My Math Forum Volume paradox -- Banach–Tarski

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April 6th, 2017, 09:03 PM   #31
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Quote:
 Originally Posted by Loren What part does porosity play?
Don't understand the question. Do you mean the nature of the nonmeasurable piece(s)? Nonmeasurable sets are very strange. They can't be visualized, only proved to exist via nonconstructive methods.

 April 16th, 2017, 08:45 PM #32 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 435 Thanks: 28 Math Focus: Number theory Porosity is referenced on one of the aforementioned websites. I was curious whether it had anything to do with density. Take a ball of radius R. Slice it into alternating spherical shells of infinitesimal thickness. Reconstruct the odd or even shells into one ball each of radius R.
April 17th, 2017, 10:22 AM   #33
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Quote:
 Originally Posted by Loren Porosity is referenced on one of the aforementioned websites. I was curious whether it had anything to do with density.
Porosity has no meaning at all. It's from the Wiki article on Banach-Tarski. As I've mentioned, IMO the proof sketch there is very good; but the surrounding handwavy exposition is quite misleading. This is a prime example.

What they are trying to say is that one of the pieces must be nonmeasurable. I might use words like "weird" or "hard to visualize" for nonmeasurable sets, but porous wouldn't be a word I'd use. A nonmeasurable set is porous in the sense that it contains no contiguous intervals; but you could say the same for the rationals or the irrationals, both of which are porous in exactly the same way, yet perfectly measurable.

I'd ignore the word porosity. It's a Wikipedia error.

Quote:
 Originally Posted by Loren Take a ball of radius R. Slice it into alternating spherical shells of infinitesimal thickness. Reconstruct the odd or even shells into one ball each of radius R.
There are no infinitesimals in the real numbers, so this doesn't actually make sense despite its intuitive appeal.

But there's no problem generating paradoxes with cardinality. We can decompose the counting numbers 1, 2, 3, ... into the evens and the odds; and each of those can be bijected to the original set of natural numbers.

We could do the same thing with a ball in 3-space, using a cardinality argument to make two balls out of one.

What's amazing about the Banach-Tarski theorem is that it shows that we can do this not with a mere cardinality argument -- after all, cardinality is a very weak and counterintuitive measure of size -- but with rigid motions, or isometries.

That is, we can decompose the ball in 3-space into as few as five pieces, move the pieces around space via rigid, distance-preserving motions, and end up with two balls, each the same size as the original.

Last edited by Maschke; April 17th, 2017 at 10:24 AM.

 April 17th, 2017, 03:23 PM #34 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 435 Thanks: 28 Math Focus: Number theory What is the fourth (lower right-hand), unlabled set on the Cayley graph at https://en.wikipedia.org/wiki/Banach...sition_F_2.svg Is the Cayley graph of 2-, 3- or other dimensions? Can Banach–Tarski be understood by means of fractals? Are isometries restricted to the Cayley graph in a nonmeasurable but continuous way?
April 17th, 2017, 03:52 PM   #35
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Quote:
 Originally Posted by Loren What is the fourth (lower right-hand), unlabled set on the Cayley graph at https://en.wikipedia.org/wiki/Banach...sition_F_2.svg
Actually the labels in that picture are off, I've been meaning to point this out on the Talk page for that article. The idea is the library of Babel. If you take all the words generated by two letters and their inverses starting with 'a', such as

$ab^2a^{-5}$ or $a^{12}b^{47}$ and so forth, if we hit them all on the left with $a^{-1}$, we end up with every possible word except the ones that begin with a single 'a'. To get to this point I need to do some more exposition. I have a post on the free group on ONE letter, which is a helpful prereq to the free group on two letters. I'll post that soon.

Perhaps if I get time in the next few days I'll answer your direct question in more detail. The point is that the set of "words" made up of two letters and their inverses is paradoxical, it can be split into subsets that can be rotated back to form two copies of the original set. This is the fundamental idea that we can then lift into 3-space in the next part of the proof.

Quote:
 Originally Posted by AplanisTophet Is the Cayley graph of 2-, 3- or other dimensions?
The picture shown is the graph of $F_2$, the free group on two letters. It's all we need for Banach-Tarski. There are of course free groups with 3 generators, 4, 5, and even infinitely many generators, but we don't need them for B-T.

You've motivated me to continue with my exposition so I'll start talking about free groups soon.

Quote:
 Originally Posted by AplanisTophet Can Banach–Tarski be understood by means of fractals?
Heck if I know. I don't think it can be "understood" at all. The best we can do is try to follow the logic. All proofs involving the axiom of choice are like that. I've never seen anything describing any connection to fractals but that doesn't mean there isn't some connection somewhere. I don't personally think this is a productive line of inquiry else someone would have a web page about it.

Quote:
 Originally Posted by AplanisTophet Are isometries restricted to the Cayley graph in a nonmeasurable but continuous way?
Isometries are just rigid motions. If I take an interval of real numbers and slide it one way or the other along the line, that's an isometry.

The Cayley graph doesn't have anything to do with isometries. In this step of the proof we're only showing that $F_2$ is paradoxical. Then in the next step, we show that the isometry group of 3-space contains a copy of $F_2$ and use that fact to get the B-T paradox.

