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April 2nd, 2017, 01:47 PM   #21
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F sub 2 seems to be a fractal, which I know somewhat, since it is visual, nondifferentiable and self-similar (and simple to generate).

Last edited by Loren; April 2nd, 2017 at 01:50 PM.
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April 2nd, 2017, 02:09 PM   #22
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F sub 2 seems to be a fractal, which I know somewhat, since it is visual, nondifferentiable and self-similar (and simple to generate).
That's an interesting idea. I think you're right. It's precisely the self-similarity that makes it paradoxical. (Paradoxical is a technical term for the condition in the next paragraph). But then it turns out that the collection of rigid motions of 3-space contains a copy of $F_2$. that's the heart of the proof. The fractal nature of $F_2$ induces very strange behavior in Euclidean 3-space. That's what Banach and Tarski worked out in 1924.

Now the next step for us, if you have interest, is to very carefully, line by line, go through the Wiki explanation of how $F_2$ is paradoxical. It is the union of five of its proper subsets. It's also the union of one of those subsets, unioned with another subset "rotated" by a constant; and it's like that in two different ways. A set that can be so decomposed is called paradoxical.

I propose working through this idea until is it crystal clear to all of us that $F_2$ is a paradoxical set.

To that end, we must walk through the symbology line by line. I'm writing up an explanation, which I'll post in the next day or two. It will be simpler and more detailed than the Wiki exposition.

Meanwhile feel free to ask any questions that you have.

Last edited by Maschke; April 2nd, 2017 at 02:15 PM.
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April 2nd, 2017, 08:58 PM   #23
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Does the rigid "rotation" in 3-space have anything to do with the Levi-Civita symbol/function yielding antisymmetry (paradoxical here?) for odd dimensions? Just a wild guess.

https://en.wikipedia.org/wiki/Levi-Civita_symbol

The axes for rotations in 3-space are 1-space singular -- "unmeasurable" perhaps?

I like your last two posts, Maschke. Getting to be understandable. I will try to keep up. Thank you for your offer. Back to Wiki B-T.
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April 2nd, 2017, 11:18 PM   #24
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Does the rigid "rotation" in 3-space have anything to do with the Levi-Civita symbol/function yielding antisymmetry (paradoxical here?) for odd dimensions? Just a wild guess.
Seems like something completely different. But let's talk about rigid motions, that's a nice place to start.

* The simplest case is the real line. What's a rigid motion in one dimension? Just a translation left or right by a constant. A function $f(x) = x + k$ where $k$ is some constant, positive for a right shift and negative for a left shift. And a shift by $0$ leaves everything where it was.

Example Take the closed unit interval $[0,1]$. If I shift it left to, say, $[-5, -4]$, that's a rigid motion. I haven't changed its length.

On the other hand if I multiply every real number by $2$, that maps $[0,1]$ to $[0,2]$. That's a stretch. It's not a rigid motion.

It's like if we put a metal stick on the real line and moved it around. It can only go left or right by some constant $k$. No shrinking or expanding.

Another type of rigid motion in the plane is reflection through a point. The interval $[0,1]$ reflected in the origin becomes $[-1,0]$.

* How about the plane? Clearly translating a set of points is a rigid motion. It's like adding a constant vector to each vector in the plane. A translation in two space is still a function $f(x) = x + k$ but now $x$ and $k$ are $2$-vectors instead of real numbers but everything else is the same.

There are also reflections in a line, for example mirroring the plane across the line $y = x$.

But wait there's more! More dimensions give you more freedom. Translations and rotations aren't the only rigid motions in two space. There are rotations, too. I can rotate the plane through and angle around a particular point. In two-space every rigid motion is a composition of a translation and a rotation.

A rigid motion is also called an isometry and since that's fewer letters to type I'll use it sometimes.

In dimensions one and two, composition of isometries is commutative. In one dimension if I shift by $a$ and then by $b$ it's the same as if I'd done it in reverse order. In the plane, order also doesn't matter when you're composing translations and rotations. Throwing in reflections is commutative too.

* Dimension three. Now this is where it gets interesting. Just as in the plane, the isometries consist of translations and rotations around lines and reflections in planes. But we can rotate three-space around any straight line in space. We can rotate an object through some angle around the $x$-axis or the $y$-axis or the $z$-axis or any straight line at all. And because of all this rotational freedom, composition of isometries in three space is not necessarily commutative. And it's exactly because of this fact that there is a Banach-Tarski paradox in three space and not in the plane.

Last edited by Maschke; April 2nd, 2017 at 11:28 PM.
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April 3rd, 2017, 11:55 AM   #25
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Quote:
Translations and rotations aren't the only rigid motions in two space. There are rotations, too.
Did you mean to say "There are reflections, too"?


Is what you are describing in three dimensions a particular symmetry group, or does a group have to be commutative?

