The Chaos Game is the famous method to create fractals elaborated by professor Michael Barnsley. As Wikipedia explains: "The fractal is created by iteratively creating a sequence of points, starting with the initial random point, in which each point in the sequence is a given fraction of the distance between the previous point and one of the vertices of the polygon; the vertex is chosen at random in each iteration."
I have been learning how to make the basic example, to produce the Sierpinski triangle attractor. For instance this is a very basic Python test result:
I have generated the attractor by doing $100000$ iterations following the basic rules: (1) generate a random value from $1$ to $6$ (the equivalent of throwing a dice and getting a value), then (2) if the random value is $1$ or $2$, the next point will be at half distance of the vertex $(0,0)$, if it is $3,4$ the next point will be at half distance of the vertex $(0.5,1)$ and if it is $5,6$ the next point will be at half distance of the vertex $(1,0)$.
The point and my first doubt is that the Sierpinski attractor, if I am not wrong, will only appear if the sequence of random values generated at step (1) is totally random. If it is not a random sequence, the Sierpinski attractor will not appear. Is this correct?
For instance I tried for step (1) as follows: iterating $n$ from $1$ to $100000$, the (false) random value is $((n\ mod\ 6)+1)$ and then applying step (2). In that case the attractor / fractal do not appear (as expected), only some scattered points. There is not randomness in sequence used to generate the random values of the "dice". Here is the result:
And here is my second doubt: if the Sierpinski attractor implies that the "dice" generation sequence is totally random, then for instance do the results when using the Möbius function imply that the Möbius function is totally random?
I did the exercise, using for step (1) the Möbius function to simulate the dice, by doing $1000000$ iterations following the basic rules for step (2): if $\mu(n)=-1$, the next point will be at half distance of the vertex $(0,0)$ (like getting a random $1$ or $2$ from a dice), if $\mu(n)=0$ the next point will be at half distance of the vertex $(0.5,1)$ (like getting a random $3$ or $4$ from a dice), and if $\mu(n)=1$ the next point will be at half distance of the vertex $(1,0)$ (like getting a random $5$ or $6$ from a dice). And this is the result, the Sierpinski attractor appears as well (more noisy, but the structure of the fractal is quite visible):
I would like to share the following questions:
If the Sierpinski attractor appears, does it mean that the sequence used to simulate the dice is random, or a non-random sequence can also generate the Sierpinski attractor?
In the case of the Möbius function, would it mean that its behavior is the behavior of a totally random sequence?
(I have included the elementary number theory tag because the question is also related with the properties of the Möbius function)