I had about the conservation of mass. How can you magic up something out of thin air, even theoretically?

The odd thing is that I was largely alone in freaking out about Banach-Tarski. The mathematicians I knew didn't really freak out about it because they were accustomed to living in strange, bizzarro worlds that had no bearing on our physical reality. The physicists I knew were equally untroubled because they were accustomed to dismissing mathematicians. Banach-Tarski didn't apply to the real world, they'd say, because matter isn't arbitrarily divisible. We have atoms, after all.

Well yes, we *do* have atoms, and no one ever claimed that these kinds of bizarre decomposition would ever be possible in practice. But that wasn't good enough for me. It was dodging the issue. It's all too easy to imagine a universe in which matter *is* infinitely decomposable, but the laws of conservation of mass should still apply. As a physicist, I had been trained to view mathematics as a reliable tool for predicting the behavior of the physical world. If mathematics predicted nonsense like this, even in theory, then mathematics was a ass. There *had* to be some flaw in the reasoning, I figured.

Anyway, I have struggled with this paradox on and off over the years, and have finally, recently, come to an uneasy truce with it. I'd like to share my journey with this problem over the next weeks and months. It makes me regret that I never enabled categories on this blog, though. That decision made sense at the time, since pretty much everything I write here is either political in nature or else defies categorization altogether. Still, it would be cool if I could classify all these coming posts as "Math blogging," so that everyone besides Adam (thanks for the inspiration to math-blog) and Tami and a handful of others could ignore it.

But since I don't have categories, just look for the words "Banach-Tarski" in the headline in days and weeks to come. If you're math-averse, you might want to avoid those. More on this later.

## Comments

Don't worry about categories; I've never had any patience for them.

But you'll still get traffic from people searching for this stuff on Google. It's been pretty cool, actually, finding out that blogging about math not only

doesn'tmake all your readers go away, but in the long run, attracts new ones!Go Long Tail!

I look forward to being utterly confused by the math of this paradox.

Posted by: Adam | March 7, 2006 09:21 PM

Well, on the bright side you always have the out of using alternative set theories that don't rely on the axiom of choice. :)

Posted by: Matt McIntosh | March 8, 2006 12:36 AM

That's an interesting theory, though I can't see how it doesn't violate a basic Law of Thermodynamics - that matter and energy must be balanced/equal on both sides of the equation.

I don't understand how a mathmetician can postulate that you can creat a larger mass out of smaller one, without first theorizing something like, "given we can create mass/matter," in lieu of the ubiquitous a priori - "assuming a perfect vacuum," that seems to precede so many math theorems.

Are all the Laws of thermodynamics trumped by quatum mechanics?

Posted by: JMK | March 8, 2006 12:55 PM

JMK, you're confusing the model with the real world. Banach-Tarski requires that matter be infinitely divisible, which is not a property of matter in the real world. Hence speaking of it "violating" any physical law is just apples and oranges.

And no, the laws of thermodynamics are compatible with QM and hold in every context we know of -- even black holes.

Posted by: Matt McIntosh | March 8, 2006 03:29 PM

I'll take your word for that Matt, but that confuses the hell out of me.

The laws of thermodynamics balance the matter/energy on both sides of every reaction/equation, so that's a tough concept for folks like me to grasp.

Posted by: JMK | March 8, 2006 04:00 PM

Matt--is there any real-world predictive power to the theory?

Posted by: Adam | March 8, 2006 05:59 PM

One of the "tricks" of the paradox is that the pieces are infinitely complex... and not even continuous, in general. In other words, they can be an arbitrary collection of discrete points.

Posted by: BNJ | March 8, 2006 06:09 PM

there is no gap in reasoning with the banach-tarski paradox. some points

1:it has been proven in ZF set theory that the banach-tarski paradox is equiconsistant with the Hahn-Banach extension theorem. What this means is that if banach-tarski is in reality false so is the hahn-banach theorem. Take a look at a book on functional analysis and see what the consequences for hilbert spaces would be and by extension the mathematical model for quatum mechanics. The Hahn-Banach theorem is often describe as a conerstone of analysis. None of this involves choice so alternate set theories can't resolve the issue. my friend has a problem with ZF's power set axiom but he is to intelligent to be taken seriously. this work was done by dr. matt foreman he was at uci. if my memory serves me the proof uses forcing arguments.

2:the standard text on BT stan wagon's book presents the material in a way that renders the use of the axiom of choice mysterious. there are other proofs which use graph theory (bipartite matching theorems) that sort of clarify the issue.

