# Mandelbrot set

• Named after Benoit B. Mandelbrot .
• A fractal generated by iterating: $z_{n+1} = z_n^2 + c; z_0=0$, and plotting how fast it diverges to infinity for different values of the complex number $c$ (speed represented as colours). The black set represents the "prisoner" points that do not diverge: it is the Mandelbrot set .

# Julia set

• Named after Gaston Julia (1893--1978).
• A fractal generated by iterating: $z_{n+1} = z_n^2 + c$, and plotting how fast it diverges to infinity for different values of the complex number $z$ (speed represented as colours) for a set value of $c$. The black set represents the "prisoner" points that do not diverge: the border of this set is the Julia set .
• Values of $c$ that lie within the Mandelbrot set result in connected Julia sets; values of $c$ from outside result in disconnected Julia sets. We can draw an array of Julia sets for various values of $c$, and map out the Mandelbrot set.

## Mandelbroids and Julioids (Java applet, JDK 1.3)

 If you had a Java browser, you would be able to alter the parameters of the Mandelbrot and Julia set, try exponents other than 2, and try other equations.
• Mandelbrot : $k=2$ gives the classic Mandelbrot set
• Julia : $k=2$ gives the classic Julia sets. Some interesting values for $c$ are $-1$, $i$, and $-0.5+0.5i$
• Left-click to zoom in
• Right-click to zoom out
• iter : more iterations may be needed when zoomed in
• Redraw : with current parameter settings
• Reset : to selected "special place" settings

### Mandelbrot

• Other related Mandelbrots can be generated by iterating $z^k+c$, for different values of $k$.
• For higher integer $k$, each Mandelbrot 'bud' becomes $k-1$ buds.
• For non-integer $k$ (try $k=1.5$), the picture has discontinuities. We are taking roots, which process gives several solutions, but we are looking at only one of them.
• Other related "Mandelbroids" can be generated by iterating other complex functions, such as $\sin z$, $e^z$, $z^z$, or combinations of these.
• For the Mandelbrot set, divergence is assured once $|z|>2$. For other Mandelbroids the divergence condition is arbitrarily set as $|z|>5$.

### Julia

• Other related "Julioids" can be generated by iterating $z^k+c$, for different values of $k$, or iterating other complex functions.

## Complex arithmetic

$$z = x + iy = r \exp i \theta \\ z^n = (r \exp i \theta)^n = r^n \exp i n \theta\\ e^z = e^{x+iy} \\ z^z = (r \exp i \theta)^z = \left( \exp (\ln r + i \theta) \right)^{x+iy} = \exp (x\ln r - y \theta) \exp i(y\ln r + x \theta) \\ \sin z = \frac{1}{2i}\left( e^{iz} - e^{-iz}\right) = \frac{-i}{2}\left( e^{ix-y} - e^{-ix+y}\right) = \frac{1}{2}\left( e^y \exp i\left(\frac{\pi}{2}-x\right) - e^{-y} \exp i\left(\frac{\pi}{2}+x\right) \right)$$