A regular $ n$-gon is inscribed in the unit circle centered in $0$.We want to build an "almost" geodesic dome upon it this way: on each side of the $n$-gon we build an equilateral triangle whose dihedral angle with the $n$-gon is such that the distance from $0$ of the upper vertex's projection on the unit circle plane is $(1-1/n)$; over this "first floor" we connect all the upper vertices and we have a new smaller $n$-gon inscribed in the circle with radius $(1-1/n)$, so we repeat the process and this time the distance from $0$ of the upper vertices' projection on the unit circle plane is $(1-2/n)$ and so on.If the limit $n \rightarrow \infty$, which function fits the border of the dome? What is the maximum height?
I tried to solve this way: $n$ fixed and $r < n$, each upper vertex coordinate is
$ ( \dfrac r{n}, \sum_{m=1}^r\sqrt{( \sqrt{3}(1-\dfrac m{n})^2 \sin(\dfrac \pi{n}) )^2-\dfrac1{n^2}} )$
With $n = 100 $ this is the plot made with Mathematica, but I wasn't able to find that function
