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Situation 1: A regular $n$-gon is inscribed in a circle. As $n$ increases without bound, the area of the $n$-gon approaches the area of the circle and the perimeter of the $n$-gon approaches the circumference of the circle.

Situation 2: Consider a $1$ by $1$ square with one side labeled South and the other labeled North East and West as in a map.

A path is constructed from the Southwest corner to the Northeast corner.

If the path runs east on the south side for a distance $\frac{1}{2^n}$, then goes north for the same distance, then east again for distance $\frac{1}{2^n}$. And so on. Then the total length of the path is $2$

As $n$ increases without bound the area under the path and above the south side of the square approaches the area under the diagonal, but the length of the path remains $2$ and does not approach the length of the diagonal.

Why is there a difference?

Richard
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Jim H
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2 Answers2

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I believe the difference you are witnessing is somehow akin to the one between strong and weak minimisers in the Calculus of variations.

In your first case, the $n$-gon converges to the circumference in a very peculiar fashion: indeed, if you consider one side of the $n$-gon and the circumference bewteen the side's end, you will notice that the "slope" of the two curves converges too:and so does the length.

In the second case, this is not true. While one intuitively sees the "piecewise" line converges to the diagonal, the slopes are not getting any closer: then the length of the "piecewise" path does not converge to the one of the diagonal. The norm under which one can claim converges occur is very different: given an $\epsilon$, one can find an $n$ such that the maximum distance between the diagonal and the "piecewise" path is less than $\delta$, yet the length does not converge. A similar problem is seen by sailing between point $A$ and $B$, in the wind's direction. The boat has to sail at fixed angle with respect to the wind, and whatever tacking strategy employs, the distance is constant. By tacking "infinitely" often, one can stay as close as desired to the $AB$ line, yet the travelled length will not vary.

An aedonist
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As $n \to \infty$, don't you get a countable set instead of the continuous diagonal, so the set would be discontinuous, as no irrational number by definition can be included?

gt6989b
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