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A regular $n$-gon contains a regular $m$-gon, where $n$ and $m$ are coprime, with no sides coinciding.

What is the maximum number of contact points between the $n$-gon and the $m$-gon?

(I'm not asking for the maximum in terms of $n$ and $m$; I'm asking for the absolute maximum.)

I made up this question. My guess is that the maximum is four, and that when this maximum is attained the polygons share a line of symmetry. (I made a simple example showing that four is attainable.)

I made a desmos graph of a regular $6$-gon containing a regular $7$-gon. You can use the sliders to expand/rotate/translate them.

I posted a similar question on MSE.

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  • $\begingroup$ Sliding to $R=1{.}13$, $T=-0{.}27$, $p=-0{.}035$, $q=-0{.}003$, it looks like there are 5-ish contact points. I haven’t tried to verify this other than by eyeballing. $\endgroup$
    – Emil Jeřábek
    Commented May 26, 2023 at 13:16

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