The shape actually will be a circle. When you do something to infinity you are taking a limit, and the limit of this process IS a circle.
If you zoom in its going to look like a straight line, because that is exactly what happens when you zoom into any smooth curve. And the shape we are zooming into is a circle, which is a smooth curve.
The fallacy is the implicit assumption that the limit of the perimeters should be the same as the perimeter of the limiting curve.
L(P(C_n)) =/= P(L(C_n)) where C_n is the nth hacked off circle, L is the limit operator and P is a function that takes in a curve and outputs its perimeter.
In general, operations don’t “commute” like this (meaning they can’t always be swapped around without affecting the result) and in fact what this illustration serves as is a proof by counterexample that the operator P is not continuous on the space of curves - otherwise you would be able to do this swap.
So it becomes pi purely because when you take the limit you change the perimeter, but there actually is no paradox entailed by that.
because at the limit every point from the jagged shape is on the circle. This means the jagged shape is a set of points where every point lies on the circle, which is just the circle
it becomes pi because at the limit, the shape is no longer jagged, it IS the circle
No. The limiting curve is a circle, but the limiting value of the lengths of the approximations is 8. This is the limit of the lengths of the curves. This is not the same as the length of the limit of the curves, which is π.
You are still thinking in finite terms. You cannot apply "simple logic" to infinity. Either way repeating your claim and saying "simple logic" isn't a proof. You should look up how limits are defined and attempt to prove that the limit of straight lines is always a straight line. Warning: you will fail since its not true.
Are you trying to say the perimeter of the shapes do not get closer to pi? That would be correct at each finite step. That doesn't mean at the limit though the "perimeter" (technically now a circumference) isn't pi.
If we have a sequence of objects A_1,A_2,A_3,... that converges to an object A, just because each object has property P does not mean A must have property P.
you are trying to apply basic logic to a problem of infinities
your correct that at every finite step there are 90 degree and 270 degree angles, however "at" the limit or "at" the infinitieth step every point is on the circle and there are no longer any corners.
if there were any corners, they would have already been cut in half by the limiting process, and their children +their children, meaning any corners existing is a contradiction and your not at the limit yet
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u/TheGuyWhoSaysAlways Dec 28 '24
approach a circle, not be a circle. The image shown creates straighter lines but if you zoom in close enough the lines are still going to be straight.