[ogogmad]

Question: Do fractals have projective duals? In the projective plane RP^2, the dual of a point is a line, and vice versa. Do fractals like "Burning Ship" and "Mandelbrot" have projective duals, where each point in the set is replaced with a line? What would these look like?

[ronald_raygun]

Mmm. I don't think that's a super useful way to think about it. The way I've always conceptualized it was that if you look at fractals, they are determined by some complex polynomial. So you could look at the space of polynomials, and each point in there , would be its own fractal. For the right space of polys thats a Hilbert space, and so by Riesz representation theorem it would have a dual space, but I dont think thats super useful here

[Jaxan]

I think Julia sets are perhaps what you are looking for.