5 - Points in Sierpiński-like Fractals

Summary

The Sierpiński Triangle (see Figure 5.1) is one of the most recognized fractals. It has many well-known properties and there are many different ways to define it. For example, one can define the Sierpiński Triangle as what is left after removing certain sets of points; see, for example, [2, p. 180]. Another definition of the Sierpiński Triangle, and the one we will use, is that the Sierpiński Triangle is the fixed point of an Iterated Function System (IFS). See [1] for a complete introduction to Iterated Function Systems; a brief description covering what is needed for our purposes is given below. Defined this way, an interesting question is exactly which points (x, y) are in the triangle since the IFS definition does not make it readily apparent. However, it is not too difficult to show that for the version of the Sierpiński Triangle that has corners at (0, 0), (0, 1) and (1, 0), the points (x, y) in the Sierpiński Triangle are those points that have a binary expansion with a certain property. Specifically, points (x, y) that have binary expansions such that x and y do not both have a 1 in the same position are in the Sierpiński Triangle [9]. In this paper, we extend this idea to the whole class of “Sierpiński-like” fractals, developing a method for generating similar rules for which points (x, y) are or are not in the fractal.