A new scaling along the spike of the Mandelbrot set

Abstract Using the zeros of a family of real polynomials, we locate a sequence of midgets along the spike of the Mandelbrot set. By this method we can readily find midgets (tiny copies of the Mandelbrot set) of high cycle number; we include a picture of a 200-cycle midget magnified by a factor of 5.68 × 10 238 . The ratios of distances between the cardioid centers for successive midgets in this sequence exhibit an asymptotic scaling, as does the ratio of head-to-center distances. We present evidence supporting a pattern of scalings for some “generalized Mandelbrot sets.”