[Mandelbrot set]

Mandelbrot set

Julia 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



Complex arithmetic

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