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In the definition of regular polygon at the Wikipedia, there is this statement about the limit of a n-gon:

"In the limit, a sequence of regular polygons with an increasing number of sides becomes a circle, if the perimeter is fixed, or a regular apeirogon, if the edge length is fixed."

My question is:

I am trying to understand how to imagine an apeirogon versus a circle, but that definition (regarding edge length) and the visual representation found in the Wikipedia or Wolfram is not very clear to me.

Indeed looking for information in other websites, I found also something called "infinigon", but it seems that there is not formal definition (or I did not find it).

This topic could be very basic but I did not find former questions at MSE, and I am interested in understand the difference. Thank you!

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    $\begingroup$ Pick a line segment $AB$ in the plane. Construct an equilateral triangle one of whose edges is $AB$. Now construct a square. Then a hexagon. Then a dodecagon. Imagine what a 100-gon on $AB$ would look like. Can you see a pattern? $\endgroup$ – Rahul Jul 11 '15 at 6:59
  • $\begingroup$ @Rahul thank you and sorry for my delay writing here. About your comment, what is the meaning of "on AB"? does it mean " construct a 100-gon one of whose edges is AB?" in that case I think I understand: the limit of that sequence of polygons would be the apeirogon, and as the segment is fixed, it will never become a circle if we go to the limit, because the segment will not become a point. Is that right? :) $\endgroup$ – iadvd Jul 11 '15 at 12:26
  • $\begingroup$ Yes, you've got it. $\endgroup$ – Rahul Jul 11 '15 at 16:47
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    $\begingroup$ @Rahul cool! very appreciated! please if you have a second if possible may I ask you to convert your comment in an answer? it was very useful and I can close the question :) again thank you! $\endgroup$ – iadvd Jul 12 '15 at 0:37
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This is an extension to Rahul's comment.

Grab a polygon, say a regular triangle, and you start bending all of the edges in half to give your more edges. If you keep doing it you get a circle. Notice when you bend an edge it turns it into two shorter edges. But you didn't add anything new to it. So a circle technically have no sides or infinitesimal sides.

While apeirogon, you don't bend it, that's not ok. Instead, you cut it at one of the vertices, and then glue an edge of the same length to your cut. So effectively replacing each vertex with an edge of the same length. Notice while the length of each side did not change, you end up with a way bigger, because of new stuff, shape, with actual infinite sides...and infinite size = perimeter. Hence the difficulty visualizing it.

Circle is the limit with the short possible side without changing what you have to begin with (think of increasing number of sides is just a consequence). Apeirogon is limit of the number of sides you add. They are different limits in that sense, roughly speaking. So my answer focuses more on how the two differ, rather than just what they are.

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  • $\begingroup$ thank you for the extension to the comment :) certainly the apeirogon is difficult to catch, intuitively the circle seems to be the "limit" of a polygon, but once you think about the way of constructing it, clearly there is a big difference. $\endgroup$ – iadvd Jul 21 '15 at 3:58
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    $\begingroup$ Indeed. I've found, myself included, that most people have a really hard time imagine infinity in general. The idea of infinite sides and vertices, which is a natural way of interpreting "limit of a polygon", people immediately think of things with finite size and shape, so you get the circle, which is perfectly ok answer because it does certainly have infinite vertices. But if you think about it, what do we mean by side? It's a straight segement with a fixed length, usually non-degenerate. So a circle is not good enough. This is the charm of Math, we find flaws in intuition, and improve it. $\endgroup$ – Darth Alpha Jul 22 '15 at 5:24
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Just to close the question I am adding Rahul's comment here (thank you!):

Pick a line segment AB in the plane. Construct an equilateral triangle one of whose edges is AB. Now construct a square. Then a hexagon. Then a dodecagon. Imagine what a 100-gon on AB would look like. Can you see a pattern?

My comment: the limit of that sequence of polygons would be the apeirogon, and as the segment is fixed, it will never become a circle if we go to the limit, because the segment will not become a point.

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