Banach-Tarski!: Fun with some very weird math

The Banach-Tarski paradox is one of the many places where higher-level math starts to sound like a stoned conversation in a Freshman college dorm room.

Imagine a ball. Now imagine cutting that ball up into a finite number of pieces. Six, maybe. Or five. The Banach-Tarski paradox proposes that you could take those pieces and, without stretching or expanding them in any way, use them to form two balls identical to the first. Basically, you've just created mass out of nothing. That is, to put it mildly, not supposed to be able to happen. Thus, the part about the paradox.

WTF, you may ask? It might help to know that Banach-Tarski isn't talking about real, physical balls. Rather, it deals with theoretical, mathematical spheres. Unlike a real ball, which only has so many atoms, a theoretical sphere can be divided up into an infinite number of pieces. Comparing different explanations of Banach-Tarski that I found online, the one that made the most sense to me stared off with this detail, and was written by "The Writer" a contributor to kuro5hin.org. He or she put together a layman's analogy that lowers the "WTF!!?" to a nice, calm, "wtf?"

So here's my proposed "intuitive" rationalization of it. I'll do it by way of an analogy with a physical sphere.

Let's forget for the moment the mathematical sphere S, which has infinite density. Let's consider a real, physical sphere B (for "ball"), also of radius 1. B is identical to S except that it consists of a finite (albeit large) number of atoms. The way these atoms are laid out in B is called the crystalline structure of B. (I.e., if you take B, or any physical object for that matter, and look at it under an electron microscope, you will see the atoms laid out in a fixed, regular pattern. That's called its "crystal lattice".) Usually, the crystalline structure is a simple geometric relationship between neighbouring atoms.

Notice that although the geometric relationship between atoms define its crystalline structure, the precise distance between atoms may vary. This leads to materials of different densities.

Now, we perform the equivalent of a Banach-Tarski decomposition on our physical sphere B: we "atomize" B into four spherical clouds of atoms, let's call them C1, C2, C3, and C4. (We'll ignore the central atom in B, just as in the mathematical version of this decomposition.) Let's assume that each of these clouds are sparse enough that they are gaseous, no longer solid by themselves (imitating the immeasurability of the mathematical pieces of S). Furthermore, let's say that the atoms in each of these clouds are laid out in a regular pattern, so that if we rotate C1 by some angle G, and put it together with C2 in the same spherical region, the atoms in both clouds line up into the same crystalline structure as B, except that now the distance between atoms is greater (to account for the missing atoms now in clouds C3 and C4). Similarly, assume we can do the same with C3 and C4: we just translate them away from the original spherical region of B so that they don't interfere with C1 and C2, and reassemble them into another sphere.

Now, we have successfully built two (physical!) spheres with the same radius as B, using only material from B itself. Each of the two spheres have the same crystalline structure as B. The only difference between these spheres and B is that they each have only half the density of B.

Anyway, this very long introduction to a mathematical concept is necessary so that you can enjoy the funny video at the top of this post. In it, an onslaught of infinitely multiplying oranges overrun the University of Copenhagen mathematics department while students cavort to a parody version of Duck Sauce's "Barbara Streisand." It is weird and wonderful (and possibly a little NSFW in parts) and I wish I knew more about the awesome people who made it.

Video Link

Thank you, samurai!

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  1. I’m confused. The BB post says that you create two balls identical to the first–mass from nothing!–but the excerpt says that you create two balls of half the density of the first. The former is remarkable; the latter, not. What am I missing?

    1. The excerpt involves physical spheres made of matter, and is indeed unremarkable. If you perform the same operation on a mathematical sphere made out of points, you’ll also end up with two spheres that are each half as dense. The difference is, mathematical spheres have infinite density, and half as dense as that is still infinitely dense.

      1. There are ways of talking about the density of a mathematical sphere as a finite number — you just have to come up with a definition of density that’s a little more sophisticated than point-counting, and the Banach-Tarski sets are an example of a limitation on that definition.

