• Named after Benoit B. Mandelbrot.
• A fractal generated by iterating: z_(n+1) = z_n² + c, z₀ = 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.

• Named after Gaston Julia (1893--1978).
• A fractal generated by iterating: z_(n+1) = z_n² + 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.

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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

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Complex arithmetic