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Figure 1

Consider the red path from A that zigzags to B, which takes $n$ even steps of length $w$. The path length of the route $P_n$ will be equal to:

$ P_n = P_x + P_y = \frac{n}{2}\times w + \frac{n}{2}\times w = n \times w $

But $\frac{n}{2}\times w = 1$ beacuse it is the length of one of the sides of the triangle so:

$P_n = 2$

Which will be true no matter how many steps you take. However in the limit $n \to \infty, w \to 0$ the parth length $P_\infty$ suddenly becomes:

$P_\infty = \sqrt{1^2 + 1^2} = \sqrt{2}$

Due to Pythagoras. Why is this the case? It seems the path length suddenly decreases by 0.59!

  • 0
    I think it is because $P_\infty$ isn't REALLY the limit of the $P_n$'s, it is calculated separately. The reason the value seems to drop is because $P_n$ is calculated for a path which only travels parallel to the sides of the triangle, while $P_\infty$ is calculated with the hypotenuse, so there is a fundamental distinction between these values. I don't think there is any contradiction here.2012-02-28
  • 1
    [This](http://math.stackexchange.com/questions/12906/is-value-of-pi-4) question is highly relevant.2012-02-28
  • 1
    A close related question, with some nice pictures, was discussed [here.](http://math.stackexchange.com/questions/43118/how-to-convince-a-layman-that-the-pi-4-proof-is-wrong)2012-02-28
  • 0
    The length of the red path is the sum of all the vertical and horizontal lengths. Each "step" has length $2w$ (a vertical side and a horizontal side). If there are $n$ steps, the total length is $n\cdot2w=2$ (since $w=1/n$). In the limit, as the number of steps becomes infinite, the red path length is 2. But you shouldn't expect this to be the same as the length of the hypotenuse of the big triangle, because the sum of the lengths of the sides of one step differs from the length of the hypotenuse of the step by a factor of $\sqrt2$.2012-02-28
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    As $n$ increases, the steps approximate the area of the triangle, not the perimeter.2017-07-04

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