# 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.

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