It seems so ridiculously counterintuitive, how could it possibly work?
In: Mathematics
The short answer is that the sphere is split into shapes that are so complex that they don’t have a volume. I don’t mean there volume is 0, I mean there volume just isn’t definable at all.
There is no simple explanation of the proof unfortunately. This paradox isn’t possible in 2D sk any proof must rely on properties of 3D that differ from 2D.
It’s couterintuitive because it’s really about some topology thing that isn’t really anything anyone deals with in common life
You treat the shape as an infinite cloud of points. You divide it up into at least 5 fractal clouds of infinite points. You can then reassemble those fractals into two spheres. The secret is that “having enough stuff” isn’t relevant because infinity.
∞ = ∞/5 = 2*∞
The proof is just that you can fill all the space and create a spherical surface.
On the x axis, the number of points in the interval (0, 1) is the same as the total number of points in the intervals (0,1) and (1,2). If this statement is already confusing/counterintuitive, you’ll need to learn the basics about cardinality of infinite sets before proceeding any further. (Can one infinite set be “bigger” than another one? How can we tell?)
If you already know why the above is true and you struggle to get some intuition about Banach-Tarski, here’s a much simpler construction that may be easier to digest and may help you.
Imagine the unit circle centered at (0, 0). Start on the circle at (1, 0). Mark the point red. Now walk around the circle counter-clockwise forever and each time you walk a unit of distance, mark the current point red again.
We now have a countably infinite set of red points. (Mathematically speaking, they are the points at x radians from the origin, for all non-negative *integer* x.)
Recolor the first 100 of the red points green. Then, color the rest of the circle (i.e., everything that’s neither green nor red) blue. We have now divided the circle into three mutually disjoint parts: red, green, and blue.
Now move the green part aside. After you do that, take the red part and rotate it around the origin by 100 radians clockwise. Once you do that, you now have a full unit circle consisting of a red and a blue part only. And in addition to that circle you now also have 100 green points set aside. In other words, you reassembled the circle and yet you have leftovers: a small part of a second circle.
In 3D and using more involved “shapes” you can do a similar thing and have more leftovers — enough to assemble a second full ball.
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