I was playing with a geogebra applet that shows regular $n$-gons of radius $1$ with their diagonals. For example, here is the $12$-gon with its diagonals:
For any value of $n$, when I shrink the image, the image becomes darkened (due to the thickness of the lines), but sometimes there remains a faint single white ring, indicating a ring of exceptionally large cells. For example:
Now here's the interesting thing: The radius of the white ring always seems to be approximately $1/e$. (I used Perfect Screen Ruler.)
For some $n$-values, I cannot perceive a single white ring; I guess it still exists but is not perceivable due to limitations in pixelation and/or visual acuity.
I can formalize my conjecture as follows:
In a regular $n$-gon of radius $1$ with it diagonals, if $d_n=$ distance between the centre and the centroid of one of the cells with the greatest area (excluding the centre cell when $n$ is odd), then $$\lim\limits_{n\to\infty}d_n=\frac{1}{e}$$
Question: Is my conjecture true?
(This question was inspired by another question: Distribution of areas in regular $n$-gon with diagonals, as $n\to\infty$.)