A regular $n$-gon has $n$ axes of symmetry. I was wondering about necessary and sufficient conditions on the polygon for the converse to be true. That is, what kind of shape would I need such that I can say "This has $n$ axes of symmetry, therefore it is a regular $n$-gon."
At first I thought convexity was sufficient, but this isn't true, since a rhombus has $2$ axes of symmetry and it is certainly not a "regular $2$-gon", although there is no such thing as a $2$-gon, so maybe convexity is sufficient with the assumption that $n\geq 3$. Convexity is certainly necessary, since a $5$ pointed star has $5$ axes of symmetry and is not a regular pentagon.
Does anyone have any insight into this problem?
You could have a polygon whose sides have alternating lengths $a$, $b$, $a$, $b,\ldots$ but is nice and symmetric. Here's a concrete example: the octagon with vertices $(\pm2,\pm1)$ and $(\pm1,\pm2)$.