7

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:

enter image description here

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?

  • I think I've seen somewhere the second problem about the inverse and I vaguely remember the value 12 as limit. – ypercubeᵀᴹ Oct 29 '14 at 19:37
  • @ypercube What tools is needed? I like this problem but I don't have idea how can I tackle this problem..:/ –  Oct 29 '14 at 19:38
  • What do you mean by "explode?" Is it a relationship between the $n$th circle and the $n+1$-th circle? – John Oct 29 '14 at 19:47

1 Answers1

10

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

heropup
  • 135,869
  • 1
    I was about to post the same kind of answer, except that I wasn't so sure that I could assert that the infinite product converges. (Your MathWorld link settles the question. :) – Blue Oct 29 '14 at 19:56
  • 1
    @r9m No; the apothem is $r_n$ which is adjacent, and the hypotenuse is $R_n$, thus it is correctly written $\cos \pi/n$. – heropup Oct 29 '14 at 19:57
  • @heropup yes yes ! my bad ! (+1) nice :) – r9m Oct 29 '14 at 19:58