The fractal character of the boundary of the Mandelbrot structure is a result of the nonlinearity of the series´ construction rule. It is not dependent on its specific square law rule.
To demonstrate this, a modified series is calculated with variable power law in the nonlinear term
zn+1= znk+ c; z0 = 0; k≧ 1, a rational number
c is the complex number in the plane, for which convergence or divergence of the series is determined.
The calculation is more transparent when the complex number is written in polar coordinates.
z = x + i y = r (cosφ + i sinφ) = r eiφ
r = √(x2 + y2); φ = arctg (y / x)
z k = r k eiφ = r k (cos (kφ) + i sin (kφ))
In the simulation a slider is used to vary the rational number k between 1 and 10. In the number field an arbitrary rational number can be input, for example an exact integer or a very high number such as 1000. k = 2 delivers the common "apple man" of the Mandelbrot set.
The default position when opening the simulation and after Reset is n = 10.
Be patient! This is a number crunching calculation, and it may take several seconds or even minutes to see the result after a change of k or after Reset, depending on the quality of your computer. When you assume that the computer has been hooked up, close the simulation and start it anew.