An "almost" geodesic dome

Question

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

$$ \left( \frac {r+1}{n}, \sum_{m=0}^r \sqrt{ \left( \sqrt{3}\left(1-\frac mn\right)^2 \sin\left(\frac\pi n\right) \right)^2-\frac1{n^2}} \, \right)$$

With $n = 100 $ this is the plot made with Mathematica, but I wasn't able to find that function

Answer 1

The expression $$f_n(x):=\sqrt{\Big(\sqrt{3}\Big(1-\dfrac x{n}\Big) \sin\dfrac \pi{n} \Big)^2-\dfrac1{n^2}}$$ is defined for real $x\ge0$ iff $$x\le x_n:=n\Big(1-\frac1{a_n\sqrt3}\Big),$$ where $a_n:=n\sin\frac\pi n\to\pi$ (as $n\to\infty$). Also, the function $f_n$ is decreasing on the interval $[0,x_n]$. So, $0\le f_n\le f_n(0)=O(1/n)$. So, the sum, $\sum_{m=0}^r f_n(m)$ (corrected according to my comment), in your displayed expression differs from the integral $$I(n,r):=\int_0^r f_n(x)\,dx$$ for integers $r\in[0,x_n]$ only by $O(1/n)$.

Using the substitution $v=(1-x/n)a_n\sqrt3$, we get $$I(n,r)=\frac1{a_n\sqrt3}\int_{(1-r/n)a_n\sqrt3}^{a_n\sqrt3}\sqrt{v^2-1}\,dv \\ =\frac{G(a_n\sqrt3)-G((1-r/n)a_n\sqrt3)}{a_n\sqrt3},$$ where $$G(v):=\frac{1}{2} v \sqrt{v^2-1}-\ln \left(\sqrt{v-1}+\sqrt{v+1}\right),$$ an anti-derivative of $\sqrt{v^2-1}$.

It follows that, for each $$t\in(0,T),\quad \text{where }T:=1-\frac1{\pi\sqrt3},$$ the vertex $(\frac{r+1}n,\sum_{m=0}^r f_n(m))$ tends to $$(t,F(t))$$ as $n\to\infty$ and $r/n\to t$, where $$F(t):=\frac{G(\pi\sqrt3)-G((1-t)\pi\sqrt3)}{\pi\sqrt3}.$$

Here is the graph $\{(t,F(t))\colon0<t<T\}$: