Deterministic Fractals, or How to Draw Entertaining Patterns

Suppose the gray bricks are atoms , whereas the white ones are just cavities. Can we work out how many atoms there are in the system  

Similar calculations lead to the dimensionality log2 3 1.58 for gasket 

Thus Sierpinski gaskets are a simple model of objects with a fractional dimensionality. Of course, the naive ideas of length, width, and height cannot possibly help us when we try to determine the dimensionality of these gaskets. Prom the point of view of these naive concepts, the noninteger dimensionality is about as absurd as the answers with a fractional number of people that some careless primary school pupils are known to come up with occasionally. But the gaskets do have the fractional noninteger dimensionality  

Prom what we have said, by the way, it follows that, if the system is situated on a plane, its dimensionality is df 2. Similarly, in a three-dimensional space, df 3, and so on. (If one fractal is placed on another one, di 2 ) 

It is very easy to write a computer program to draw Sierpinski gaskets. All you really need to do is design a subroutine that composes the very first shape out of the elementary bricks. Then you can just keep calling this routine at each subsequent stage. In other words, you use the idea of matryoshkas, little traditionally Russian dolls that you put into one another. This principle works not only for Sierpinski gaskets, but for many other patterns as well. Some of them are very beautiful, and they all are selfsimilar. 

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