    2. indeed. as a corollary: i feel like if the process is repeated then the number of spheres would increase to N where N was the number of atoms in the original sphere and each sphere would have 1 particle occupying the volume of a sphere with radius 1. 

      i think i’m tripping on the idea of continuous, solid spheres as opposed to spheres  comprised of a countably large number of discrete elements. perhaps the key lies in the rotational symmetry condition mentioned earlier? this seems to be beyond my reckoning.

    3. If you have balls that are infinitely divisible into points, you create two balls identical to the first. If you have finitely divisible balls that are actually made of atoms, any approximation of this that you attempt will give you two balls of half the density of the first.

      Fundamentally, what’s going on is that giving a consistent mathematical definition of concepts like “mass” and “density” in continuous settings is really, really hard. The Banach-Tarski paradox is an example of the kind of problem you run into when you try to do it.

  2. Filmed in the maths department of the University of Copenhagen … I spent many a day there waiting for my girlfriend to finish her lectures :)

  3. And this is why engineers dream of killing mathematicians. The engineer would argue that reducing the density does not leave you with the same crystalline structure and that you can’t just force a solid to half its density. The astrophysicists would argue that your infinitely dense sphere is a blackhole beyond the event horizon and information can’t escape so there’s no point in discussing it further

    1. The decomposition isn’t differentiable or even continuous so it really doesn’t have anything to do with real world solids. Anyway, why should infinite density bother you so much? Don’t engineers love point particles? Do you get equally upset by ideal gases? That’s the same infinite density that is being discussed.
      Also, what’s with the murderyness?

    2. the actual construction does not entail reduction in density; rather, it uses  *extremely* weird sets of points that are not meant to be analogues of any physical object, but rather are simply designed to test the limits of abstractions like “set of points” .  The quoted excerpt is somewhere between misleading and completely wrong.  There some bits of mathematics that are difficult to explain in an intuitive way; Banach-Tarski (and non-measurable sets, and the axiom of choice) are simply not amenable to one paragraph explanations.  No need to kill any mathematicians….

  4. Minor correction: That’s not a German math department. The signs look like a nordic language to me – Swedish or Norwegian maybe.

  5. The BT paradox depends on “the axiom of choice (AC)” – a mathematical principle that is usually very usefull but can lead to strange stuff. Reasoning using the AC can lead to situation where you can prove that something exists without having any definition or idea of what it is.

    “Constructive mathematics” only allows things that have been explicitly constructed and the BT paradox doesn’t arise there. Constructivism also disallows the law of excluded middle (either P or not-P must be true). Disallowing this in similar to the every day reasoning that “either God is all-knowing or not” is not necessarily true because God might not exist.

    So essentially the BT construction is not “constructive”.

  6. Note that by “The Writer”‘s description, you create volume out of nothing without changing the mass. The Banach-Tarski Theorem belongs is a frustrating beast known as an “existence theorem” because it posits the existence of something without giving a clue about how to create it.

    One other mathematical quibble: It’s important to distinguish between a ball, which is solid, and a sphere, which is the zero-thickness shell. B-T deals with balls.

  7. Mathematician here– my two cents… :)

    The Banach-Tarski theorem is impossible to relate to the physical world. I think that physical illustrations, such as the one presented here, are more confusing than anything, as indeed has already been seen in the comments.

    The truth is that there is no analogue of this idea in the physical world. Imagine splitting the number line into two pieces: the rational numbers and the irrational numbers. Both pieces are “infinitely dense” within each other. The Banach-Tarski decomposition is far more complicated than this one, but even this relatively simple decomposition has no real analogue in the physical world.

    B-T belongs in the realm of the logical, not the physical. As another commenter suggests, its ultimate moral is: the axiom of choice, and all its non-constructibility, is truly weird.

  8. “Paradox” is caused by combining sigma calculus (designed for very close approximations of discrete entities via approaching them) with infinite-anything (which is by definition unapproachable).

  9. Maybe I’m missing something but this seems exceptionally underwhelming. My understanding is that “if you have an infinity and divide it, you still have infinity.” So… yeah, by definition, right? What’s so special?

