No hope to reach the center

From the figure, we can see how the radii of two consecutive circles relate: $r_n=r_{n-1}\cdot \cos\left(\dfrac{\pi}{N}\right)$ .

Applying the recurrence from the start, with $r_0=1$ , we get:

$r_{\infty}=\displaystyle \prod_{N=3}^{\infty}\cos\left(\dfrac{\pi}{N}\right)\approx0.114942$

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I evaluated the infinite product using Wolfram Alpha .

See this ............