The Mandelbrot Set ( or "Mset" ) is shown in the image below.   It has been processed to emphasize the "dendrites", which are actually much thinner.   Most of them are too thin to be seen in a normal image.   The boundary is extremely complex.   Zooming in on parts of this boundary region, outside the set itself, can produce very elaborate images when used with a suitable color gradient bar.

The Mandelbrot Set with Dendrites Emphasized

Expanded View of Negative Spike of the Mset


Dendrites Emphasized       Normal Color Pattern

The two views above are of the largest Mini-M on the spike.   The dendrites are invisible in the image with a normal color gradient, but their presence can easily be inferred from the colored spikes in this image.   Compare the two views, which have the same image pattern and are to the same scale.

There is a very close relationship between a "zoom" image into a region outside the Mset boundary and a zoom into the Julia set image having a C-value in the same area.   The next two images illustrate this.   Note that points on the Mset X-Y plane correspond to values of "C" for the Julia sets.

Mandelbrot Set Image

Zoom into Mset near the Mini-M shown above.   It is located above and to the right of the cusp of the large Mini-M which is at C=(-1.75, 0).

Julia Set Image

Zoom into the Jset with a C-value in the center of Mset zoom above.

Julia Set Image

Zoom-out to show a larger scale view of the central part of the Jset.   The zoom above is at the center of this image, although the zoom factor is so large that the zoom image is only a small dot and connot be seen at this resolution.   This is still only the central part of the overall Jset, however.   The full set is long and thin.

The Mcube and Jcube Sets

Many variations on these sets can be obtained by using different generating functions.   A cubic function, as shown below, is the only one that will be considered here:

where z is a point on the X-Y plane, and C is a constant with both x and y components, C_x and C_y.

The Jcube sets, and the corresponding Mcube set, are directly analogous to those obtained using a quadratic function.   The images produced have a different type of symmetry, of course.   The Mcube set, for example, is symmetrical with respect to a reflection about the Y-axis as well as to a reflection about the X-axis.   The image shown on the right has the dendrites emphasized.

The Mcube Set has a less complex boundary than the Mandelbrot Set with fewer interesting images.   The Jcube sets, however, produce images that are quite comparable to the Julia set images.

The images below give an expanded view of the top on the Mcube set with a normal color gradient, and a deep zoom to a Mini-M3 island.