# Is this a valid example of a non-euclidean Sierpinski attractor?

I am learning the basic concepts about the Chaos Game (I did a previous question about the same topic here), the method to create fractals elaborated by professor Michael Barnsley.

The basic example (euclidean-cartesian) requires three attractor vertices, $$ABC$$, and 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."

The sequence of points $$(x,y)$$ will produce the pattern of the Sierpinski attractor: When I saw that construction, I wondered if there was another way of obtaining an attractor with two variables and three attractor points, and thought about polar coordinates. I did not find any references, so I did my own version of the Chaos Game for polar coordinates.

In this version, the points are $$(\theta, r)$$, (the angle in radians and the radius). And the three attractor points are $$A=(0,0)$$,$$B=(\frac{5\pi}{4},1)$$ and $$C=(\frac{7\pi}{4},10^4)$$.

This is the result for $$10^7$$ iterations: The Sierpinski attractor appears, but there is a distortion due to the use of polar coordinates. In the case of the polar coordinates the point $$(\theta,r)$$ has two options to arrive to the attractor point: using the clockwise angle or the counterclockwise angle. Only selecting the shortest path (in this case the smallest angle of both options) the Sierpinski attractor is shown. It looks like a (sorry if I am not using the appropriated words) "non-euclidean" Sierpinski attractor.

This is another example locating the attractor points in the same axis: $$A=(0,0)$$,$$B=(\pi-\frac{\pi}{8},1)$$ and $$C=(\frac{7\pi}{4},10^4)$$. I would like to share the following questions:

1. According to the results, can be said that the Sierpinski attractor in the polar coordinates pattern is a "non-euclidean" version of the attractor? or it is only a "euclidean" distortion of the original attractor?

2. Are there papers or web references about the Chaos Game in non-cartesian coordinates? and non-euclidean versions of it?

All the references I have found are euclidean-cartesian, and I think it would be interesting finding other "flavors" of the Chaos Game. Thank you!

Update

As requested, here is how the algorithm works:

1. Start from a random point in polar coordinates $$(\theta_1,r_1)$$ (angle and radius, the angle is in radians).

2. Generate a random value in $$\{1,2,3\}$$.

3. If the value is $$1$$, the attractor point is $$(\theta_a,r_a)=(0,0)$$. The next point of the sequence is a point $$(\theta_2,\frac{r_1}{2})$$ where $$\theta_2$$ will be the angle at half distance from the current angle $$\theta_1$$ to the attractor angle $$\theta_a$$ using the shortest path. For instance if the current point has $$\pi \lt \theta_1 \le 2 \pi$$ the shortest path to $$\theta_a = 0$$ is counterclockwise.

4. If the value is $$2$$, the attractor point is $$(\theta_a,r_a)=(\frac{5\pi}{4},1)$$. The next point of the sequence is a point $$(\theta_2,r_2)$$ where $$\theta_2$$ will be the angle at half distance from the current angle to the attractor angle using the shortest path (as explained in point 3) and $$r_2$$ will be a radius with length $$r_1+\frac{r_a-r_1}{2}$$ if $$r_1 \lt r_a(=1)$$ or length $$r_1-\frac{r_1-r_a}{2}$$ if $$r_1 \ge r_a(=1)$$ (means that the new radius $$r_2$$ is half the way from $$r_1$$ to the radius $$r_a$$ of the attractor).

5. Finally if the value is $$3$$, the attractor point is $$(\theta_a,r_a)=(\frac{7\pi}{4},10^4)$$ and the calculation of the new point $$(\theta_2,r_2)$$ is the same way as in step (4).

6. Iterate from step (2) as much as possible -now the current point is $$(\theta_2,r_2)$$-.

(*) The graph is made converting $$(x,y)=(cos(\theta)\cdot r,sin(\theta)\cdot r)$$

• It is not really clear what your algorithm is doing. Could you give a concrete example? Such as how would your algorithm determine which point comes after (0,5) assuming that it is going towards C? Jul 22, 2015 at 8:48
• @BSteinhurst thanks for having a look! I have added the algorithm at the end of the question. I hope it will be easy to follow. Just let me know any doubts. :) Jul 22, 2015 at 9:24

What you have is the Chaos Game on the metric space $S^{1} \times [0,\infty)$ with the metric $d((\theta_1,x_1),(\theta_2,x_2)) = d_{S^1}(\theta_1,\theta_2) + |x_1-x_2|$. The distance in $S^1$ is given by the smallest angle measure between $\theta_1$ and $\theta_2$ (this is actually a scaled Euclidean metric on the unit circle itself.
Since the metric space you have defined is not the Euclidean metric on $\mathbb{R}^2$ it is fair to say this is a non-Euclidean implementation of the Chaos Game. I would be careful to say that you went beyond just using polar coordinates here when you implicitly changed how you define "half the distance." In $\mathbb{R}^2$ given any two points the actual distance between them is the same whether in Cartesian or polar coordinates.