Trying to understand the limit of regular polygons: circle vs apeirogon (vs infinigon?)

Question

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!

Answer 1

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.

Answer 2

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.