A C.A.T. fractal image is an image that is identical to a combination of transformations of itself by a number of contracting affine transformations. If you have trouble understanding this this example may help. Such images can be generated by the Random Iteration Method, which consist of starting from an initial point and repeatedly randomly selecting one from a number of. As each point is calculated the corresponding pixel on the screen is plotted. Over time, usually a second or two, this will build up a fractal image. The pixel is colored according to how many times it has been visited. This kind of fractal is often called an I.F.S. (Iterated function system) fractal, but it is perfectly possible to create I.F.S fractals using other kinds of transformation.
In Spirofractal 5, several other algorithms can be used to generate these images. Usually these give similar results, only more slowly, but in some cases the resulting image is quite different. This is because the random iteration method is not good at finding parts of the image that are very rarely visited by random iteration.
In order to generate a pattern you have to have at least two transformations, because repeatedly applying any single C.A.T. will always tend to a fixed point (in the same way as if you start from a number then add half the number, then half again and so on on the sum tends to twice the original number).
Selecting two transformations completely at random will often produce an image consisting of nothing but a few straight lines, and in general the images won't have symmetry. However, some of these images can be surprisingly intricate.
Careful choice of the transformations, so that there is an underlying geometric
relation between them, often yields beautiful patterns, some of which are
shown here.
The simplest kind of symmetry arises from taking one base transformation and adding more transformations that combine it with rotations about the origin through the angles the corners of a regular polygon make with the x-axis. This gives rise to images with rotational symmetry, but which are not the same as their mirror image.
Combining the same base transformation with different polygons' symmetries can make a big difference to the image. This image uses the same transformation as the one above, but with the symmetry of a triangle rather than a square. I think this particular image causes an optical illusion which make it quite hard
to see the symmetry here.
Images with a high degree of symmetry look less like fractals and more like Spirographs. This image still only has rotational symmetry, but it is also possible to add extra transformations which make the image completely symmetrical. This can be seen below.
Spirofractal also has many possibilities for creating images which are almost, but not quite symmetrical. These images can be very interesting.
This spiral effect is what you get if you shrink the transformations by a fixed proportion as you go round the polygon.
This fan like effect is what happens when the transformations are rotated only by half the "correct" amount as you move round the corners of the polygon. Notice the small fans appearing on the feathers of the large fan.
You can also miss out some of the transformations needed for full symmetry...
... so you end up with something like this pair of images. This can give you a
better understanding of how the different transformations fit together.
Another possibility is to use a set of transformations that normally make a symmetric image, but to bias the process of selecting between them, so that some transformations are selected more often than others. Notice how even this aspect of the image is reproduced in the scaled down versions.
It is also possible to add transformations which will give the effect of repeated copies of the image receding into the distance, or reflected endlessly
in a pair of mirrors.
On to Sierpiński room