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Description
I've been interested recently in trying to represent or visualise the changes the Mandelbrot function goes through, and came across this interesting twist on the traditional formula. Random values for c are chosen, excluding those that fall inside the set. Rather than mapping c to an image and assigning it a colour based on the behaviour of z, we plot each iteration of z as it takes it's path towards escaping.

In particular, very close to the boundary of the set, some starting values result in very long paths before finally escaping. This image represents around 500,000 of these paths, between 10 and 10,000 steps long. I also used a colour lookup table and some postprocessing that strongly favoured the longer values, meaning that the rarest, longest paths are highlighted. Their rareness also makes them stochastic and asymmetric; where regular patterns and whirls appear - for example, regular strings of dots that curve outward from the cardioid slightly right-of-centre, up or down - these are the result of single paths. Note also that the colour for an individual path is defined by it's length, with the green-yellow-red spectrum corresponding to paths from 6000 to 10,000 steps long.

As a point of reference, I also overlayed a render of all the values of c for this dataset - effectively the first step of each path - using the same LUT (i.e. with longer paths represented more brightly) but hue shifted to blue and violet to contrast with the geen-yellow-red of the Nebulabrot trace. Again, these extremely long escaping values are constrained close to the boundary of the set, effectively tracing a sharp outline of the Mandelbrot fractal.

Description
I've been interested recently in trying to represent or visualise the changes the Mandelbrot function goes through, and came across this interesting twist on the traditional formula. Random values for c are chosen, excluding those that fall inside the set. Rather than mapping c to an image and assigning it a colour based on the behaviour of z, we plot each iteration of z as it takes it's path towards escaping.

In particular, very close to the boundary of the set, some starting values result in very long paths before finally escaping. This image represents around 500,000 of these paths, between 10 and 10,000 steps long. I also used a colour lookup table and some postprocessing that strongly favoured the longer values, meaning that the rarest, longest paths are highlighted. Their rareness also makes them stochastic and asymmetric; where regular patterns and whirls appear - for example, regular strings of dots that curve outward from the cardioid slightly right-of-centre, up or down - these are the result of single paths. Note also that the colour for an individual path is defined by it's length, with the green-yellow-red spectrum corresponding to paths from 6000 to 10,000 steps long.

As a point of reference, I also overlayed a render of all the values of c for this dataset - effectively the first step of each path - using the same LUT (i.e. with longer paths represented more brightly) but hue shifted to blue and violet to contrast with the geen-yellow-red of the Nebulabrot trace. Again, these extremely long escaping values are constrained close to the boundary of the set, effectively tracing a sharp outline of the Mandelbrot fractal.