Bugs #11

Bugs Fractal 85

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.

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