In 3-space, isn't a final state/orientation dependent on the number and order of rotations?

You are doing well, Maschke.

Aside: is the set of these operations in 3-space of higher cardinality than that of 2-space?
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April 3rd, 2017, 02:09 PM   #26
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Quote:
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Did you mean to say "There are reflections, too"?
Yes, thanks.

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Is what you are describing in three dimensions a particular symmetry group, or does a group have to be commutative?
Yes, the set of isometries of Euclidean $n$-space is an example of a group. I wrote up a post about that. A group need not be commutative, or Abelian as it's called in honor of Niels Henrik Abel. His story is far less well known than that of Galois, but just as tragic. Abel was the first person to prove the unsolvability of the quintic (before Galois) and made many other fundamental contributions to math, before dying broke and unknown at age 27.

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In 3-space, isn't a final state/orientation dependent on the number and order of rotations?
Exactly right. The isometry group in dimension 1 and 2 is Abelian. In dimensions 3 and above, it's nonabelian. That's what makes the paradox work as von Neumann noted in the Wiki article.

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You are doing well, Maschke.
You too Loren! We're making great progress. We now know what is an isometry group (or will after my next post). Once we show that $F_2$ is paradoxical [to be defined] and that the isometry group in 3-space contains a copy of $F_2$, we can extend the paradoxical decomposition to the unit ball in 3-space to prove the theorem.


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Aside: is the set of these operations in 3-space of higher cardinality than that of 2-space?
Hmmm. We know that $n$-space has the same cardinality regardless of $n$, which was a big surprise to Cantor and everyone else. The sets of bijections of each $n$-space must all have the same cardinality. Isometries are distance-preserving bijections. I can't think of any reason that would cause a problem but you never know. So I'd guess the isometry groups all have the same cardinality but I don't have a proof offhand. (You'd have to characterize all the distance-preserving maps and count them).

Last edited by Maschke; April 3rd, 2017 at 02:18 PM.
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April 4th, 2017, 04:36 PM   #27
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Isometry groups

I'm going to push ahead with this but no hurry with anything. Since you mentioned isometry groups earlier I wanted to just provide some background.

If you think about the one, two, and three-dimensional cases (and the n-dimensional case if you have a good imagination), composition of isometries has some important properties.

* The composition of two isometries is another isometry. If I make a rigid (length-preserving) motion and then do another one, I get their combined rigid motion. We say that the collection of all isometries in a given dimension is closed under composition.

* Composition of isometries is associative. This follows from the fact that isometries are just functions, and composition of functions is associative.

* There's always an isometry that doesn't do anything. Translation by zero, for example. "Doing nothing is doing something," if you want to be philosophical

* For any isometry, there's another one that "undoes it." If I shift by $3$ units I can shift left by $-3$ and end up back where I started. If I rotate around the origin in the plane by $\frac{\pi}{2}$ radians, I can then rotate by $\frac{3 \pi}{2}$ and return to where I started.

Any set of objects, along with an operation that combines any two objects and results in a third, and that obeys these rules, is called a url=
https://en.wikipedia.org/wiki/Group_(mathematics] group. When groups are taught in abstract algebra, most of the examples are numeric (the integers, the reals, etc.) or else permutation groups (as the standard example of nonabelian groups). For our purposes we will be interested in geometric groups: groups of isometries, rotations, and so forth. The application of group theory to geometry is a big theme of twentieth century math and physics. Geometry is understood in terms of the groups of transformations that preserve various properties.

If you learned any group theory you remember that a subgroup is a group contained in some other group. Consider the isometry group of the 2-plane, consisting of all the translations, rotations, and reflections.

Now think about just the rotations around the origin. These are the familiar rotations of the unit circle. If you compose two such rotations you get another one. So the collection of all rotations about the origin is a subgroup of the isometry group.

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

I want to give one more example, since it relates to something we'll eventually need. We just observed that the rotations about the origin in the plane are a subgroup of the isometry group of the plane. Consider a rotation through an angle that's a rational multiple of $2\pi$, say $\frac{2\pi m}{n}$. Clearly if we compose this rotation with itself $n$ times we'll come back to where we started. On the other hand if we rotate through an irrational multiple of $2\pi$, we'll never come back to where we started. This is why we'll need to carefully choose our rotations in 3-space for the proof of Banach-Tarski. We need to make sure that no combination of these rotations ends up being the identity.

Next post, the free group on one letter, and then the free group on two letters.
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April 5th, 2017, 10:13 AM   #28
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What do you mean by "is called a url=
https://en.wikipedia.org/wiki/Group_(mathematics] group."?
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April 5th, 2017, 10:24 AM   #29
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What do you mean by "is called a url=
https://en.wikipedia.org/wiki/Group_(mathematics] group."?
Markup error.

group
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April 6th, 2017, 08:52 PM   #30
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What part does porosity play?
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