Julia Set Images
An Introduction to the Creation of Julia Set Images
This section will be concerned primarily with the effects of the color gradients used in the creation of Julia sets. The first two images below use only 2 colors, one for the interior of the set and one for the exterior. The next two use a normal multi-color gradient. The constant term is C=(0.32, 0.043) for both the images on the left and C=(0.32, -0.043) for the ones on the right.
The isolated points of the "Cantor dust" set, in the image on the left above, are invisible. However, by reducing the maximum number of iterations from 1024, in the colored image, to 256 for the black and white image on the right, their approximate location and density variation can be observed. A normal color gradient would make it harder to see these small clusters of black pixels. Note: These are not the points of the actual dust set, many of which are contained within each of the small black regions.
The density of the dust cloud of points is highest in the neighborhood of the sink points. It is the density variations in this cloud of invisible points that causes the complex patterns seen in NC Jsets when they are created using a multicolor gradient. Such sets produce more intricate and more interesting patterns than the connected Jsets.
Interior points are usually colored black, by default, although any solid color could be used. Colors can be assigned to each point outside the set by storing the number of iterations required to escape from that point to the region of known divergence. Although this is any magnitude greater than 2.0, it is usually chosen to be considerably larger, 4 or 8 for example. These colors can be assigned arbitrarily. If MI=256, and you want a different color for each step, that would require a table with 256 color entries.
Creating such a table, or "palette", is a lot of work, so a number of "preset" color palettes are provided by most programs. For a larger MI, these colors could simply be repeated as often as necessary. If the palette is restricted to 256 colors, or less, the image could be created as a GIF or PNG file type, which would save some storage space since only 8 bits are required per pixel.
There are obviously many variations on how these color presets can be designed. One possibility, that allows easy visualization of the result, would be to use a "color bar", as shown below.
Rainbow Color Bar Variations
Such a color bar can be modified in a number of ways, as shown above. It can also be repeated a specified number of times. The iteration points ( 1 to MI ) are uniformly distributed along the bar ( or group of bars ). The total number of colors selected is never more than MI, and may be much less if colors are repeated.
A typical group of color bars are shown in the image to the right. A large number of such preset bars may be supplied. Certain programs may also provide a way for users to design and store their own color bars or palettes.
Images with colors that change rapidly from one pixel to the next may become too complex, which makes the patterns hard to see. In most color bars, the colors vary smoothly along the bar, with only a small number of abrupt changes.
These images are usually created with 24-bit per pixel RGB colors and compressed as TIFF or JPEG files. They can be converted to 8-bit "paletted color" files later, if desired, with some loss of color variation.
All of the color contours approach circles as you move away from the the set. Another way of thinking about these contours is by analogy with the electric fields between two charged plates. Consider a connected Julia set to be the cross-section of a long metal rod perpendicular to the X-Y plane. Surround this with a metal cylinder with a radius large enough so that the color coutours have become circular, as in the figure below. Connect a 1000 volt battery between the Jset rod and the outer cylinder. The closed equipotential lines surrounding the Jset rod correspond exactly to the color contours based on counting the number of iteration steps as a point moves away from the boundary of the set.
If the set rod were to emit electrons, they would be attracted to the plus 1000 volt outer cylinder, moving slowly at first, and speeding up as they approach the cylinder. Initially, from valleys shielded by the set boundary, the electrons would move very slowly, speeding up as they get further away. This corresponds to the color contour strips becomming wider, as successive iterations produce larger jumps.
The figure below, using a striped color bar, provides a better view of the color contour lines, or "equipotentials". It is only approximately quantitative, however, since the stripes are not quite equally spaced.
The point z=(0,0) on the X-Y plane is a point on every connected Julia set. The origin, z(0,0), is also the "center of gravity" of all Julia sets, connected or not. It is not, however, a point on the NC Julia sets.
The points of all Julia sets lie within a radius of 2.0 from the origin, regardless of the size of the C-parameter. For magnitudes of C larger than two, the corresponding Julia sets become relatively simple low-density "dust" sets and are of little interest.
All Julia sets are symmetrical with respect to a 180 degree rotation about the origin. Such a rotated set is identical to the original set.
The Julia set for a given constant, "C( Cx, Cy )", has a very simple relationship to the set obtained by replacing the Cy by its negative value, -Cy. The second set will be the mirror image of the first. Such an image could also be obtained by flipping the original Julia set about the vertical axis, or, equally well, by flipping it about the horizontal axis. The fact that flipping the set about either axis gives the same result is a consequence of the rotational symmetry of these sets.
The spiral arms, rotating about a sink-point in a set, reverse their direction of rotation from CW, clockwise, to CCW, counter-clockwise, when the sign of the y-component of C is changed.
One result of these symmetries is that you only need to explore the patterns of the Julia sets in the upper half of the X-Y C-plane. The corresponding patterns obtained for negative values of Cy, although different, are easily deducible from those for positive Cy values.
