[Sinidir]

Can someone correct me if im wrong?

What i see here is a splitting of the set of points in the sphere? However the set of points in the sphere is not really the sphere. A point has no volume so no matter how many you add together you don't get something with a volume. This seems more akin to splitting the natural numbers into odd and even numbers which are all equally large.

The language that i see in this article and elsewhere however is suggesting that we actually duplicated the sphere (doubled the volume).

This seems incorrect.

[thaumasiotes]

> The language that i see in this article and elsewhere however is suggesting that we actually duplicated the sphere (doubled the volume).

> This seems incorrect.

It isn't incorrect. You're right that the number of points in the sphere does not equate to the volume of the sphere. But the Banach-Tarski theorem does in fact let you double the volume. It is considered to be of interest because it does the following:

1. You have a ball.

2. You cut the ball into 5 pieces in a very clever way.

3. You move the pieces around.

4. Now you have two balls, each the same size as the first.

The key, interesting part of this is in step 3, where we only use translations and rotations. Those preserve volume. (By contrast, it's easy to scale a ball of radius 2 to become a ball of radius 3, but that's not a volume-preserving transformation.) The part of the process that doesn't preserve volume is actually step 2, where we cut the ball into pieces. People find it unintuitive that this step doesn't preserve volume.

You can also cut your ball into several pieces and move the pieces around such that you end up with a much larger ball.

[joiguru]

"cut the ball into 5 pieces" is not the best description. A better one is: 2a. Split the ball into infinite pieces 2b. Divide the infinite pieces into 5 groups

[thaumasiotes]

> 2a. Split the ball into infinite pieces 2b. Divide the infinite pieces into 5 groups

Huh? The ball is already composed of infinite points. So in 2a you recognize that the ball exists, and then in 2b you cut it into pieces. But it seems superfluous to mention 2a separately.

[joiguru]

In regular natural language, cutting the ball into 5 pieces implies cutting 5 contiguous pieces.

If I say a cake is cut into 5 pieces, no person will consider that each piece contains parts from all parts of the cake.

[pfortuny]

“Cutting” it is certainly not: none of those sets is given by the zeroes of a continuous function (they would be measurable, and they cannot be).

So the paradox breaks down when you start to realize that you are not CUTTING but “choosing some points” and rearranging them. The fact that this rearrangement can be done with Euclidean moves is the surprise.

[thaumasiotes]

Come to think of it, the fact that two spheres contain the same number of points as one sphere does would seem to be closely related to why it's possible to produce two spheres from one sphere just by rearranging the points.

You can obviously produce a large sphere from a small sphere by rearranging the points, as long as you're willing to handle one point at a time -- that's what scaling is. But that requires an uncountably infinite number of translations. The Banach-Tarski theorem says we can do the same thing in only a finite number of translations.

[throwaway81523]

It is deeper than that. There is no way to do a similar duplication of a 2-dimensional disc. Why is it different in 3 dimensions? That is a property of the transformation group (rotation and translation) rather than 3-space itself.

[karmakaze]

This comment and the parent sheds some light on what makes this interesting. It didn't occur to me that we were allowing rotations and translations and that scaling is excluded. The paradox isn't about getting more from less as they're all comparably infinite, but rather being able to arrange them to be so.

[thaumasiotes]

> Why is it different in 3 dimensions? That is a property of the transformation group (rotation and translation) rather than 3-space itself.

Can you be more specific? Rotations and translations also exist in 2-space. It seems difficult to argue that this difference between 2-space and 3-space is "not a property of 3-space".

[throwaway81523]

What I mean is if you pick different transformation groups you can get different spaces where the "paradox" occurs. Likewise you can get rid of the paradox in 3-space by picking a different group.

[impendia]

In your example, I would say that it's addition that's failing.

Addition is defined as an operation with two inputs. You can't add more than two things, unless there is some particular rule that lets you.

If you have finitely many things, then this rule is the associative law: add them pairwise in whatever order, and you are guaranteed to get the same result.

To add infinitely many numbers, you need to talk about limits. Formally, when you say something like

1 - 1/2 + 1/4 - 1/8 + 1/16 - ...

you mean: look at the sum of the first two; then, look at the sum of the first three; then, look at the sum of the first four; and so on -- this sum converges to a limit, which is 2/3.

This sum is "absolutely convergent", which means you get the same result no matter how you order the summands, but some infinite sums change if you reorder things!

With points on the sphere the situation gets even worse, as there is no way to "list them in order". These sets are "uncountable", which means don't even try to sum any function defined on them.

To say approximately the same thing using technical jargon, one has countable additivity for Lebesgue measure on the reals, but uncountable additivity does not hold.

https://mathworld.wolfram.com/CountableAdditivity.html

[ajuc]

> A point has no volume so no matter how many you add together you don't get something with a volume

Not true. If you add uncountably many infinitesimal objects they can add up to noninfinitesimal object, that's how integration works in math, it's pretty confusing cause there's many kinds of infinity and they allow some unintuitive things to happen, but if they didn't worked we couldn't move (see Zeno paradox).

Banach-Tarski is formally correct, you add a finite number of sets with uncountably many points in each so you can get something with volume (depending on how they are positioned).

And yes - a line in math is just a set of points, same with a sphere (but it has 0 volume cause a sphere is just the "skin" without the insides) and a ball (which is what Banach-Tarski talks about). In fact every geometric object is just a set of points.

[stan_rogers]

Points, though, as traditonally defined, are not infinitessimal. They are literally zero in extent, having only a defined location.

[BeetleB]

> A point has no volume so no matter how many you add together you don't get something with a volume.

You're on the right track.

The Banach-Tarski paradox requires accepting that non-measurable sets[1] exist. A non-measurable set is a set with a an inspecifiable volume. Note: That's non-measurable - not 0. It means you have a quantity of something, whose volume is not 0, but it's also not any other number.

Once I realized that the paradox requires it, all the WTF aspect went away. Of course - if you can accept quantities for which you cannot specify a volume, you can probably accept about anything.

[1] https://en.wikipedia.org/wiki/Non-measurable_set

[ouid]

"no matter how many you add together", is where this argument breaks down in ZFC. The sphere is indeed the union of all of the singletons consisting of its points, all of which are measure zero. Banach-Tarski is mainly considered "weird" because it describes a partition into so few pieces, and they are rearranged via rigid motions only. It is trivial to come up with bijections between compact finite dimensional manifolds, (https://en.wikipedia.org/wiki/Space-filling_curve). For another example of the axiom of choice wreaking havoc on the notion of measure, see https://en.wikipedia.org/wiki/Vitali_set .

[thaumasiotes]

> "no matter how many you add together", is where this argument breaks down

Interpret it as "adding more points will not necessarily increase the volume, no matter how many points you add". There are plenty of measure-0 sets containing as many points as the continuum does.

[nmaleki]

I recently made a video on the zeroth dimension (what you call a point) and how it relates to higher dimensions. https://youtu.be/u1MUrVBQTyE

I'll be using spatial dimensions as a conceptual framework to tackle this exact issue in future videos.

[pacifist]

A sphere in the mathematical sense is the 2D shell on the surface of the sphere you're thinking of. So, no height, no volume.

[pacifist]

My bad. Should have RTFA. We're talking about 3D here.