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I had this strange notion some time ago, and I recently wrote a blog post about it, as a mere curiosity. I don't really consider it a "serious" mathematical question; but out of interest, I wondered if someone on this site could shed some light on what principle might be underlying the idea.

Basically, I envisioned a "pseudo-triangle" consisting of two straight edges and one jagged "edge" (not really an edge, since it's jagged, but I'm calling it that anyway):

Pseudo-triangle with a 4-step jagged edge

The above shape has 4 steps, its area is 10, and its perimeter is 16. Now let's increase the number of steps to 8:

Pseudo-triangle with an 8-step jagged edge

This shape has an area of 9 and a perimeter of 16. Now, without me having to write out a formal proof, I think it's pretty clear that as the number of steps increases, the area will approach 8 while the perimeter will remain constant at 16. And the resulting shape will look like this:

Pseudo-triangle with N steps (approaching infinity) along its jagged edge

Ultimately, there's nothing really "mysterious" about this; the shape above is not a triangle, and so it shouldn't be surprising that it doesn't have quite the same properties as a triangle. However, it does approach the same area as an analogous triangle; and, more to the point, it just seems odd.

Is there a concept in mathematics that describes this phenomenon (for lack of a better word)? That is, the effect of some kind of mathematical entity (e.g., a shape) converging to what resembles another entity but differs from it in a critically important and counter-intuitive way (in this case, having a completely different perimeter)?

If it seems that I'm having trouble articulating this question, that's because I am. But hopefully someone out there can see what I'm getting at and shed some light on the issue for me.

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    What do you mean for one shape to converge to another? I suggest you study the concept of *limits* more in-depth.2011-03-25
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    Maybe this will help: http://math.stackexchange.com/questions/12906/is-value-of-pi-42011-03-25
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    @Alex: I realize that I used some terms in a very informal sense. I can remove my usage of that particular word ("converge"), but I don't *think* I have an insufficient understanding of limits.2011-03-25
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    @Eivind: That is indeed very helpful! Looks like basically the same idea.2011-03-25
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    @Dan: The problem does lie in the "convergence" of the shape to the triangle, as the perimeter of the shape is actually not changing and so its perimeter does not converge to that of the triangle.2011-03-25
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    @Alex: Right, I gotcha there. My question is really about what the mathematical description of this "illusive convergence" is, if there is any. Is there a term for it? Has anything really interesting been said about it, aside from oddities like mine or jokes like the one Eivind linked to?2011-03-25
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    @Dan: Well, it depends on the metric you apply to the space of "shapes" in question. This is what determines how limits behave in the space. The metric you have been using is essentially the area of the symmetric difference of the two shapes, which is actually a psuedometric. In order to convert it to a metric you have been considering shapes with a distance between them of zero equivalent. While this works fine for talking about area, this equivalence relation does not preserve information about the boundary of the shapes, and so you cannot talk about the perimeter of an equivalence class.2011-03-25
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    @Alex: Suppose you restrict the "shapes" in question to be simple polygons. The area of the symmetric difference does define a metric on that space, right? And the corrugated triangles do, in fact, converge to a triangle in that metric. OP's observation is just that the perimeter is a discontinuous function on that space.2011-03-25
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    @mjqxxxx: Yes, restriction to simple polygons does make area of symmetric difference a metric for which the perimeter function is discontinuous.2011-03-25
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    The same subject was discussed in "Is value of $\pi=4$" at http://math.stackexchange.com/questions/12906/is-value-of-pi-4/12907#129072011-03-26

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