Are my expectations true?
My expectation 1 : There exist some intersection points, which are not on the center of the regular $n$-gon, of three or more diagonals when you draw all diagonals of a regular $n$-gon if and only if $n\ge 12$ is an even number.
My expectation 2 : There is no intersection points of seven or more diagonals when you draw all diagonals of a regular $n$-gon for any $n$.
My expectation 3 : There exist some intersection points of six diagonals when you draw all diagonals of a regular $n$-gon if and only if $n$ is a multiple of $30$.
If these are famous, please let me know some information about them.
Example : Let $f_i(n)$ be the number of intersection points of $i$ diagonals when you draw all diagonals of a regular $n$-gon. $$f_2(18)=1512,f_3(18)=216, f_4(18)=54, f_5(18)=54.$$ The sum of these points is $1837.$
See Poonen and Rubenstein, The number of intersection points made by the diagonals of a regular polygon, available at http://math.mit.edu/~poonen/papers/ngon.pdf