How come Hausdorff dimension of Mandelbrot is lower than Lorenz attractor?

Question

I am not good at topology but it seems Hausdorff dimension is a complexity indicator.

If so, the shape of Mandelbrot set looks significantly more complicated than Lorenz attractor locus.

But how come the boundary of Mandelbrot set has such a low Hausdorff dimension of 2.0 and for a simple Lorenz attractor it is about 2.06 ?

https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension

Answer 1

One thing is that since the boundary of the Mandelbrot set lies in a plane, its Hausdorff dimension is as high as it can be and, since the Lorenz attractor lives in space, its dimension can exceed two.