Op Art and More 

Structures in Fatou sets

Fatou and Julia sets are determined via the functions shown above by iteration at each image point (pixel). The results of the iterations either converge to a stable numerical value or grow to infinity. The boundary between these two zones becomes the Julia sets assigned. Those areas with converging values that are not a contiguous area are called the Cantor set. The Cantor sets increasingly split into unlimited many individual structures, these structures are referred to as Fatou dust. The Julia area is here called those areas with converging values which are enclosed by the Julia set. The Julia set itself is therefore shown in the boundary area between the converging and growing values. In the graphics, the converging and growing values are represented by different colors, so that the border areas can be recognized. For more information about the generation of the graphs, see the menu item "Fractals".

On this page the focus is on the representation in the structures of the Fatou sets, the Julia area as well as their boundary regions. For better comparability of the properties of the Julia sets, 50 iteration steps have been used everywhere on the videos of this page unless otherwise mentioned.

Areas with converging values are shown here with shades of blue as well as pale yellow. These colored areas represent those areas where the applied formula yields stable values at a given number of iteration steps and thus are part of the Julia area. In the border area to the Julia sets one finds partly pale yellow but also strongly blue colored areas, which indicate that these areas would become unstable with further iteration steps. Only after a limit value formation with iterations going to infinity could one clear demarcation between the Julia set and converging and find diverging values. Areas with divergent values whose values grow to infinity become shown with reddish and yellowish hues.

The following 2 images show how the apparent boundary of the Julia set shifts within 5 iteration steps (from 48 to 53, where the constant terms of the formula "c" were not varied here). The image section is identical for both images. The apparent boundary region of the Julia set between converging and diverging values is indicated by the pale yellow and yellowish gray regions and the light blue regions, with some of the blue stripes extending deep into the border area of diverging values, showing the boundary region between the converging and diverging areas to be very dynamic. The 2 images show the displacement of the boundary region. Even some of the deep-blue, seemingly stable regions turn out to be unstable when iterated further. In the upper right quadrant, comparing the 2 images, one can see how the boundary area of converging and diverging values appears to slowly stabilize.

The growth of numerical values into infinity is shown here with yellow and reddish signatures are shown. Differently than with the fractals, which are represented on the preceding pages, where the coloring indicates at which iteration step a value slips into infinity, here, on the other hand, the intensity with which the numerical values at the respective pixel location of the pixel move towards infinity. The speed increases from black-red over red over black-yellow to yellow. The intensity corresponds to a  velocity vector, which is displayed here  in the respective pixel as a point-shaped color value.

The different representation of the Julia- and Fatou sets with their intensity values compared to the conventional representation as equipotential surfaces with the point in time (iteration step) at which the slide into infinity begins, sometimes lead to different mapping results, as shown in the 2 following images, which show the same area with these 2 different types of representation after 50 iterations each.


Some structures in the upper image with colors indicating at which the iteration depth at which values slip to infinity, deviate from the velocity vectors of the lower image. These structures are not visible in the lower image, while now other detailed information becomes visible in the lower image.

In the videos, in addition to enlargements and reductions, changes in the shape of the Julia- and Fatou sets become visible in the form of movements. This is based on the fact that the actually constant term "c" in the formulas is subjected to a time-dependent variation c(t). In the process, sudden changes in shape or colors sometimes occur in the videos. If one excludes those events which occur as a consequence of limitations of the chosen color space, it would have to be determined for the rest by limit value investigations whether these sudden changes are caused by the relatively coarse grid of the pixels and by the limited number space of the floating point numbers used for the calculation or whether the change of the constants leads to discontinuity points in the formulas. As an example for a possible discontinuity the beginning of the video sequence of the 2nd video can be considered. In the two following pictures only the real part of the actually constant term "c" was increased starting from the value 0 by the value 0.1E-66, which already leads to a massive change in the undefined black area, so that I assume that with the formula (z³ +c)/z, which is the basis for all following pictures here, every change of "c" in relation to 0 leads to a discontinuous jump, which one could probably also prove with the help of a curve discussion. The black central area shows the zone where the Julia area immediately grows to infinity as a consequence of a zero division, while the blue areas in the lower picture represent stable zones.

At a magnification of nearly 10 billion times of the central nucleus in the last image, self-similar shapes with new structures appear. See the following 2 images.


The gap between the two deep blue zones slowly closes as one iterates deeper. The following image shows a detail magnification of the gap at a magnification of about 2.5²³ after 4000 iterations.


The small dots, which approach with deeper iteration the deep-blue area on the upper left as in a line with the limit 0, show themselves again fractal structures (see the following picture). What are the consequences with respect to the Julia area? Light and dark blue surfaces provide stable values for the iterations, where here only the intensity of the variability is shown. The values of the light blue and dark blue areas are only marginally different from 0 (e.g. less than 10^-44). The intensity vectors differ only by their sign. In any case, the shapes remind me of water fleas (Daphnia) viewed from above.


