Multibrot set

The sets for other values of d also show fractal images[7] when they are plotted on the complex plane.

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 resulting set rises vertically from the origin in a narrow column to infinity.

The first prominent bump or spike is seen at an exponent of 2, the location of the traditional Mandelbrot set at its cross-section.

[8] All the above images are rendered using an Escape Time algorithm that identifies points outside the set in a simple way.

Much greater fractal detail is revealed by plotting the Lyapunov exponent,[9] as shown by the example below.

Multibrot 3 at the bottom-left of the main part.
Detail of Multijulia 8.
Multibrot 4.
Multibrot exponent 0 - 8
z ↦ z 2 + c
z ↦ z 3 + c
z ↦ z 4 + c
z ↦ z 5 + c
z ↦ z 6 + c
z ↦ z 96 + c
z ↦ z 96 + c detail x40
z ↦ z −2 + c
z ↦ z −3 + c
z ↦ z −4 + c
z ↦ z −5 + c
z ↦ z −6 + c
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.
Enlarged first quadrant of the multibrot set for the iteration z ↦ z −2 + c rendered with the Escape Time algorithm.
Enlarged first quadrant of the multibrot set for the iteration z ↦ z −2 + c rendered using the Lyapunov exponent of the sequence as a stability criterion rather than using the Escape Time algorithm. Periodicity checking was used to colour the set according to the period of the cycles of the orbits.