3: Foreman and Doughetry poved a version of BT which runs as follows: it is possible to cut a pea into i think seven pieces and move them using rotations and translations, rigid motions, to form the sun. What's new? well the proof is in ZF using no choice of any form. the pieces form an OPEN set whose complementis nowhere dense in the pea and the same is true of the sun. What this means is that if you deleted the portions of the pea which are not used and the sun which are not filled up you couldn't see the difference. The original proof is very complicated.

4:the BT paradox is really about the uniqueness of the lebesgue integral and the possibility of extending it to an averaging functional on the entire powerset of a space loosing countable additivity but keeping invariance under some group action. The point is that this is not possible on the sphere. I wouldn't get overly worked up about geometric/mechanical metaphors like the tired pea-sun thing. physicist never tire of trying for mechanical explantations in realms where they don't apply (wave/particle duality?). I never know what it means to say that space is not "infinitely divisible" this may or may not be true i just don't know what it means. Again it is the metaphor thing. Am i to believe that the sqrt of 2 does not really exist where do you draw the line. All mathematicians mean by the continuum is something like dedikend cuts or equivalence classes of cauchy sequences. The possibility of uniqueness and extension theorems based on them.study the construction of the exponential functions to see what i mean. For sure we do not have infinite resolution in experimentation. As far as predictive power I read a paper linking hadron physics to BT but the argument was flawed because it relied on specifics of the wagon proofs which are not needed. I would venture the idea that BT like phenonmena are probably behind alot of the non-sense explanations in physics. what is so apriori about the sigma-algebra of lebesgue measurble sets anhyway?

I havn't thought about this for years so some of the specifics may be wrong.

Posted by: nathan | April 1, 2006 03:37 PM

there is no gap in reasoning with the banach-tarski paradox. some points

1:it has been proven in ZF set theory that the banach-tarski paradox is equiconsistant with the Hahn-Banach extension theorem. What this means is that if banach-tarski is in reality false so is the hahn-banach theorem. Take a look at a book on functional analysis and see what the consequences for hilbert spaces would be and by extension the mathematical model for quatum mechanics. The Hahn-Banach theorem is often describe as a conerstone of analysis. None of this involves choice so alternate set theories can't resolve the issue. my friend has a problem with ZF's power set axiom but he is to intelligent to be taken seriously. this work was done by dr. matt foreman he was at uci. if my memory serves me the proof uses forcing arguments.

2:the standard text on BT stan wagon's book presents the material in a way that renders the use of the axiom of choice mysterious. there are other proofs which use graph theory (bipartite matching theorems) that sort of clarify the issue.

3: Foreman and Doughetry poved a version of BT which runs as follows: it is possible to cut a pea into i think seven pieces and move them using rotations and translations, rigid motions, to form the sun. What's new? well the proof is in ZF using no choice of any form. the pieces form an OPEN set whose complementis nowhere dense in the pea and the same is true of the sun. What this means is that if you deleted the portions of the pea which are not used and the sun which are not filled up you couldn't see the difference. The original proof is very complicated.

4:the BT paradox is really about the uniqueness of the lebesgue integral and the possibility of extending it to an averaging functional on the entire powerset of a space loosing countable additivity but keeping invariance under some group action. The point is that this is not possible on the sphere. I wouldn't get overly worked up about geometric/mechanical metaphors like the tired pea-sun thing. physicist never tire of trying for mechanical explantations in realms where they don't apply (wave/particle duality?). I never know what it means to say that space is not "infinitely divisible" this may or may not be true i just don't know what it means. Again it is the metaphor thing. Am i to believe that the sqrt of 2 does not really exist where do you draw the line. All mathematicians mean by the continuum is something like dedikend cuts or equivalence classes of cauchy sequences. The possibility of uniqueness and extension theorems based on them.study the construction of the exponential functions to see what i mean. For sure we do not have infinite resolution in experimentation. As far as predictive power I read a paper linking hadron physics to BT but the argument was flawed because it relied on specifics of the wagon proofs which are not needed. I would venture the idea that BT like phenonmena are probably behind alot of the non-sense explanations in physics. what is so apriori about the sigma-algebra of lebesgue measurble sets anyway?

Posted by: Anonymous | April 1, 2006 03:38 PM

there is no gap in reasoning with the banach-tarski paradox. some points

1:it has been proven in ZF set theory that the banach-tarski paradox is equiconsistant with the Hahn-Banach extension theorem. What this means is that if banach-tarski is in reality false then so is the hahn-banach theorem. Take a look at a book on functional analysis and see what the consequences for hilbert spaces would be and by extension the mathematical model for quatum mechanics. The Hahn-Banach theorem is often describe as a conerstone of analysis. None of this involves choice so alternate set theories can't resolve the issue. my friend has a problem with ZF's power set axiom but he is to intelligent to be taken seriously. this work was done by dr. matt foreman he was at uci. if my memory serves me the proof uses forcing arguments.

