Sorry, I can't visualize your procedure because there are too many possible variants.

Absolutely any n sides? Not necessarily connected?Now to make a polygon with 2(n+1) sides, we take the 2n-polygon which has already been divided into rhombuses. We take any n sides

This looks suspicious because if those lines are not drawn at a precise distance, then the resulting figure will not all have sides equal, i.e., will not be a regular polygon.we draw n lines parallel to those n sides, at some distance

There are so many ways to "join them with the 2n-polygon". It is not cleat what you mean.And then join them with the 2n-polygon with a line at each end.

Could you draw a picture how you transform a hexagon into an octagon?