If you cut up the sphere's surface into pieces, the combined surface area will remain the same. If you then reassemble them in a different configuration into two spheres both the same size as the original, the surface area will be twice as much.
I don't see how that could be true. What am I missing here?
ETA: thanks for all the explanations. The most succinct answer seems to be because it assumes the surface is made of infinitely many points, and infinity breaks math. 2*inf = inf.
One more reason why it makes no sense to treat infinity as a number.
Take a (solid) sphere ... there are uncountably infinitely many points.
Now for each point colour it read, green, blue, yellow, or purple. Do this completely randomly.
Then every point will have nearby, arbitrarily close, other points of every colour. The colours will be deeply intertwungle, with the points of each colour just being a "cloud", spread throughout the entire volume.
This is a better mental image of what the "pieces" are like.
Now, if you do this colouring in a very, very special way, the red points can be rotated around to match exactly the green points. Well, OK, so the green points are a rotation of the red points. Not a problem.
But you can also arrange it so that if you rotate the red points a different way then they will exactly match the blue points. Well, OK, so the blue points are also a rotation of the red points.
The utterly, utterly bizarre result is that the above can be true, and we can also have a third rotation of the red points that will match all the green points and the blue points AT THE SAME TIME!
At its heart, that how the B-T theorem works. There are details, and arranging that this happens in also non-trivial, but at its heart, this is what's happening.
The purpose of the theorem is to show that the concept of "volume" cannot be applied to arbitrary sets of points in 3D. If you want more details about that, I wrote a blog post about this many years ago. It was my project for my BSc (Hons) way back in 1982.
The core of the paradox is that that the intuition that the volume of a bunch of disjoint sets obey the law
V[A ∪ B ∪ C ∪ ...] = V[A] + V[B] + V[C] + ...
is only guaranteed if you have a countable number of sets[1]. If you split a sphere into an uncountable number of pieces in the right way (which requires the Axiom of Choice) you can break this rule without being inconsistent with measure theory.
Completely wrong. From the Wikipedia article: "Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball."
There is absolutely no issue with uncountability here. The issue is with the particular shape of the parts, where V is not reasonably definable.
You first split the sphere into an uncountable number of subsets, then group these into a finite number of subsets, whose measure sum to twice the measure of the original set.
(Un)countability is at the core of most of the counterintuitive results of measure theory, exactly because of the the third property of measure.
To be clear, the construction given here violates the finite additivity property of measure. It's got nothing to do with the countable/uncountable additivity property.
IIRC correctly first you split the surface into an uncountable partition then you use the axiom of choice to "color" each point of each partition of a color and then define your finite number of subsets as all the points of each given color.
(Not an explanation of how it works, but just of how it could be true)
IIRC it follows a principle similar to the Hilbert hotel[0]. The idea is that you can split an infinity into an infinity[1] of infinite parts.
for a simplex example imagine all the points (n,m) where n,m are natural numbers, lets call this set P. we can split P into two sets:
- Q defined by all the (n,m) in P where n<m
- R defined by all the (n,m) in P where n>=m
Now you can "bend" Q by mapping (n,m) into (n, m-n-1) and R by mapping (n,m) into (n-m,m).
These bent version of Q and R are both identical to the initial P set.
This was a very informal and messy proof, but the core idea is the same: split the set (like a sphere surface) into many sets, manipulate (rotate) each one taking advantage of their infinity, recompose them as needed.
I do not think I am able to legibly comunicate the idea behind Banach-Tarski, but hopefully this gives some intuition
Essentially the difficulty arises from attempting to assign a measure (area) to every single subset of the sphere, where you say that rotations need to preserve this measure. The paradox can be viewed as a proof that you cannot assign a measure to every subset of the sphere in a consistent way.
The way measure theory resolves this is by showing that if you restrict to appropriate subsets, called measurable subsets, you can get all the nice properties you would expect.
It turns out that basically everything is measurable. In fact the existence of a non-measurable set is independent of ZF. This means that you need the axiom of choice, which was used here in the Banach-Tarski paradox, in order to construct a non-measurable set. So measure theory doesn't really lose a great deal by restricting in this way, which is why it gives such a great theory of integration.
The pieces are fractal and exceptionally complex. It’s related to the idea that a point has zero length, but a line (an infinite collection of points) has nonzero length.
It also highlights a discrepancy between our physical intuition of space and the way we model space mathematically.
It comes down to the fact that not every set can be reasonably assigned a volume. Once you restrict yourself to sets where volume is meaningfully defined, the paradox immediately disappears and your reasoning becomes completely valid.
It can be proven using the axiom of choice, which allows for the construction of non-measurable sets, i.e., collections of points that do not have a volume in the ordinary sense, and whose construction requires an uncountable number of choices.
So you chop up a sphere (which has a volume V1) into a finite number of collections of points (which don't have a volume), then assemble the collections of points into a new sphere (which has a volume V2 != V1).
The trick is to cut it up to an extent that you can't meaningfully assign area (or rather volume) any more.
