Is the boundary of the Mandelbrot set jagged or smooth? As the title states, I am wondering if the boundary of the Mandelbrot set is jagged or smooth. If it is jagged, is there some algorithm to find the vertices of any one of them? Are there an infinite number of vertices? I am aware that we do not know whether or not the Mandelbrot set is actually computable (I believe it hinges on the truth of a conjecture). Does this mean that we cannot compute the exact co-ordinates of a point on the boundary (with the exception of trivial points like $c=-2$)? Thanks in advance!

It is certainly not smooth. In fact, Shishikura showed in '94 that the boundary is itself a fractal with Hausdorff dimension $2$ which informally, is very jagged, and is the same dimension as the Mandelbrot set itself.

See:

> Shishikura, M. "The Boundary of the Mandelbrot Set has Hausdorff Dimension Two." _Astérisque_ , No. 222, 7, 389-405, 1994.