2:the standard text on BT stan wagon's book presents the material in a way that renders the use of the axiom of choice mysterious. there are other proofs which use graph theory (bipartite matching theorems) that sort of clarify the issue.

3: Foreman and Doughetry poved a version of BT which runs as follows: it is possible to cut a pea into i think six pieces and move them using rotations and translations, rigid motions, to form the sun. What's new? well the proof is in ZF using no choice of any form. the pieces form an OPEN set whose complement is nowhere dense in the pea and the same is true of the sun. What this means is that if you deleted the portions of the pea which are not used and the sun which are not filled up you couldn't see the difference. The original proof is very complicated.

4:the BT paradox is really about the uniqueness of the lebesgue measure and the possibility of extending it to an averaging functional on the powers et of a space loosing countable additivity but keeping invariance under some group action. The point is that this is not possible on the sphere. I wouldn't get overly worked up about geometric/mechanical metaphors like the tired pea-sun thing. physicist never tire of trying for mechanical explantations in realms where they don't apply (wave/particle duality?). I never know what it means to say that space is not "infinitely divisible" this may or may not be true i just don't know what it means. Again it is the metaphor thing. Am i to believe that the sqrt of 2 does not really exist where do you draw the line. All mathematicians mean by the continuum is something like dedikend cuts or equivalence classes of cauchy sequences. The possibility of uniqueness and extension theorems based on them.study the construction of the exponential functions to see what i mean. For sure we do not have infinite resolution in experimentation. As far as predictive power I read a paper linking hadron physics to BT but the argument was flawed because it relied on specifics of the wagon proofs which are not needed. I would venture the idea that BT like phenonmena are probably behind alot of the non-sense explanations in physics. what is so apriori about the sigma-algebra of lebesgue measurble sets anhyway?

i am not sure i know how to post this so sorry if this is the third time

Posted by: nathan | April 1, 2006 03:42 PM

there is no gap in reasoning with the banach-tarski paradox. some points

1:it has been proven in ZF set theory that the banach-tarski paradox is equiconsistant with the Hahn-Banach extension theorem. What this means is that if banach-tarski is in reality false then so is the hahn-banach theorem. Take a look at a book on functional analysis and see what the consequences for hilbert spaces would be and by extension the mathematical model for quatum mechanics. The Hahn-Banach theorem is often describe as a conerstone of analysis. None of this involves choice so alternate set theories can't resolve the issue. my friend has a problem with ZF's power set axiom but he is to intelligent to be taken seriously. this work was done by dr. matt foreman he was at uci. if my memory serves me the proof uses forcing arguments.

3: Foreman and Doughetry poved a version of BT which runs as follows: it is possible to cut a pea into i think six pieces and move them using rotations and translations, rigid motions, to form the sun. What's new? well the proof is in ZF using no choice of any form. the pieces form an OPEN set whose complement is nowhere dense in the pea and the same is true of the sun. What this means is that if you deleted the portions of the pea which are not used and the sun which are not filled up you couldn't see the difference. The original proof is very complicated.

4:the BT paradox is really about the uniqueness of the lebesgue measure and the possibility of extending it to an averaging functional on the powers et of a space loosing countable additivity but keeping invariance under some group action. The point is that this is not possible on the sphere. I wouldn't get overly worked up about geometric/mechanical metaphors like the tired pea-sun thing. physicist never tire of trying for mechanical explantations in realms where they don't apply (wave/particle duality?). I never know what it means to say that space is not "infinitely divisible" this may or may not be true i just don't know what it means. Again it is the metaphor thing. Am i to believe that the sqrt of 2 does not really exist where do you draw the line. All mathematicians mean by the continuum is something like dedikend cuts or equivalence classes of cauchy sequences. The possibility of uniqueness and extension theorems based on them.study the construction of the exponential functions to see what i mean. For sure we do not have infinite resolution in experimentation. As far as predictive power I read a paper linking hadron physics to BT but the argument was flawed because it relied on specifics of the wagon proofs which are not needed. I would venture the idea that BT like phenonmena are probably behind alot of the non-sense explanations in physics. what is so apriori about the sigma-algebra of lebesgue measurble sets anhyway?

Posted by: nathan | April 1, 2006 04:27 PM

the above is a drunken rampage

Posted by: nathan | April 8, 2006 04:05 PM

4/1/06

Posted by: nathan | April 8, 2006 04:06 PM