 April 17th, 2017, 05:15 PM #36 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 435 Thanks: 28 Math Focus: Number theory Is AplanisTophet an anagram?
April 18th, 2017, 12:28 AM   #37
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Quote:
 Originally Posted by Loren What is the fourth (lower right-hand), unlabled set on the Cayley graph at
Ok first realize that this discussion is really two more lengthy posts away in my exposition. My next post will be about the free group on ONE letter; then after that we'll talk about the free group on TWO letters, and all this will make sense. So this current post is getting a little ahead of the story, but you asked a about the Cayley graph and this is really the heart of the proof anyway so I'll explain this now.

To begin with, the picture of the Cayley graph on the Wiki page is MISLABELED. I will fix this for you here.

In what follows I'm using this picture of the same graph but drawn a little differently on this page: https://en.wikipedia.org/wiki/Free_group. It's the same picture but a little simpler.

So now first, here is the picture with the properly labeled quadrants.

Say we start in the center at the identity word $e$. (All undefined terminology is defined in the Wiki pages and will be defined in the next two articles on free groups, as I say I'm a little ahead of myself here).

Now the first letter of our word might be $a$, $b$, $a^{-1}$, or $b^{-1}$. This corresponds to branching to the right, up, left, or down subgraphs, respectively.

Now say our word starts with $a$. What are the possible choices for the second letter? They can be $a$, which would result in $a^2$; or $b$, resulting in $ab$; or $b^{-1}$, resulting in $ab^{-1}$.

But what CAN'T be the second letter is $a^{-1}$; because that would reduce to $aa^{-1} = e$. That's why once you make your initial move into one of the four quadrant subgraphs, from then on there are only three possible next choices. [Did I say that clearly?]

So now all our non-identity words fall into four classes, representing the east, north, west, and south subgraphs. These are:

* $S(a)$, the set of words starting with $a$;

* $S(b)$, the set of words starting with $b$;

* $S(a^{-1})$, the set of words starting with $a^{-1}$;

* $S(b^{-1})$, the set of words starting with $b^{-1}$.

Reasonable, yes?

So we can write: $F_2 = \{e\} \cup S(a) \cup S(b) \cup S(a^{-1}) \cup S(b^{-1})$.

Now, the magic happens. This is the mathematical version of the tower of Babel.

Suppose we take every word in $S(a^{-1})$ and hit it on the left by $a$. In other words we're "translating" it by $a$, which simply means applying $a$ to every word in $S(a^{-1})$. What happens?

* Any word that began $a^{-1}bx$ where $x$ is any continuation, now becomes $bx$. In other words $S(b) \subset aS(a^{-1})$.

* Likewise any word that began $a^{-1}b^{-1}x$ now becomes $b^{-1}x$. So $S(b^{-1}) \subset aS(a^{-1})$.

* The word $a^{-1} \in S(a^{-1})$, so $aa^{-1} = e\in aS(a^{-1})$.

* If a word was already in $S(a^{-1})$, let's say $a^{-5}b$ for example, hitting it on the left with $a$ simply reduces the exponent to $a^{-4}b$. So all the words starting with $a^2$ and higher exponents end up in $aS(a^{-1}$. And the words starting with $a^{-1}b$ or $a^{-1}b^{-1}$ we already covered, they end up in $aS(a^{-1})$ as noted.

So what did we just prove? That $F_2 = aS(a^{-1}) \cup S(a)$. We took one of the four quadrants, translated it, and it ate two of the other quadrants! That's the heart of the proof right there.

Now I didn't draw a picture, but in the exact same way we can see that $F_2 = bS(b^{-1}) \cup S(b)$.

So we can start with the decomposition $F_2 = \{e\} \cup S(a) \cup S(b) \cup S(a^{-1}) \cup S(b^{-1})$, translate two of the pieces, and end up with TWO copies of $F_2$.

Now all that remains is to translate this magic to Euclidean 3-space, which we'll do later by showing that the isometry group of 3-space contains a copy of $F_2$.

This is the heart of the proof. I'd just add that none of this is completely obvious the first time through, you have to work at this a bit to see what's going on with all the notation and such, and play around with some examples to see how $aS(a^{-1})$ works with the translating.

Quote:
 Originally Posted by Loren Can Banach–Tarski be understood by means of fractals?
Now that I've gone through this in detail I see your point. This has something to do with the fractal nature of the quadrants. Tower of Babel again, each sub-library contains a copy of all the other sub-libraries. But this isn't used directly in the proof, it's sort of lurking in the background.

Last edited by Maschke; April 18th, 2017 at 12:41 AM.

April 23rd, 2017, 08:38 PM   #38
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Maschke,

Quote:
 Now all that remains is to translate this magic to Euclidean 3-space, which we'll do later by showing that the isometry group of 3-space contains a copy of F2.

 April 27th, 2017, 09:31 PM #39 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 435 Thanks: 28 Math Focus: Number theory Isn't "doubling the cube," proved impossible in 1837, a form of Banach–Tarski?
April 28th, 2017, 05:45 AM   #40
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Quote:
 Originally Posted by Loren Isn't "doubling the cube," proved impossible in 1837, a form of Banach–Tarski?
That shows that a particular compass and straightedge construction is impossible. Different problem. You can't construct (with compass and straightedge) a line of length $\sqrt[3]{2}$. Of course there's no problem having a cube that's double the volume of some other cube. You just can't do a classical construction.

https://en.wikipedia.org/wiki/Doubling_the_cube

Last edited by Maschke; April 28th, 2017 at 05:51 AM.

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