April 2nd, 2017, 01:47 PM  #21 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 464 Thanks: 29 Math Focus: Number theory 
F sub 2 seems to be a fractal, which I know somewhat, since it is visual, nondifferentiable and selfsimilar (and simple to generate).
Last edited by Loren; April 2nd, 2017 at 01:50 PM. 
April 2nd, 2017, 02:09 PM  #22  
Senior Member Joined: Aug 2012 Posts: 2,409 Thanks: 753  Quote:
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.  
April 2nd, 2017, 08:58 PM  #23 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 464 Thanks: 29 Math Focus: Number theory 
Does the rigid "rotation" in 3space have anything to do with the LeviCivita symbol/function yielding antisymmetry (paradoxical here?) for odd dimensions? Just a wild guess. https://en.wikipedia.org/wiki/LeviCivita_symbol The axes for rotations in 3space are 1space 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 BT. 
April 2nd, 2017, 11:18 PM  #24  
Senior Member Joined: Aug 2012 Posts: 2,409 Thanks: 753  Quote:
* 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 twospace 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 threespace 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 BanachTarski paradox in three space and not in the plane. Last edited by Maschke; April 2nd, 2017 at 11:28 PM.  
April 3rd, 2017, 11:55 AM  #25  
Senior Member Joined: May 2015 From: Arlington, VA Posts: 464 Thanks: 29 Math Focus: Number theory  Quote:
Is what you are describing in three dimensions a particular symmetry group, or does a group have to be commutative? In 3space, 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 3space of higher cardinality than that of 2space?  
April 3rd, 2017, 02:09 PM  #26  
Senior Member Joined: Aug 2012 Posts: 2,409 Thanks: 753  Yes, thanks. Quote:
Quote:
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 3space contains a copy of $F_2$, we can extend the paradoxical decomposition to the unit ball in 3space to prove the theorem. 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 distancepreserving 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 distancepreserving maps and count them). Last edited by Maschke; April 3rd, 2017 at 02:18 PM.  
April 4th, 2017, 04:36 PM  #27 
Senior Member Joined: Aug 2012 Posts: 2,409 Thanks: 753  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 threedimensional cases (and the ndimensional 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 (lengthpreserving) 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 2plane, 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 3space for the proof of BanachTarski. 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. 
April 5th, 2017, 10:13 AM  #28 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 464 Thanks: 29 Math Focus: Number theory 
What do you mean by "is called a url= https://en.wikipedia.org/wiki/Group_(mathematics] group."? 
April 5th, 2017, 10:24 AM  #29  
Senior Member Joined: Aug 2012 Posts: 2,409 Thanks: 753  Quote:
group  
April 6th, 2017, 08:52 PM  #30 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 464 Thanks: 29 Math Focus: Number theory 
What part does porosity play?


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