eli5:if there’s an infinite number of triangles in a circle starting from Centrepoint to circumference then why cant we make up 2 circles in the real world from a single circle?

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eli5:if there’s an infinite number of triangles in a circle starting from Centrepoint to circumference then why cant we make up 2 circles in the real world from a single circle?

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Because infinitesimally small triangles don’t actually exist in real life.
Nor do true circles, it’s all jagged polygons of atoms.

Because infinite triangles doesn’t mean you can magically pull a sphere out of thin air? How’s that reasoning supposed to work?

Do you mean that, because we can infinitely divide something, we should be able to reassemble those infinitely small things into two things? Because we can, but those two things will each be half the size of the original.

Put simply, infinite infinitely small things can have finite sizes, and in fact infinities can have different sizes.

Think of it like a preservation of mass in physics: if your circle is weighing 1ton, and you split it into 2, then both half will be weighing 0.5 tons, and there is no way around it. And even if you’re splitting indefinitely, if you keep track of the “weight” of each of the infinite triangles, then they must sum at 1ton.

Well, that’s a little more complex, since triangles that are infinitely small have weight “zero but not really zero”, so you need a more complex notion of measure to be able to talk about the weight of infinitely small triangles, but you can still do it.

In ELI5: instead of trying to weight a single infinitely small triangle, you never look at triangles alone and instead look at set of triangles and measure the weight of those sets.

But in the end, you still finish with the same issue: even if you split your infinitely many triangle into two infinities, by looking at their weight you will notice that the first infinity weight 1ton while the new infinity your created weight 0.5 ton each, so not enough to make two circles of the same size. (Though you could try to make two smaller circles, and then it become a much more complex problem)

[Small note: when you add a new notion to a problem that was not present before, like the weight here, you must check that this notion is compatible with everything you’re doing, otherwise you’re just cheating by adding constraints that weren’t there. And here “moving triangle around” will preserve the weight, so that’s fine. If you were to allow to scale-up or scale-down the triangles, then this “weight” argument would fall apart, but I’m assuming that in your question you were not allowing that.]

If you are talking about copying an object as in the [Banach-Tarski paradox](https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox) – we say that we can’t do it because the proof of the paradox is non-constructive (and a constructive one cannot exist). We have no way of figuring out which points we would need to move to turn one circle into two circles, even if we had an imaginary machine that could select infinitely many infinitely small points and move them around. The proof simply says that such sets of points exist.

Not all infinities are equal. The total number of infinitely small triangles in two similarly sizes circles is always two times the number of triangles in one circle.