Can a ball be reassembled into two identically-sized copies of itself?
The answer is surprisingly yes!
What is this Theorem?
The Banach-Tarski paradox is perhaps the most astonishing theorem in modern mathematics, being highly counterintuitive. Essentially, it states:
A sphere can be broken off into a finite number of pieces and reassembled to make two copies of itself, identical in size! (Fig 1)
Essentially, one can think of this in similar terms from my post on Hilbert's paradox. Consider the set of all natural numbers {1,2,3, • • • }. Now, we can divide this set up into two equally-sized sets (both uncountably infinite), by separating even and odd numbers. How come the set of only even numbers is equal in size to the set of all natural numbers? The equality of sizes of sets is established by a one — to — one correspondence, which you can accomplish through 1→ 2,2 → 4, 3 → 6, • • •, n → 2n. Since the sphere we are dealing with is not an actual sphere of atoms but a mathematical one defined as an infinite collection of points, we have a similar analogy.
Infinity and its Surprises
Before we proceed, let's pause and think about infinity. Consider a hyper-dictionary which contains every possible word the English language can assemble, meaning all possible sequences of letters. How would we go about organizing this dictionary? The chief editor of the dictionary suggests 26 volumes, one for each starting letter. One of the assistant editors suggests that they can fit everything into one volume, as the reader can simply remove the "A" or first letter of each word in Volume A to get every single volume's worth of words. Now, let's touch on another thing. Consider a circle of radius 1. Now, remove one point from the circumference of that circle. (Fig 2)
Can you move the points in the circle in such a way to fill that point? It turns out the answer is yes! Consider labelling the points on the circumference, starting from a random starting point and then labelling a new point every time you cross a distance 1. Similar to the case of Hilbert's hotel, we can move point 1 to the empty point, point 2 to point 1, point 3 to point 2, and so on. After doing this, no empty points are left because they are never repeated! Weird.
A Simplified Explanation - 1
Back to Banach-Tarski. Let's start by considering a sphere, simply an uncountable collection of points ordered in such a way to produce a ball-shape (we'll discuss how this deviates from our conception of objects later). Now, we will try to name each point on the surface of this sphere by assigning each point a sequence of moves to get there from a starting point, each of a specific length. For example, we can call point A in Fig 3 "UR," after the sequence of events "Right, Up." 
Notice that we apply moves from right to left, which we will see the importance of, later on.
We have four total moves we can do: Left, Right, Up, and Down (L,R,U,D, respectively). Of course, instances of LR, RL, UD, or DU are not permissible as they will cancel out. We can then imagine listing each possible sequence of moves and colouring points based on the last move we make. However, starting with only one point is inevitably going to make us miss some points. To avoid this, we will select all points that we miss to be starting points too (which we will colour distinctly). Alongside our hyper-dictionary of moves, this would name every point. Except for poles.
A Simplified Explanation - 2
The problem with these poles is that they can be counted and named more than once. Adding any subsequent moves can preserve the point's location. We know that every sequence has two, so we will just count these out and colour them another colour. We now have six total decompositions of the sphere's surface: starting points, poles, and one for each move. Since we are dealing with a solid sphere, we can extend each point into a line ending at the centre of the sphere, making seven total pieces with the centre (it is not included in other pieces). Now comes the really neat part.
Take the "left" piece. Based on our definition, it is simply the set of all points on the surface that had a sequence ending with a "left," i.e.
L, LL, •••, LU, LUU, ••• , LD, LDD, ••• , LUL, LULL, •••
Let's rotate this piece to the right. This is just an additional R at the end of the sequence, which we write as the first letter (remember, we perform sequences from right to left). But if we add the R, we can cancel the R and L
, L, •••, U, UU, •••, D, DD, ••• UL, ULL, •••
We now have every move that ends with L, U, D, and a blank move in the beginning, which is all starting points! We now have turned our left piece into a four pieces with a simple rotation. Just like the hyperwebster!
A Simplified Explanation - 3
Adding the right piece, poles, and centre to our rotated left piece, we have ourselves a whole new sphere! But, there's still three pieces left over. The up, down, and starting points pieces. Let's take a look at the "Up" piece. For this piece, however, instead of rotating all points, we will rotate all points that aren't a pure string of U's.
U, UU, UUU, • • •, UL, ULL, • • •, UR, URR, • • •, ULU, ULUU, • • •
Turns into
U, UU, UUU, • • •, L, LL, • • •, R, RR, • • •, LU, LUU, • • •
Which gives us the Up, Left, and Right pieces! We will now combine this rotated Up piece with the clown pieces and starting points to get a whole new sphere! That is, except the poles. And the center point too! To fix this, recall the circle with the missing point. The number of holes in the sphere is countably infinite, being made up of just the poles and the centre point. We can fill them all just like we did in the example of the circle.
Wow, that was a marathon! We just took one ball and made it into two. Almost seems like magic. The implications of this for both abstract mathematics and science can be huge. The paradox relies critically on the axiom of choice, an initially controversial axiom of set theory, and provokes thought into the nature of mathematics. Okay, it might be important mathematically, but could this paradox have any physical meaning?
Real World Examples
If matters were infinitely divisible, which it is not according to our most cutting-edge theories, then it might be possible. Even then, the pieces we defined are so exotic that they don't have a well-defined notion of volume (or more generally, measure) associated to them. As much as the Banach-Tarski paradox is about weird set-theoretic geometry, it is about fundamental notions we have about mathematics and breaking them clown. Paradoxes like this one best embody mathematical exploration because they invite us to lands which challenge our current knowledge and add to it more than anything else.
There has been increasing interest in the application of this paradox to particle physics, particularly the manner in which particles collide. Throughout history, there have been countless examples of things that "didn't make sense" that eventually turned into the most useful science we know. From negative numbers to imaginary numbers, the abstraction of many mathematical concepts always caused controversy. Nonetheless, these numbers find application in almost every scientific field now. Beautifully, mathematics often finds its way into theories of the physical world within a few centuries. Could the Banach-Tarski paradox be just another example of that?
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