Notation of set on $R^2$ (attractor of Cantor dust)

Question

I am studying fractal geometry, and pretty much confused by the notation/meaning below:

Quote: The Cantor dust is easily seen to be the attractor of the four similarities on $R^2$ which give the basic self-similarities of the set:

$S_1(x,y)=(x/4,y/4+1/2)$, $S_2(x,y)=(x/4+1/4,y/4)$,

$S_3(x,y)=(x/4+1/2,y/4+3/4)$, $S_4(x,y)=(x/4+3/4,y/4+1/4)$

Question: What do these set notations mean? Can x,y be any value as wish? How is this an attractor of Cantor?

Thanks a lot,

Answer 1

Okay, I got it, after nearly trying to put my head against wall. :)

It turns out to be simple.

The Cantor dust definition gives $x \in [0,1], y \in [0,1]$

Hence:

$S_1(x,y)=([0,1/4],[1/2,3/4])$

$S_2(x,y)=([1/4,1/2],[0,1/4])$

$S_3(x,y)=([1/2,3/4],[3/4,1])$

$S_4(x,y)=([3/4,1],[1/4,3/4])$

Union of these sets defines exactly the attractor of Cantor dust.