Halley Filaments

The Halley fractal is a graph of the behavior of Halley's Method, (for finding roots of an equation), it is similar in a way to the Newton Fractal which is like a graph of the behavior of the Newton-Raphson Method, and similar to Nova too, in that it also has a 'relaxation' parameter, that can be bumped up, which makes the dynamics in the lake become very dense and detailed.

Halley Fractal was introduced to me By Kerry Mitchell, Damien Jones and Ron Barnett in a very interesting thread on the UltraFractal mailing list. I asked about the immense detail that is generated by Nova Fractal and Negative-Fractional-Exponent Mandelbrot (negmandel) and they put me onto Halley.

It also renders very quickly considering the amount of complexity it produces - you don't need many iterations to get sub-pixel fractal detail everywhere.

The first few show the basic structures of the Halley Fractal, with its intricate mathematical field of balance and gentle curves:









I spent some time investigating this quite alien looking area:

















I was super impressed by the look of these glowing filaments, which are still some of the closest to flame-fractal looking that I've seen in an orbits fractal:



























Flipping things around a bit gave a more electric feeling:











Glowing beams of light:









And a bunch of work using filamentous orbit traps: