Playing with the Chaos Game (II): visualizing a complex division transformation of a 4-gon fractal lattice

There is a very nice way of representing the points generated by The Chaos Game, by applying complex division. All my previous post about The Chaos Game can be found here.

For instance, let us start with the typical lattice result of The Chaos Game when there are four attractor points (4-gon) located in the corners of a square. To obtain a lattice like the one in the image below, the ratio applied is 0.4 (you can see some other combinations at Wolfram's website). Let us call the four attractor points a1, a2, a3 and a4. As it can be seen in the image, the lattice points tend to be symmetric in the x and y axes. The four attractors are located in the corners of the global square.

 

Now instead of representing those points as they appear in the image above, let us imagine that the points are complex points in the complex plane, z=a+bi, and so are the four attractors, that we will call z1, z2, z3 and z4. It is possible to apply the following transformation: divide each attractor by every lattice point generated by The Chaos Game, this is: d1(z) = z1/z, d2(z) = z2/z, d3(z) = z3/z and d4(z) = z4/z for each z in the lattice points. The resulting numbers d1(z), d2(z), d3(z) and d4(z) are also complex numbers. Let us call the transformed lattice to the set of all the points d1(z), d2(z), d3(z), d4(z) obtained applying the complex division to the whole lattice. The result is as follows:

 

The properties of the complex division and the symmetry of the original lattice points made the result very interesting!