    The implication seems to be that if this were analogous to something in the real world with mass, then you’d create something out of nothing. But it ISN’T analogous to the real world. And if it was, you’d have something with infinite mass. That’s like saying, “If a car was a cupcake, it would be delicious, and wouldn’t that be amazing, something delicious that you can drive.” But a car isn’t a cupcake. And if it was, you couldn’t drive it. So it’d just be… a cupcake which, while delicious, rarely warrants a “WTF?”

    1. There are just as many points in two spheres as in one sphere. You’re right: from that perspective, it’s not amazing.

      The amazing part comes from the phrase “without stretching”.  You can divide the original sphere and then rearrange the parts *without stretching them* to get two spheres. In other words, even though there was no stretching, the volume of the reassembled pieces is different from the volume you started with.

      Here’s another statement of the theorem that might help solidify its strangeness. You can take a solid ball of radius 1, chop it into a finite number of pieces, and rearrange the pieces *without stretching them* to assemble a solid ball of radius 100!

      1. I suppose I still don’t understand, then. When I read the post and saw words like “atomize” and “4 spherical clouds of atoms” I’m imagining a sort of abstract dividing. Like “Say you have a cloud of X particles. Now split that cloud into 2 clouds of  X/2 particles.” So we’re not talking about “chopping” the way you “chop” an apple into slices, right? We’re just “pulling” half the ‘atoms’ out in the same shape…

        This is how I understand it, please explain where I went wrong:
        You take a ball of infinite density and “chop” it into two simply by pulling every other “particle” out into an identically sized, but half-as-dense ball (like you would ‘create’ two squares of identical size if you pulled the black tiles off the white tiles on a chessboard).
        But we’re assuming the ball is infinitely dense. So you produce two balls with half the density out of a single ball that has an infinite density. And now you have twice the balls without reshaping or “stretching” (which I take to mean: changing the density).

        If that’s about the right idea, I still don’t think that’s amazing. What’s “amazing” is the properties of infinity which we assume into this ‘paradox.’ It seems like this paradox is just a manifestation of the property whereby half of infinity is STILL infinity. Which, to go back to my cupcake analogy would be like saying “Imagine an infinite number of cupcakes… that’s a lot. Those are pretty amazing cupcakes to be so many.” No, we’ve ASSUMED into the thought the property that makes it amazing. It’s not the cupcakes that are amazing, it’s the concept of infinity.

        1. I agree with the posters who find that whole “atom cloud” analogy rather counterproductive.

          Specifically, the baffling thing here has nothing at all to do with density (much less so in a mathematical sense) and everything with volume: You decompose a sphere of volume 1 into 5 parts, move and turn those parts (but don’t stretch or reshape them in any way) and get a sphere of volume 2, without anything missing. It certainly is counter-intuitive, at very least at first. I definitely was baffled when I first saw that in a lecture.

        2. unregular,

          I think I see your vision. But if you split the sphere into two as you outlined, your spheres would still have lots of little “holes” in them. After all, suppose you removed just one point from a solid sphere. You’d get a sphere with a hole in it, which is a different mathematical object from the original sphere. And you’re thinking of removing *lots* of points.

          Now, you could “squish” the resulting holy spheres up a little bit to fill in the holes. But what’s amazing here is that no squishing (or stretching, etc.) is required of the pieces– you can assemble two complete spheres, with no holes in them, with only rigid body motions. And each of the resulting spheres is completely identical to the original. They do not contain any holes.

          If this still doesn’t amaze you, think about the other version I mentioned: you can take a sphere or radius 1, chop it up, reassemble the pieces with rigid body motions, and get a solid sphere of radius 100.

          1. 1) if you remove every other ‘particle’ in a ball with infinite density, you don’t have any ‘holes’ because it’s infinitely dense. Just like Infinity – Infinity/2 = Infinity. Right? In an infinitely dense ball, if you reduce the density by half by removing half the material in an even distribution, you’d have 2 balls of identical radius to the original which still have the same density (and thus ‘mass’… and thus you would have ‘created’ ‘mass’).

            2) I still don’t get it. Are we talking about chopping the ball up like you could do with a knife to an apple (except, of course, the infinitely dense thing)? Or is our “chopping” more abstract like I was imagining?

