# When is it possible to find a regular $k$-gon in a centered $n$-gon?

For $$n \geq 3$$, say that a centered $$n$$-gon with $$L$$ layers is given by the origin, $$(0,0)$$ together with the points $$\left\{\alpha\zeta_n^j + \beta\zeta_n^{j+1}\ \middle\vert\ 0 \leq j < n, 1 \leq \alpha \leq L, 0 \leq \beta \leq L - \alpha\right\},$$ where $$\zeta_n$$ is a primitive $$n$$th root of unity.

For example, here are the cases when $$(n,L) = (3,4)$$, $$(n,L) = (4,4)$$, $$(n,L) = (5,3)$$, and $$(n,L) = (6,3)$$:

### Question

For a given integer $$k \geq 3$$, what are the conditions on $$n$$ and $$L$$ such that it is possible to find $$k$$ points that form a regular $$k$$-gon in the centered $$n$$-gon with $$L$$ layers?

Besides the hexagon in the centered $$3$$-gon, are there any examples of $$k$$-gons in a centered $$n$$-gon where $$k\nmid n$$?