Fractal n-gons and their Mandelbrot sets

We consider self-similar sets in the plane for which a cyclic group acts transitively on the pieces. Examples like n-gon Sierpinski gaskets, Gosper snowflake and terdragon are well known, but we study the whole family. For each n our family is parametrized by the points in the unit disc. Due to a connectedness criterion, there are corresponding Mandelbrot sets which are used to find various new examples with interesting properties. The Mandelbrot sets for n > 2 are regular-closed, and the open set condition holds for all parameters on their boundary, which is not known for the case n = 2.