            3) I can see how if you sliced the ball into an infinite number of slices (like apple slices), you could make a ball of half the slices and a second ball of the OTHER half and then have 2 balls still with infinite density and the same radius as the first ball. But we’re talking about a finite number of pieces, right?

            4) Can we explain this is a more simple (perhaps simplistic) fashion? I get the whole “it’s like this except with infinity” concept, it’s just the vision I have in my head just isn’t right, apparently.

    2. I think there are two issues. (1) why is this surprising, and (2) why is it of practical importance? The answers (to me) are different.

      (1) surprising

      Ok, so this isn’t surprising because “infinity is strange”. As an example, we can take an infinite line of cup cakes and make two infinite lines of cup cakes. I can even tell you how: number all the cupcakes, put all the even ones in one line, and the odd ones in the other line. If you try and do all the even ones at once you’ll never get on to the odd ones, but that’s not a problem because we can interleave them i.e. cupcake 1 to odd line, 2 to even line, 3 to odd line, etc.

      Now actually try doing the same thing with the sphere, and like the cupcake, actually say how we’re going to chop it up to make two other spheres. Or try doing this with something simpler, like a square. Maybe we could start by chopping the it into three parts (not necessarily equal), and labelling one as “part 1”, another as “part 2”, and leaving the remaining part to be chopped up again. Then we could iterate. But how exactly do we chop them up on the first go, or on future goes? Remember that you’re only allowed a finite number of parts, and they must fit together to make two more complete squares of the same size. Even after the first cut, no matter how you do it, it’ll look hopeless. 

      Perhaps the main part is that you can’t just choose “every other point”, because there are infinitely many within any little area, so this doesn’t make sense. If I give you a point, how do you choose “the next one”?

      (2) important

      Surely any set of points in space has a volume? It could be zero (e.g. if it’s completely flat) or infinite, but it can only be one number. What’s more if two parts make up a whole the sum of the volumes equals the final volume (unless they overlap, in which case their sum can be even bigger than the final volume). 

      Now consider the pieces in the Banach-Tarski theorem. They’re bounded, so their volume is finite, and there are a finite number of them, so we should just be able to write down their volumes in a list and forget that they came from anything to do with infinities. These numbers should add up to the volume of one sphere, and should also add up to at least the volume of two spheres. This is definitely a problem.

      You might say, “well all volumes in maths must be infinite then, because they have an infinite number of points in them”. But at some point (no pun intended) physicists are going to want to use maths to model the real world, and if we tell them we can only model infinite volumes, we aren’t doing our jobs!

      The solution is that we only consider the volumes of certain collections of points that are not complicated enough for this type of problem to happen. The precise description of what types of sets are allowed is called measure theory.

  10. A couple of comments which might make it a bit clearer why this isn’t just about decomposing one infinite set into two of the same cardinality (which is easy): 

    i)  There is no equivalent of the B-T theorem for disks in the plane.
    ii) If you don’t allow all possibly rigid motions (what other posters call “without stretching”) when moving the bits of the balls around, but restrict to a smaller group of motions, the theorem disappears.

    In fact, (ii) is the reason for (i): space has many more ways of moving things around rigidly than does the plane.  Once your group of motions contains a thing called the “free group on two generators”, you can do a paradoxical decomposition.

    When I teach the B-T Theorem, I like to include the following fairly easy corollary: given any two regions of space each of which (a) is bounded and (b) contains at least one ball-shape volume (no matter how small), there exists a way to decompose one into a finite number of pieces and reassemble those pieces to form the other.  For example, the coffee mug on my desk can be decomposed and reassembled in this way to form the Milky Way galaxy.  Abstractly speaking, of course.

  11. IANAP but concerning thought experiments about the theory’s practical use in the physical world, what about Bose-Einstein condensates?
    Surely there are plenty of possibilities for a system that can be thought of as having been (physically) duplicated?

  12. There are a couple articles relating this to the Hotel paradox, which is more accessible and helps give the intuition behind it.

  13. Everyone is missing the most remarkable thing about this video. There’s a guy in the men’s room dancing in front of a urinal … BAREFOOTED!! Brave? Or just stupid? That’s the real conundrum here.

