Multibrot set

The example d = 2 is the original Mandelbrot set. The examples for d > 2 are often called multibrot sets. These sets include the origin and have fractal perimeters, with (d − 1)-fold rotational symmetry.

z ↦ z2 + c

z ↦ z3 + c

z ↦ z4 + c

z ↦ z5 + c

z ↦ z6 + c

z ↦ z96 + c

z ↦ z96 + c detail x40

When d is negative the set appears to surround but does not include the origin, However this is just an artifact of the fixed maximum radius allowed by the Escape Time algorithm, and is not a limit of the sets that actually have a shape in the middle with an no hole (You can see this by using the Lyapunov exponent [No hole because the origin diverges to undefined not infinity because the origin {0 or 0+0i} taken to a negative power becomes undefined]). There is interesting complex behaviour in the contours between the set and the origin, in a star-shaped area with (1 − d)-fold rotational symmetry. The sets appear to have a circular perimeter, however this is just an artifact of the fixed maximum radius allowed by the Escape Time algorithm, and is not a limit of the sets that actually extend in all directions to infinity.

z ↦ z−2 + c

z ↦ z−3 + c

z ↦ z−4 + c

z ↦ z−5 + c

z ↦ z−6 + c

An alternative method is to render the exponent along the vertical axis. This requires either fixing the real or the imaginary value, and rendering the remaining value along the horizontal axis. The resulting set rises vertically from the origin in a narrow column to infinity. Magnification reveals increasing complexity. The first prominent bump or spike is seen at an exponent of 2, the location of the traditional Mandelbrot set at its cross-section. The third image here renders on a plane that is fixed at a 45-degree angle between the real and imaginary axes. [8]

Multibrot rendered with real value along horizontal axis and exponent along vertical axis, imaginary value fixed at zero

Multibrot rendered with imaginary value on horizontal axis and exponent on vertical axis, real value fixed at zero

Multibrot rendered with exponent on vertical axis along a plane angled 45-degrees between the real and imaginary axes.

ESCAPE TIME ALGORITHM
=====================

for each pixel on the screen do
    x = x0 = x co-ordinate of pixel
    y = y0 = y co-ordinate of pixel
  
    iteration := 0
    max_iteration := 1000
  
    while (x*x + y*y ≤ (2*2) and iteration < max_iteration do
        /* INSERT CODE(S)FOR Z^d FROM TABLE BELOW */
        iteration := iteration + 1
  
    if iteration = max_iteration then
        colour := black
    else
        colour := iteration
  
    plot(x0, y0, colour)

The complex value z has coordinates (x,y) on the complex plane and is raised to various powers inside the iteration loop by codes shown in this table. Powers not shown in the table can be obtained by concatenating the codes shown.

Wikibooks has a book on the topic of: Fractals

Wikimedia Commons has media related to Multibrot sets.