Bugs #11

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 $ a*\pi*i$.

Recall the definition of the parameter version of bug: $ bug(x+y i) = x + a * sin(y/a) i$. Then, a little algebra,

$ \begin{matrix} bug(z+a \pi i) & = & bug( x + (y+a \pi)i) \\ & = & x + a *sin(y/a+\pi)i \\& = & x - a * sin(y/a) i \\ & = & \overline{bug(z)} \end{matrix} $

If you are a little rusty the definition of complex conjugate is $ \overline{x+yi} = x-yi$, and a well known trig identity is $ sin(\theta+\pi) = -sin(\theta)$.

Also recall that near z=0 for large a it is almost true that bug(z) = z. More formally, for any $ \epsilon, R > 0$ if a is large enough then $ |bug(z)-z|< \epsilon \text{ whenever } z < R$. Now we can also say $ |bug(z+\pi i)- \overline{z}|< \epsilon \text{ whenever } z < R$

There is a little more to be done, I will skip the details. If you track what is going on in the iteration of $ bug(z)^2+c$ near $ c = a * \pi i$ you will see that it behaves like the tricorn, $ \overline z^2+c$ near $ c = 0$.

Given the periodicity of the sin function, the same is true for c near $ c = n * a * \pi i$ for any integer n. Even n looks like a Mandelbrot set, odd n looks like the Tricorn.