This is a zoom into yesterday’s image, getting a closer look at the tricorn. The parameter is still a = 2.5. The image center is at .
Recall the definition of the parameter version of bug: . Then, a little algebra,
If you are a little rusty the definition of complex conjugate is , and a well known trig identity is .
Also recall that near z=0 for large a it is almost true that bug(z) = z. More formally, for any if a is large enough then . Now we can also say
There is a little more to be done, I will skip the details. If you track what is going on in the iteration of near you will see that it behaves like the tricorn, near .
Given the periodicity of the sin function, the same is true for c near for any integer n. Even n looks like a Mandelbrot set, odd n looks like the Tricorn.