Convergences: Mandelbrot sets
Mandelbrot sets belong to the group of fractals. This requires a function zn+1 = f(zn) that contains a parameter.
After each iteration, a check is performed to see whether the result lies outside the specified decision limit. If this is the case, the result is plotted at the location of the initial value z0. Otherwise, the calculation is performed up to the specified maximum value n.
For Mandelbrot sets, the parameter p is varied across all displayable points in the window, while the starting point z0 is kept constant.
Unlike many other graphics, the settings and graphics are located in the same window.
Only functions containing the variables z and param are permitted, whereby the parameter param is mandatory. Enter the desired function in the input field for the function, or select one of the existing functions by clicking on the arrow to the right of the input field.
Complex numbers are permitted for the starting point. The decision limit, on the other hand, is the distance from the starting point, i.e. an amount. Therefore, only real numbers greater than zero are permitted here.
The slider for the iterations determines how many operations are to be performed until it is determined that a certain point does not converge. This has a significant influence on the duration of the calculation – but good results can be achieved even with the smallest setting.
Optionally, a monochrome display can be selected instead of the coloured one. In this case, dark and light grey tones alternate.
After clicking on the Create graphic button, CGRAPH displays the Mandelbrot fractal. This is a computationally intensive operation, but CGRAPH uses all available processor cores for the calculation, so that the result is still available immediately.
The image shows the classic apple man of the function f(z)= z2 + param. The default values can be viewed in the tree on the left. Since the parameter is part of the graphic function here, only the starting point can be changed. An explanation of how to change the starting point graphically can be found in the notes on changing the starting point in the graphic display.
The values can be changed retrospectively. The graphic is adjusted automatically, at the latest when the Create graphic button is clicked.
The display of the axes is also part of the tree, as the graphics for this graphic type are independent of each other and can be set separately.