Paradox curve lengths limit

Asked Mar 03 '20 at 05:34

Active Mar 03 '20 at 05:34

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A possible similarity link

Consider a sequence of curves as follow:

$C_1$ is a semicircle drawn on the segment $[0,1]$, as in figure bellow; $C_2$ is the union of semicirocles of diameters $[0,1/2], [1/2,1]$; \, $C_3$ is the union of semicircles of diameters $[0,1/4],[1/4,1/2],[1/2,3/4],[3/4,1]$ etc....

Obviously, this sequences of curves tend to a limit, which is the segment $[0,1]$.

Moreover, the lengths of this curves are \begin{align*} l(C_1)& = \frac{\pi}{2} \\ l(C_2)& = 2\times \frac{\pi}{4} =\frac{\pi}{2}\\ l(C_3)& = 4\times \frac{\pi}{8} =\frac{\pi}{2}\\ \end{align*} So the sequence does not tend to $l([0,1])=1$.

How do you explain this bizarreness?

Any help is welcome and thank's in advance.

asked Mar 03 '20 at 05:34

Furdzik Zbignew

1

What is bizarre about it? – Jyrki Lahtonen Mar 03 '20 at 05:40
The fact that $\pi=2$ – Furdzik Zbignew Mar 03 '20 at 07:43
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The link you gave specifically gives an explanation, namely that the length function is not continuous. That is, the length of a limit of curves is not the limit of the lengths of the curves. For some information about the history of this result, see What is the history of staircase or =4 paradox? – Dave L. Renfro Mar 03 '20 at 08:02

Paradox curve lengths limit

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