Paul N. Lee wrote:
A question was recently asked in the FractalForums concerning the quantity of midgets found closely clustered together:
http://www.fractalforums.com/mandelbrot-and-julia-set/proximity-of-minibrots...
I was wondering if anybody had some images readily available for reference purposes?? (I am too tired at the moment to recall where I have seen such renderings.)
I guess a better view (of larger midgets) could be achieved by panning to the lower right, but still: http://www.abm-enterprises.net/fractals/mandelbrotgalaxywallpaper.html Two quotes from Michael Frame's pages at Yale about fractals: http://classes.yale.edu/fractals/MandelSet/MandelBoundary/Mis.html "The Misiurewicz points are scattered throughout the boundary of M: every circle centered at every boundary point encloses infinitely many Misiurewicz points." and http://classes.yale.edu/fractals/MandelSet/MandelBoundary/2Dim.html "The boundary of the Mandelbrot set is a very complicated place, because in addition to the Misiurewicz points, every circle centered at every boundary point encloses infinitely many centers, hence infinitely many copies of the Mandelbrot set." Since we are talking about circles of any (finite) size, that's quite a complicated structure indeed! As of guessing, computing, whatsoever the number of (roughly) same size midgets in an area, that question is probably best put to someone like, say, Robert Devaney or any other of the mathematicians working in the field. The only thing I know is that midgets "shrink faster than you can look", meaning that if you zoom into an M area by a given factor, the average midget size in view goes down by about the square of that factor. Regards, Gerald