sample 3

Tutorial: Cardioid Boundary Orbits Coloring

Text and Images © 2009 Kerry Mitchell


Points on the edge of the main cardioid of the Mandelbrot set straddle two worlds. The interior of the cardioid contains points with fixed-point orbits: the orbits eventually settle down to a final value. Points immediately outside of the main cardioid are either divergent (their orbits go to infinity) or periodic (their orbits settle down into a loop).

Right on the boundary, the points can be characterized by the angle of the point relative to the cusp at (0.25, 0). 0° is right at the cusp. Increasing counter-clockwise, 120° is at the base of the disk at the top of the cardioid, 180° is at the base of the largest disk, at (-0.75,0), 240° is at the base of the disk at the bottom of the cardioid, and 360° is back to the cusp. Alternatively, the angles can be measured in radians, where 2π radians is 360° or a full circle. A full circle is also known as 1 turn. This is important because, if the angle is a fraction of 360° or of 2π radians or of 1 turn, then the point will be at the base of a disk. Then, its orbit will combine both fixed point and periodic behaviors--it will converge to a final value, but do so very slowly and through a loop of values. Below, the points for the fractions 1/4 (90°, red), 2/5 (144°, green), and 5/8 (225°, blue) are shown on the Mandelbrot set. You can see that they are at the bases of disks.

tanget points on Mandelbrot 

The periods of the orbits are 4, 5, and 8 (the denominators of the fractions), which is reflected in the shapes of the orbit diagrams. (The colors of the orbits correspond with the colors of the points, above.)

orbits of tangent points

Not all points on the boundary of the cardioid have angles that are exactly fractions of 360° or of 2π radians or of 1 turn. These points are not associated with periodic disks. Below are shown three such points, at angles of 1 radian (180°/π or about 57.3°, shown in cyan), 2 radians (2 × 180°/π or about 114.6°, shown in magenta), and 4 radians (4 × 180°/π or about 229.2°, shown in yellow). It may be hard to tell, but these points, no matter how far you zoom in, will never be at the base of a disk.

non-tanget points on Mandelbrot 

So what? Well, these points, cardioid boundary points that are not bases of periodic disks, have chaotic orbits. They wander around irregularly, never falling into a repeating pattern. Here, we see the orbits of the three above-mentioned points, in corresponding colors.

orbits of non-tangent points

One characteristic of chaos is that orbits that start out close together will move apart quickly. We can see this by making a new orbit out of the difference between two chaotic orbits and seeing where that orbit goes. Thatís what this coloring does (in addition to drawing the single-point orbits shown above). Here is the orbit that is the difference between the orbits of the points at 1.999 radians and 2.001 radians.

difference orbit

Another way to see that the two orbits differ is to form their ratio. Hereís what happens with these two points:

ratio orbit

If you can take the difference or ratio of two numbers, you may as well add and multiply them, too,

sum and product orbits

and do all sorts of other weird things.

weird orbit



Sample Image

Let's see how to create the image at the top of this page. If you get hopelessly lost (or just don't care to read any further), you can find the parameters for it in lkm3.upr as "Cardioid Boundary Orbits Sample 3."
  1. Create a new fractal using the "Pixel" formula in lkm.ufm. Load "Cardioid Boundary Orbits" from lkm3.ucl into the Outside tab. Change the dimensions of the image to 600 pixels wide by 400 high.
  2. Change the Color Density to 0.125 and clear the "Repeat Gradient" checkbox. You should now see something vaguely like the differnce image, above, but probably in different colors.
  3. Change the number of points to 4 and leave the angle units at radians. Set the four angles to: 3.9998, 3.9999, 4.0001, and 4.0002 radians.
  4. Open the gradient and remove all the control points. Insert five new points with these colors:
    IndexHue SaturationLuminence
    00 00
    1250 19063
    25025 190127
    35060 255223
    3990 0255
  5. Set weight 1 to weight 4 to 1/0 (Re/Im), -2/0, 2/0, and -1/0, respectively. Leave the functions at "ident."
  6. Increase the sampling factor to 100.
  7. Change the values on the Location tab to Center: 0.591/-0.217 (Re/Im) and Magnification to 2.17. There you are!
I hope you have fun with this coloring and if you have any comments or suggestions, please contact me.

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