Mandyn - Mandelbrot and Julia Set Dynamics


Width: Height:

Iterations: Min x: Max x: Min y: Max y:
Juila Set: Juila x: Julia y:
Colour Centre x: Colour Centre y: Click to Set: Hue is Magnitude: Magnitude Shading: Magnitude Saturation: Hue Rotation: Iteration Time (ms):

Just a little bit different from the usual Mandelbrot set plotters. The usual Mandelbrot set iterates zn+1 = zn 2 + c (z0 = 0) until either the iteration "escapes" - i.e. the magnitude of z gets over some threshold, often 4, or it until it looks like the iteration isn't going to escape. The different positions on the plot refer to different values of c.

Mandyn is a little bit different - it looks at the iteration. The idea is that the Mandelbrot set is telling something about a dynamical system, and let's see the dynamics! On each iteration it looks up zn. If zn has well and truly escaped, then it plots a shade from white to black to show when it escaped (there's a technique called Normalised Iteration Count to make this smooth). If zn hasn't escaped, then by default it selects a hue based on the argument of zn, and a saturation or value based on the magnitude; for high magnitudes, well on their way to escaping, the saturation will be low. Alternatively, Mandyn can forget the argument and plot magnitude as hue.

You can make Mandyn step through the iterations automatically, and adjust the speed. Like any good Mandelbrot program you can draw a rectangle to zoom in. There's an option for setting the "colour centre" - this effectively subtracts a complex number from zn for the purposes of calculating the colour of the pixel. "Click to Set" lets you click on the chart to set the colour centre - this temporarily disables zooming.

Finally, there's an option to plot Julia sets; these use the same iteration as the Mandelbrot, but instead of setting z0 = 0, each pixel represents a different value of z0. You have to set c yourself, using "Julia x" and "Julia y". Hint: find a nice x and y on the Mandelbrot, and the Julia set may well look a bit like the area around that. x=0.278 y=0.01 is a good place to start, it's quite interesting to turn on Auto-Iterate. Don't forget to zoom in! I find that experimenting with the colour centre is often good with Julia sets.

Peter Corbett, 2014

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