Banach-Tarski Paradox

If you haven't heard of the Banach-Tarski Paradox, prepare to go "What? Huh? Whoa! Oh." I'm going to try to break this down into a digestible brain meal.

What?

The Banach-Tarski Paradox is a geometry theorem proposed by Stefan Banach and Alfred Tarski in 1924. As per Wikipedia, the theorem is as follows: "Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of non-overlapping pieces (i.e., subsets), which can then be put back together in a different way to yield two identical copies of the original ball." Umm... what?

Here's a picture I made that I think sums up what's going on here...

Magic indeed.

In other words, if you have a sphere, there is a way to cut it up and then rearrange the pieces only by moving and rotating them that will create two identical spheres the same size as the first.

Huh?

Basically, unless you're a mathematician, reading the above Wikipedia link to figure this out will hurt your brain. Badly. So, besides the crazy math that looks like this...

I warned you.

... let's find a way to understand this.

First, forget about these spheres being physical. This "paradox" is only a paradox if we assume these spheres should act like things that exist in the real world. They don't. They're theoretical math spheres. They only exist in the minds of people with way too much time on their hands.

Second, this theorem relies on the idea of infinite divisibility. In other words, you can always cut the sphere smaller. There are no atoms. There is no limit to how small things can get. Remember, this is math. There is no such thing as a number too small.

So, what does this mean for our pretty little sphere? This means it is infinitely dense. Let's use a simple, somewhat inaccurate analogy to understand why infinite density is so important.

Say we take a physical, real-world sphere and cut it in half. Now we take a look at our half sphere. It has one flat side, and the rest is still spherical.

See that flat side? There are a bunch of teeny tiny atoms on it. Let's move some of them, while pretending we aren't stretching anything. If we pull the middle one up as far as the radius of the sphere, and move the others accordingly, we end up with a full sphere that's the same size of our first one! But, what about the density? Since this is the real-world, it's now half as dense as our original sphere.

And there's the key. In math theory world, there is infinite density. So we could do this little analogy all day and never run out of atoms (which, in math theory world, are called "points").

Still having trouble? Here's one more example. For this, let's forget the no stretching rule. Imagine we have a balloon. We blow it up and it makes a sphere with a radius of one foot. Now, we deflate it and cut the balloon in half. We take one of these halves and glue it together. Now we have a balloon made of only half the material. No one said we couldn't blow it up to the same size, but this time the balloon material itself is thinner. Chances are, though, that if we continue to repeat this, at some point we will not have enough material to blow up the balloon to the same size. But, if the balloon had infinite density, we could keep stretching and cutting it forever, continuing to make more and more balloons of the same blown up size.

Whoa!

While these analogies aren't exactly what the Banach-Tarski Paradox suggests, it demonstrates the same concept in a much simpler way. Basically, Banach and Tarski were making more and more spheres that normally would be less dense, but it doesn't matter, because in math theory world, there is infinite density.

So now that we know this, we realize the sad truth. This is absolutely the least exciting paradox of all time. This has absolutely nothing to do with the real world and only works because math theory world has crazy rules. It's like if I made up a world where gravity was optional then got surprised when my dog started peeing on the ceiling. Well, duh! The game was fixed to begin with.

Oh.

Sorry. Didn't mean ruin it for you. Basically, the Banach-Tarski Paradox is simply an exploration of the consequences of theoretical mathematic rules. If division, and therefore density, is infinite, it's possible to make two spheres from one. Now you know!