The image at the top of the post shows the first five generations of the family of curves obtained from the 'scrambler' construction that I described briefly in
the last post. These curves are generated by the equations
where
n is the generation (starting at 0), and the sign of the coefficients in the expression for
y are chosen to yield the different branches of the family. In the diagram above, if you choose all positive coefficients, you get the curves on the extreme left, while if you choose positive for the first term but negative for the rest, you get the curves on the extreme right.
The curves formed by choosing alternate + and - signs are the ones most closely related to the 'scrambler' that got this started. This choice has each circle turning in the direction opposite to the circle that preceded it, and generates the 'propeller' curves that lie in the center right of the diagram above.
One way to express this branch of the family is
GSP was used for the first few generations, but to look at this for very large
n, I resorted to writing a short (surprisingly short)
Processing program that gave these pictures for generations 0-7, and 20:
Looking at the images above and the one below, you can see that as
n goes to infinity a
fractal emerges that displays nice self-similarity along each propeller blade.
A text file with the Processing source code for drawing the fractal is
here.
Update: A post about the leftmost curve on the chart above ("brain fractal") is here.
