Let say we have an equilateral triangle and I draw its circumscribed circle, to continue we draw a square in which the previous circle is inscribed. After that we draw the circle circumscribed to the square and to continue the process we plot a regular pentagon in which the previous circle is inscribed and plot its circumscribed circle. Here is a picture:
So the question is : Can we continue this process infinitely? I would say that at some point the figure will somehow "explode". Any ideas what tools I need to understand this process?
As a remark (perhaps false) is to do the inverse processus, starting with an equilateral triangle and drawing this inscribed circle and so one. I guess that the dimension tends to $0$, right?
Best Answer
The key observation is that the circumcircle of the polygon with $n$ sides is the incircle of the polygon with $n+1$ sides; thus $$R_n = r_{n+1}.$$ But we also have the relationship $$\cos \frac{\pi}{n} = \frac{r_n}{R_n}$$ in a given regular $n$-gon, thus we have $$\frac{R_N}{r_3} = \prod_{n=3}^N \frac{R_n}{r_n}$$ and as $N \to \infty$, $$\frac{R_\infty }{r_3} = \prod_{n=3}^\infty \sec \frac{\pi}{n} \approx 8.7.$$ This suggests that were the process of circumscribing regular polygons and circles were continued indefinitely, the figure reaches a limiting, finite size. If we were to perform the reverse and continually inscribe polygons, then this too would result in a limiting incircle of positive radius, which I will leave to others to calculate.
Here is a reference from Mathworld: http://mathworld.wolfram.com/PolygonCircumscribing.html