The only real trick here is the low number of pieces, if you allow an arbitrary number of pieces it's simple to split the points into two sets and move the points into two equal sized spheres, that's just because infinity is weird that way (see Hilbert's hotel if you're not sure).
The question then becomes can you meaningfully talk about volume if you cut a sphere into a finite number of pieces? Turns out you can't.
Not really, Banach-Tarski is very similar to the hilbert hotel with odd-even rooms, the genius is in finding a way to recompose these parts using only rotations.
Not quite sure what part you disagree with, but indeed that's more or less what's happening. Though also the fact that it's a finite number of pieces is important.
The only real trick here is the low number of pieces, if you allow an arbitrary number of pieces it's simple to split the points into two sets and move the points into two equal sized spheres
I took this as saying that it is easy to split the sphere into an infinte number of (I implicitly assumed) non null sets and recombine them in two spheres, to which I disagreed that it is hard to do so using only rotations.
But I guess what you meant is that you can build a bijective mapping S^2 -> {0,1} x S^2 just like you can build one for R -> {0,1} x R.
I did not consider the extreme case of every element of your partition being allowed to be a single point.
I'm with you here, doing finite -> infinite -> finite transformations logically creates paradoxes. If those infinite points have no dimensions, how do their cumulative zeros add up to a positive number?
To the risk of sounding naive, I see this as a self-inflicted paradox, much like the immovable object and the irresistible force paradox. If it's a paradox like Schrödinger's cat thought experiment, then I'm cool with it, because it points the limits of theory by carrying it into absurdeness.
While I haven't read the proof sufficiently in depth to explain it, but these aren't physical spheres because they have infinite parts.
They're less practically offensive to our senses, but I don't think this is really any different from zooming infinitely into a fractal or the difference between countable and uncountable sets. Infinities behave strangely.
I don't know if this will explain your questions, but years ago I watched this video from Vsauce about this very topic, and IIRC it was explained quite nicely: https://www.youtube.com/watch?v=s86-Z-CbaHA
The key thing is that this is not something even vaguely physically realizable. The pieces of the sphere are not "pieces" in any physical sense - they are just subsets of points inside the sphere, that don't have a definite shape and volume.
Comments
Can someone explain this more simply?
If you cut up the sphere's surface into pieces, the combined surface area will remain the same. If you then reassemble them in a different configuration into two spheres both the same size as the original, the surface area will be twice as much.
I don't see how that could be true. What am I missing here?
ETA: thanks for all the explanations. The most succinct answer seems to be because it assumes the surface is made of infinitely many points, and infinity breaks math. 2*inf = inf.
One more reason why it makes no sense to treat infinity as a number.
The "pieces" aren't pieces such as you picture.
Take a (solid) sphere ... there are uncountably infinitely many points.
Now for each point colour it read, green, blue, yellow, or purple. Do this completely randomly.
Then every point will have nearby, arbitrarily close, other points of every colour. The colours will be deeply intertwungle, with the points of each colour just being a "cloud", spread throughout the entire volume.
This is a better mental image of what the "pieces" are like.
Now, if you do this colouring in a very, very special way, the red points can be rotated around to match exactly the green points. Well, OK, so the green points are a rotation of the red points. Not a problem.
But you can also arrange it so that if you rotate the red points a different way then they will exactly match the blue points. Well, OK, so the blue points are also a rotation of the red points.
The utterly, utterly bizarre result is that the above can be true, and we can also have a third rotation of the red points that will match all the green points and the blue points AT THE SAME TIME!
At its heart, that how the B-T theorem works. There are details, and arranging that this happens in also non-trivial, but at its heart, this is what's happening.
The purpose of the theorem is to show that the concept of "volume" cannot be applied to arbitrary sets of points in 3D. If you want more details about that, I wrote a blog post about this many years ago. It was my project for my BSc (Hons) way back in 1982.
The core of the paradox is that that the intuition that the volume of a bunch of disjoint sets obey the law
V[A ∪ B ∪ C ∪ ...] = V[A] + V[B] + V[C] + ...
is only guaranteed if you have a countable number of sets[1]. If you split a sphere into an uncountable number of pieces in the right way (which requires the Axiom of Choice) you can break this rule without being inconsistent with measure theory.
[1]https://en.wikipedia.org/wiki/Measure_(mathematics)#Definiti...
Completely wrong. From the Wikipedia article: "Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball."
There is absolutely no issue with uncountability here. The issue is with the particular shape of the parts, where V is not reasonably definable.
You first split the sphere into an uncountable number of subsets, then group these into a finite number of subsets, whose measure sum to twice the measure of the original set.
(Un)countability is at the core of most of the counterintuitive results of measure theory, exactly because of the the third property of measure.
To be clear, the construction given here violates the finite additivity property of measure. It's got nothing to do with the countable/uncountable additivity property.
Yeah but this is only possible if the sets in question are uncountably infinite.
Yes but you are not taking an uncountable union. You are taking a finite union.
IIRC correctly first you split the surface into an uncountable partition then you use the axiom of choice to "color" each point of each partition of a color and then define your finite number of subsets as all the points of each given color.
