Inspired by https://en.wikipedia.org/wiki/Herman_ring#Herman_and_parabolic_basin

with a sightly different parameter 'a' and a more interesting coloring.

\(z_{n+1}=e^{2 \pi i t}z_n^3\frac{1-\bar{a}z_n}{z_n-a}\frac{1-\bar{b}z_n}{z_n-b}\)

with
\(t=0.6141866\)
\(a=0.25+0.008i\)
\(b=0.0405353-0.0255082i\)

#fractal #fractalart #juliaset #rationalfunction #escapetimefractals #rendering #distanceestimation

I implemented Slow Mating for quadratic polynomials using the equations and hints in Chapter 5 of Wolf Jung's 2017 paper "The Thurston Algorithm for quadratic matings" https://arxiv.org/abs/1706.04177

My code is 145 lines of quite-straightforward C, vs 2249 lines of C++ with various state hidden in mutating objects for the code accompanying the paper (which admittedly does a lot more, working from angles to compute the complex points and (pre)periods that are the input to my code). I'll do a blog post next week once I've tested more cases to make sure I haven't done any big mistakes.

There were a couple of subtleties, 1. needing to use cproj() to normalize infinity's representation and avoid NaNs; and 2. in one place, converting (a - b) / (a - c) to (1 - b/a) / (1 - c/a) so that it still works when a is infinite.

Attached images are the north period 4 island mated with the west period 4 island (blue background), and 2/5 bulb mated with 1/2 bulb (turquoise background).

#Quadratic #JuliaSets #SlowMating #RationalFunction #maths #fractals