The Mandelbrot Set and Julia Sets

Scalings in the Mandelbrot Set

Hurwitz-Robucci scaling

The small copies of the Mandelbrot set sometimes are called midgets.

Some years ago, Adam Robucci, a student of mine, wanted to generate pictures of ultra-small midgets. When Adam started his project, the smallest well-known midget was Richard Voss' Avogadro's Midget, magnified by a factor of 10²³. Click here to see Avogadro's midget, in a new window. Adam wanted to find one smaller.

We devised a method to find the midgets along the real axis, the horizontal line bisecting the Mandelbrot set.

For higher n there are many n-cycle midgets, so Adam located

the left-most 3-cycle midget (there is only one 3-cycle midget),

the left-most 4-cycle midget (there is only one 4-cycle midget on the real axis),

the left-most 5-cycle midget (there are three 5-cycle midgets on the real axis),

and so on.

Eventually he produced a picture of a midget magnified by a factor of 10³⁵⁹, but had grown bored with that project. Click here to see this midget, and an estimate of how very small it is.

I suggested he compute a Feigenbaum ratio called the Robucci constant.

The sequence of ratios does have a limit, but it will never be called the Robucc constant: the limit is 4, exactly 4.

The numerical discovery is reported in Philip, Robucci, Frame, the proof in Hurwitz, Frame, Peak, and is outlined below.

Cycles and components locating components by solving fixed point equations

Locating the centers of components finding a family of polynomials whose roots are the centers of components

Renormalized polynomials relations between the graphs of f1, f2, and f3 nesr the center of the main cardioid and fn, f2n, and f3n near the center of an n-cycle midget

The Robucci Constant the scaling of distances between centers of successive left-most n-cycle midgets, alas never to be called the Robucci constant

Rescaled polynomials Henry Hurwitz's rescaling of the polynomials, gn(e)

Computing the scaling factor in the polynomial rescaling

Convergence of rescaled polynomials the graphs of successive gn(e) suggest these polynomials converge to a universal function g(e)

Universal function scaling relation the relation between g(4e) and g(e), a clue leading to the formula for g(e)

Finding the universal function g(e) = 2cos(sqrt(e))

Locating the midgets a formula to find the centers, from which the Robucci limit follows easily

Return to Scalings in the Mandelbrot set.