Chapter 40 - A new scaling along the spike of the Mandelbrot set

Abstract

Using the zeros of a family of real polynomials, a sequence of midgets along the spike of the Mandelbrot set can be located. By this method, midgets (tiny copies of the Mandelbrot set) of a high cycle number can readily be found. 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. This chapter presents evidence supporting a pattern of scalings for some “generalized Mandelbrot sets.” The Mandelbrot set, M, is the collection of complex numbers c for which the Julia set Jc of Fc (z) = z2+c is connected. There are many sequences of midgets related through scalings of size. This chapter reviews some of the known sequences and reports on a new scaling of midgets, discovered along the real axis, progressing toward c = - 2 .