One more note about the pale-yellow areas (as in Figs. 1 and 2) and to the calculation of the intensity values. For the calculation whether at an iteration step the numerical values at the location of the pixel raise to infinity, some kind of distance value of the complex number to the origin is determined. Actually, distance values are always positive or zero. If instead of the sums of the components, however, one inadmissibly also the signs for the distance calculation in a fuzzy way. way, in some cases "negative distance values" or invalid 0-values can occur, which are shown in the images in pale yellow. Large negative distance values are displayed in pure blue blue (this is sometimes difficult to distinguish visually from the rest of the blue tones). The positive distance values determined in this way deviate from the real distance values. The word distance then no longer quite fits; one could call it fuzzy distance. But independently of this, the question arises whether negative distances can occur at all in our perceived.

The physicist Brian Greene (The Fabric of the Cosmos) shows that in space units below the Planck length the conventional concepts of time and space have no more validity, but are replaced by a world of quantum fluctuations, in which space and time according to ideas of several physicists (e.g. John Archibald Wheeler) in extremely small scales of the Planck scale could run quite negatively but also cyclically or still other structures. It is surely only a coincidence, but the first two pictures on this page show a similarity with a representation of Brian Greene, in which he shows schematically on the level of the paper (2 dimensions) the emergence of new and parallel universes over the mechanism of the repeated inflation assumed by astrophysicists, whereby however space dimensions can be represented only limited. At most it would be conceivable that the principles of the formation of universes show, however, a certain relation to the self-similarity of fractals. But even the mechanism of inflation is still disputed in science. Alternative theories as for example of Roger Penrose (Cycles of Time) provide a so-called conformal cyclic cosmology CCC on the basis of the quantum gravity, in which the increase of the entropy is considered. Or, for example, a theory of Lee Smolin (Why does the world exist?), which describes an emergence of world ages from singularities in black holes based on inheritance. Surprisingly, his diagram about the world ages also has similarities with the top 2 pictures.

Fuzzy distance

(Supplement from 23.08.2018)>

With the Mandelbrot set it holds that a point belongs to the Mandelbrot set if the iteration sequence from the formula z_n+1 = z_n² + 1 yields a bounded limit lim sup <= 2 for n--> infinity. For Julia sets an assignment as a point of a Julia surface is likewise made here as a boundary value for the value z_n computed from the respective formula, if absolute value |(z_n)| <= 2.

In contrast to an infinite iteration sequence, in the calculations here as a consequence of the finiteness of the resources of the computer after a small number of iteration steps the computation is stopped which can lead to distortions. Figures 1 and 2 clearly show the difference in the boundary region of the Julia set after 48 and 53 iterations, respectively, as a result of the inaccuracy due to the small number of iterations.

For the calculation of a Euclidean distance value to the lim sup on a 2-dimensional plane, the formula corresponding to the Pythagorean theorem is usually used: SQRT((x₁-x₂)² +(y₁-y₂ )² ), which always yields values >= 0 (SQRT := square root). With the representations of structured Julia surfaces (in different shades of blue), on the other hand, this conventional distance calculation is not applied here, but only the inadmissible addition of the real part with the imaginary part of the value determined last during the iteration is carried out. This corresponds to the procedure for the distance computation similarly as with a Manhattan metric, with which here however no absolute values are used. The space of the metric is thereby spanned by the finite floating point number space of the computer.

With some calculation sequences as for example (z³+c)/z negative values can arise for the real part or imaginary part, which can result then in their direct addition also in a negative total value. The addition of real part and imaginary part is actually inadmissible for the calculation of a distance value, because the results of this distance calculation are quite inaccurate compared to the Pythagorean theorem and can also lead to negative distance values, which are usually forbidden. However, as a consequence of this fuzzy calculation (here called fuzzy distance), new self-similar fractal structures (see Figures 6-10 and also under menu item "Daphnia") can become visible within the Julia surface, distinguished by their positive and negative sign (light and dark blue). Changes of the iteration depth furthermore lead to oscillations of these virtual structures (see menu item "Daphnia").

Possibly these virtual structures are caused by inaccuracies of the floating point operations and the limitedness of the field of floating point numbers. This number field with Java uses with 8 bytes 15 significant digits in a value range of +/-4.9E-324 ... +/-1.7E+308. However, the fractal structures caused within the Julia area cannot arise only by these restrictions, but must depend in addition on the hyperbolic term of the formula (z³+c)/z, since with other formulas these fractals are not formed.


Note: The term fuzzy distance used here does not correspond to the term fuzzy distance used in mathematics by Josef Bednár or the fuzzy numbers defined by J.G. Dijkman et al.

An example of the difference between conventional equipotential surfaces and intensity values with the fuzzy distance within the Julia surface is shown in the following 2 images at c(t)=(-0.001 RE, -0.00001 IM) in a coordinate window of approximately (-1.03 RE, -0.19 IM) in the upper left and (-0.79 RE, 0.04 IM) in the lower right.

Image 10: Example equipotential surfaces

Image 11: Intensity values compared to Fig. 10

Continuation with Daphnia
jump to Daphnia