  14. IIRC, the pieces of the ball exist (per the axiom of choice), but they cannot be constructed.

  15. The key thing to realize is that the phrase “cut the sphere up into parts” is a misnomer – it isn’t really “cutting” in any sense that matches the intuitive notion of “cutting.”  It’s more like “partitioning” – we take the set of all points on the sphere and “partition” them into seperate collections of points.  This partition is horribly aberrant  – they don’t behave like nice sets of points as we’d understand them geometrically.  Fundamentally, BT is possible because of a powerful axiom, the Axiom of Choice, that allows the assertion of the existence of mathematical objects that we can’t possibly construct.  BT doesn’t ever show how to partition to yield the “paradox,” but rather asserts that there exists such a partition.  That might seem a fine distinction, but what it means, amongst other things, is that this paradox is not about any intuitive notion of a “construction.”

  16. I also agree the above explanation is interesting, but ultimately counterproductive if you’re trying to understand the theorem.

    As a philosopher and a mathematician, here’s my gloss, hopefully somebody will find it helpful:

    What’s useful from the explanation is the “cloud-like” structure of the pieces.  That’s right, but it would have to be even more “cloud-like” than a gas of atoms.  Banach-Tarski really does violate physical laws, as far as we understand them.  The reason is that our matter isn’t infinitely divisible, and it takes up non-zero amounts of space.

    “Density” isn’t really applicable to the measure we’re talking about.  The way the math goes, we were trying to get a way of measuring the actual filled-volume of things.  This isn’t just the volume taken up by their rough shape if you assume they’re solid, that’s like physical volume.  So for instance, in the physical world, you might think a hollow ball has the same volume as a solid ball of the same diameter, because they take up the same space in most situations (they displace the same amount of water, for instance).  But there’s empty space in the hollow ball, so if you subtract all the empty space, the actual volume taken up by the “filled” part of the ball is smaller.  In the physical world, density is like a ratio of the “filled” part of a solid shape to the part that is empty space.
    The kind of volume we were trying measure mathematically, in the context of Banach-Tarski was a “filled” volume.  So in this case, a small solid ball might have the same volume as a larger, hollow ball, because they have the same amount of filled space. Even though a cloud takes up a lot of space and might have large volume in that sense, if you get rid of all the spaces in between its parts it has a comparatively small filled-volume.
    The idea is that cutting a space up into pieces and moving them around, putting space between them, etc. shouldn’t change the ultimate filled-volume that you get.
    The Banach-Tarski theorem shows that won’t work if you insist on saying that every crazy, cloud-like shape whatsoever can be assigned a determinate volume.  Because in the continuous realm you can make some very strange shapes, some of these turn out simply too crazy for us to decide how much counts as filled and how much counts as empty.

    The fact that atoms have a non-zero volume all by themselves is what makes it hard in the physical realm.  One atom takes up a certain filled-space, another of the same kind takes up twice as much filled-space, and so on.  If your ball has a finite volume, it can only have a finite number of atoms, and all of the pieces you cut it into will have a finite filled-volume, and if you recompose them, you never change the amount of filled-volume you started with, even if you can make cloudy cuts and make two shapes that look like balls, their individual filled-volume will be less because they have more empty space.

    There’s a property of spaces with a volume measure that ensures this, called finite additivity.  The idea is that if you take two shapes with a defined volume and add them together, you get a shape with the sum of their volumes.  This works for any finite number of shapes.

    In the continuous realm, we can make shapes that are points and have zero volume.  If we add any finite number of individual points together, we get a shape with zero volume.

    We also have countable additivity, which tells us, roughly speaking, that we can take an infinite number of shapes with definite volume, add them together, and the volume we get is just the sum of their volumes.  So even if you take an infinite number of zero-volume points and add them together, the volume is still zero.  But the ball is just made of points and there are an infinite number of them, how does it have non-zero volume? The volume of a solid ball has to come from somewhere, right?

    To figure this out, we have to consider what we don’t have: uncountable additivity. 

