This image is based on a fractal which is similar to the one used to
create Fiery Dragon and
Steps to Infinity. This curve is also defined by two transformations, but with slightly different parameters:
xform scale 0.618 /* Dragon scales by approximately .7071 */ rotate -137.5 xform scale -0.618 rotate 137.5 translate 1.0 0.0
This image is an interpretation of the process by which the points of the plane are captured by the transformations in the description of the curve to form the actual curve.
I've assigned an initial point to each of the two transformations and I apply the transformations to their respective points and then to the results, over and over again. As I compose the transformations into longer and longer strings, the contractivity of each composite transformation increases. These "stronger" transformations capture more and more of the plane. I've illustrated this process in a movie, using cones to show the effect of the transformations. As the strength of a transformation increases, a cone under that transformation gets wider and shorter.
The upper black and white image above shows one possible colouring for the image used to construct Captivating Rhythm. Each band corresponds to a different level of "capture", increasing away from the curve. The reverse of "capture" (lower left above) is "escape" (lower right above). Notice how the arrangement of disks differs between them.
The image below shows another way to view the same data, ultimately selected for the final image. Here, I have broken up what were solid bands in the earlier image by using information about which transformation was applied at a particular time. In creating the final image, I applied a fractional Brownian motion texture and added an outline of the fractal.
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