- #1
Johnny B.
- 5
- 0
Check it out: http://imgur.com/EpYQv
Where's the trick?
Where's the trick?
mathman said:The 10 line is never quite flat. In the limit it consists of a lot of infinitesimal wiggles.
I think the only trick is the bare assertion that 6 = 10.Johnny B. said:Check it out: http://imgur.com/EpYQv
Where's the trick?
This kind of post would be greatly improved by a link...olivermsun said:Isn't this kind of the same as the rectangle-circle problem that was discussed a little while back?
Jocko Homo said:This kind of post would be greatly improved by a link...
Thank you!olivermsun said:
micromass said:Indeed, calculating the length involves taking the derivative. And a uniform convergent sequence might not have a converging sequence of derivatives.
micromass said:That is certainly not my intuition of the topic. The limit IS flat, and the functions mentioned in the example WILL converge uniformly to the flat line.
The only thing is that even uniform convergence does not imply convergence of the lengths. Indeed, calculating the length involves taking the derivative. And a uniform convergent sequence might not have a converging sequence of derivatives. That is the thing that's going on here!
wisvuze said:is this the same explanation for the pi = 4 paradox? The cutting corners method will converge uniformly to the circle, but there may not exist a converging sequence of derivatives?
wisvuze said:Cool, thanks :)
I believe you can prove that the cutting corners thing *does* converge uniformly to the circle; you can define on a quadrant-by-quadrant basis functions f_n to represent the nth cut-corner spiky thing, and C to be the original circle ( or partial circle on each quadrant). Then, you can come up with a sequence of numbers M_n, which represent the distances between C and "bigger circles" ( and also engulfing, being bigger than the spiky thing ). You can make M_n converge, and so by the weierstrass M-test, the sequence {f_n} converges uniformly to the circle.
If the convergence of the spiky things is not even uniform, then there is no hope at all right? All that says, is that for some point on your spiky thing, after some n, the point will come arbitrarily close to the smooth curve. But, the ability to draw a picture like the one linked above, or the pi = 4 picture, with ALL points looking arbitrarily closer and closer to the smooth-curve, it seems like uniform convergence is guaranteed
This paradox is known as the Gabriel's Horn paradox, and it occurs when a shape has a finite volume but an infinite surface area. In this case, the shape is created by rotating the function y=1/x around the x-axis. While the shape appears to have a finite area, its perimeter continues to increase as the function approaches the y-axis, resulting in an infinite perimeter.
Zeno's paradox is a philosophical paradox that raises the question of whether motion is possible if space and time can be infinitely divided. In geometry, this paradox can be explained by the concept of limits. Essentially, as the distance between two objects approaches zero, the time it takes for one object to reach the other also approaches zero, allowing for motion to occur.
The Banach-Tarski paradox is a mathematical theorem that states a solid sphere can be divided into a finite number of pieces and reassembled into two identical copies of the original sphere. This is possible because the pieces can be rearranged and rotated in specific ways without altering their size or shape. However, this paradox violates the Law of Conservation of Volume, which states that the volume of a closed system cannot change.
The Möbius strip paradox, also known as the Möbius band paradox, is a topological paradox where a strip of paper with a half-twist is joined to form a loop with only one side and one edge. While it may seem impossible, this paradox can be solved by cutting the strip down the middle, resulting in two linked strips, each with two sides and two edges.
The barber paradox is a self-referential paradox that asks whether a barber can shave himself if and only if he does not shave himself. This creates a paradox as the barber cannot both shave and not shave himself at the same time. In geometry, this paradox can be resolved by considering it as a logical paradox rather than a mathematical one. The statement itself is self-contradictory and therefore has no solution.