Multibrot set

In mathematics, a multibrot set is the set of values in the complex plane whose absolute value remains below some finite value throughout iterations by a member of the general monic univariate polynomial family of recursions. The name is a portmanteau of multiple and Mandelbrot set.

where d ≥ 2. The exponent d may be further generalized to negative and fractional values.

Several graphics are available but, as far as can be verified, none of these have been taken a step further to display a 3-D stack of the various stages so that the evolution of the general shape can be seen from other than vertically above.

The case of

is the classic Mandelbrot set from which the name is derived.

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

Each of the examples of various powers d shown below is plotted to the same scale. Values of c belonging to the set are black. Values of c that have unbounded value under recursion, and thus do not belong in the set, are plotted in different colours, that show as contours, depending on the number of recursions that caused a value to exceed a fixed magnitude in the Escape Time algorithm.

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.

When d is negative the set surrounds but does not include the origin. 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.

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.

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