    See, in mathematics, there are multiple types of infinity.  Trust me.  Look up Cantor’s diagonal proof if you don’t believe me.  One type is “countable” infinity, which is the infinity you get by counting, and just never stopping.   0,1,2,3, etc. forever.
    There’s also a “bigger” infinity which is uncountable, and it’s the sort of infinity of, say, the real numbers.  The infinity you get from counting forever can be restricted to different lists of numbers, like finite ones, or even infinite ones, e.g. the set of even numbers is still countably infinite.  It’s even the same size as the normal infinite set of counting numbers in the sense that you can find a way of lining them up next to each other, such that there’s one counting number for each even number.  If you were to consider ALL the possible lists of numbers taken from the countable infinity, and think of how many lists there are, that’s actually a different kind of infinity, which is provably “bigger” in the sense that you can no longer assign a number to each list in a one-to-one way, there are just too many lists.

    We have additivity for countable infinities but not for uncountable ones: we don’t know what happens when you add up an uncountable number of shapes. This is where the volume of the ball comes from. It turns out that the solid ball is made of an uncountable number of points, and we don’t have a rule that says what happens when you add up an uncountable infinity of points even if we know they all have zero volume, so we’re free to say it has a non-zero volume.  The idea is that if you take an uncountable infinity of zero volume points and add them up, sometimes you’ll break through and make it to something with non-zero volume.  But sometimes you don’t.  You can make something that’s “too sparse”, for any part of it to count as filled, such that it has zero volume even if it has an uncountable number of points (see Cantor dust).  But “solid”, non-cloudy shapes do end up with non-zero volume. So the question is: where’s the line?  When do you break past zero volume?  One way to think about the Banach-Tarski theorem, is that it says that there are shapes that are so complicated, we have no way of deciding if they manage to actually have a volume.  These shapes are so complicated, we can’t actually define them, but given certain assumptions, we can prove that they exist.  They’re very weird shapes that sit in a middle region between too cloudy to have non-zero volume and too solid to have zero volume.   So we just can’t assign a volume to them at all.  If we could, we would violate countable additivity, because we’d be able to divide balls up into a small number of pieces and then put them back together and change the volume.

    The example with atoms fails, because you’d have a finite number of atoms.  To be able to do this physically, you’d need to be able to divide the atoms up.  To imagine this, imagine you’ve divided your ball into a few pieces, which are clouds of atoms. Then you go on and divide each atom into pieces which are clouds of of atom parts, and distribute those pieces.  Then you divide each atom piece into pieces which are atom-piece-pieces.  And so on.  Forever.  When you get to infinity, keep going: you wouldn’t even have an uncountable number of pieces yet.

    So, we can’t do that.  And that’s why physical metaphors don’t work for Banach-Tarski.

  17. After a few minutes of thinking, it occurs to me that not all immesurable shapes  have to be “cloudy”.  If my understanding is correct, you could also have shapes that are solid on the inside but have veeery complicated edges.  Similar issues abound: atoms would be too coarse and you’d have to jigsaw through them…

  18. If you’re a calculus person, the above example about countable/uncountable additivity also sheds some light on how integration works.  For a strictly positive function, you can split the area under a curve up into a line segment that is associated with each point that the function is defined on.  Each of these line segments has zero volume, and that’s why when you evaluate the definite integral over a single point, you get zero.  And in fact, if you integrate over some countable number of points, you get zero.  But if you integrate over an interval of the real line, that has an uncountable number of points, and thus an uncountable number of those little line segments.  Added together, you break through, past zero, and get an actual positive measure.

    All of this is basic measure theory, which is interesting stuff for the mathematically inclined.

  19. this is how I understand it. Imagine a rigid suitcase that you have to pack with say, multiple long rectangular boxes  (say a box with 2-3 pringles cans stacked end to end inside) . If you precisely place the boxes you can fit them all in one suitcase. If you just throw them randomly in the suitcase, you need 2 suitcases

  20. The video is produced as a part of a student revue on the faculty of Mathematics at the University of Copenhagen (IMF). If you are interested in knowing more about us, we upload a couple more videos from this years show, along with videos from past years. If you want to see more look here; http://www.facebook.com/pages/Matematik-Revyen/21813913501?sk=wall&filter=2, or find our youtube channel at “Matematikrevyen”.

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