Sierpiński fractals

Sierpiński was the mathematician who discovered one of the first fractals, the Sierpiński triangle. You can find out more about it on the Spirofractal tour. (Possibly the very first fractal was the Koch snowflake curve, but this is not a C.A.T. fractal, though it is closely related to one).

The process used to construct the triangle can be generalised slightly by picking a number of fixed points and picking one of them at random as a starting point. Then we pick one of the other fixed points at random, and move a fixed proportion (less than one) of the distance between the two points, either towards the point or away from it. We repeat this indefinitely, coloring points according to how often they are visited.

In fact Sierpiński fractals are really just a special kind of C.A.T. fractal. However they are usually very recognisable. In one way they produce images that are more truly fractal than general C.A.T. fractals, because if you magnify a Sierpinski fractal, the structure of the whole is repeated in each part, without the distortions that occur in general C.A.T. fractals.

This is a randomly generated Sierpiński pentagon. Notice that whereas the Sierpiński triangle can be decomposed into its component parts, which are all identical (except half size) copies of the whole image, the same cannot be done for the Pentagon. Different scaled down copies of the fractal overlap. Fractals that do not contain overlapping parts are called strict fractals. Ones that overlap are called overlaid fractals. The classic Sierpiński pentagon is on the boundary between these two types of fractal: it is what as known as a just touching fractal. Overlaid and just touching fractals are connected - strict fractals consist of a cloud of "dust". A little overlap usually adds considerably to the interest of a fractal image, but too much makes for a shapeless splodge.

The Sierpiński triangle, and related shapes such as the Sierpiński pentagon, are a good example of how different sets of transformations can give rise to the same image: each individual transformation can be roated through suitable amounts about the image of the origin under that transformation, and it will make no difference to the final image.

Increasing amounts of symmetry tends not to yield very interesting results for Sierpiński fractals, but adding another transformation can sometimes make a remarkable change in the image as this example shows.

In Spirofractal 4 I broadened the notion of what I considered to be a Sierpiński fractal, to allow for rotation, in other words a general similarity transformation. This makes the images possible for this type of Fractal far more varied. For example the Starfish Colony is created using just two transformations of this type.

Distortions

Spirofractal can distort fractals in various ways. This usually works very well with Sierpiński fractals. These images show six different distortions applied to the same starting image.

 On to Mandelbrot room