Can a polygon with an infinite number of sides be viewed as a line?

Question

The inner angles of a polygon approach 180º as the number of sides (N) of the polygon increases.

So, if N approaches infinity, we would have a circle.

But... At infinity, we would also have a set of sides with 180º between each other, i.e., a straight line.

So, is it a circle, line or both?

This seems strange, but how could this be explained? Is this reasoning wrong?

Thanks :)

Answer 1

What happens is that the angles of the polygon tend to the tangent lines of the circle at each point.

If you'd like, the circle is a ($1$-dimensional) manifold : it locally looks like a line, but has more sophisticated global properties.

Answer 2

There is no such thing as a regular polygon with infinitely many sides. So it has neither this nor that property -- or, depending on temperament, you can say that it is (vacuously!) a straight line and a circle at the same time (in addition to being made of green cheese).

If one takes care to make appropriate definitions, one may talk about the limit of a sequence of figures that are regular polygons with ever increasing numbers of sides. In that case, the limit might well be a straight line if your sequence of polygons all have the same side length, or it might be a circle is your sequence of polygons all have the same diameter.