Banach-Tarski only splits the ball into a finite number of parts. However, the parts are not measurable, which is what requires the axiom of choice.
(Not an explanation of how it works, but just of how it could be true)
IIRC it follows a principle similar to the Hilbert hotel[0]. The idea is that you can split an infinity into an infinity[1] of infinite parts.
for a simplex example imagine all the points (n,m) where n,m are natural numbers, lets call this set P. we can split P into two sets:
- Q defined by all the (n,m) in P where n<m - R defined by all the (n,m) in P where n>=m
Now you can "bend" Q by mapping (n,m) into (n, m-n-1) and R by mapping (n,m) into (n-m,m).
These bent version of Q and R are both identical to the initial P set.
This was a very informal and messy proof, but the core idea is the same: split the set (like a sphere surface) into many sets, manipulate (rotate) each one taking advantage of their infinity, recompose them as needed.
I do not think I am able to legibly comunicate the idea behind Banach-Tarski, but hopefully this gives some intuition
[0] https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Gra...
[1] the idea is that if K and H are two infinities then size(H * K) = max(H, K) for reasonable definitions of multiplication and size https://en.wikipedia.org/wiki/Cardinal_number#Cardinal_multi...
Essentially the difficulty arises from attempting to assign a measure (area) to every single subset of the sphere, where you say that rotations need to preserve this measure. The paradox can be viewed as a proof that you cannot assign a measure to every subset of the sphere in a consistent way.
The way measure theory resolves this is by showing that if you restrict to appropriate subsets, called measurable subsets, you can get all the nice properties you would expect.
It turns out that basically everything is measurable. In fact the existence of a non-measurable set is independent of ZF. This means that you need the axiom of choice, which was used here in the Banach-Tarski paradox, in order to construct a non-measurable set. So measure theory doesn't really lose a great deal by restricting in this way, which is why it gives such a great theory of integration.
The pieces are fractal and exceptionally complex. It’s related to the idea that a point has zero length, but a line (an infinite collection of points) has nonzero length.
It also highlights a discrepancy between our physical intuition of space and the way we model space mathematically.
It comes down to the fact that not every set can be reasonably assigned a volume. Once you restrict yourself to sets where volume is meaningfully defined, the paradox immediately disappears and your reasoning becomes completely valid.
This sentence seems crucial:
So you chop up a sphere (which has a volume V1) into a finite number of collections of points (which don't have a volume), then assemble the collections of points into a new sphere (which has a volume V2 != V1).You chop up a sphere of volume V1 into non-measurable pieces, and then re-assemble those pieces into 2 spheres, each of volume V1.
VSauce did a nice video on this, starting at 15m25s with the sphere re-assembling:
https://www.youtube.com/watch?v=s86-Z-CbaHA&t=15m25s
The trick is to cut it up to an extent that you can't meaningfully assign area (or rather volume) any more.
The only real trick here is the low number of pieces, if you allow an arbitrary number of pieces it's simple to split the points into two sets and move the points into two equal sized spheres, that's just because infinity is weird that way (see Hilbert's hotel if you're not sure).
The question then becomes can you meaningfully talk about volume if you cut a sphere into a finite number of pieces? Turns out you can't.
Not really, Banach-Tarski is very similar to the hilbert hotel with odd-even rooms, the genius is in finding a way to recompose these parts using only rotations.
Not quite sure what part you disagree with, but indeed that's more or less what's happening. Though also the fact that it's a finite number of pieces is important.
I took this as saying that it is easy to split the sphere into an infinte number of (I implicitly assumed) non null sets and recombine them in two spheres, to which I disagreed that it is hard to do so using only rotations.
But I guess what you meant is that you can build a bijective mapping S^2 -> {0,1} x S^2 just like you can build one for R -> {0,1} x R.
I did not consider the extreme case of every element of your partition being allowed to be a single point.
I'm with you here, doing finite -> infinite -> finite transformations logically creates paradoxes. If those infinite points have no dimensions, how do their cumulative zeros add up to a positive number?
To the risk of sounding naive, I see this as a self-inflicted paradox, much like the immovable object and the irresistible force paradox. If it's a paradox like Schrödinger's cat thought experiment, then I'm cool with it, because it points the limits of theory by carrying it into absurdeness.
While I haven't read the proof sufficiently in depth to explain it, but these aren't physical spheres because they have infinite parts.
They're less practically offensive to our senses, but I don't think this is really any different from zooming infinitely into a fractal or the difference between countable and uncountable sets. Infinities behave strangely.
I don't know if this will explain your questions, but years ago I watched this video from Vsauce about this very topic, and IIRC it was explained quite nicely: https://www.youtube.com/watch?v=s86-Z-CbaHA
The key thing is that this is not something even vaguely physically realizable. The pieces of the sphere are not "pieces" in any physical sense - they are just subsets of points inside the sphere, that don't have a definite shape and volume.
The point is that the pieces do not have volume because not all sets have a volume when you assume the axiom of choice.