I was playing with the Turtle module in Python and decided to try plotting the Fibonacci series with the following scheme where $f_n$ is the $n^{th}$ Fibonacci number:
- Rotate the turtle $k (f_n \operatorname{mod} m)$ degrees counter clockwise for $k,m \in \mathbb{N}$
- move forward one unit
- $n \to n+1$
- GOTO $1$
I was honestly expecting random squiggles, but instead got these:
for various values of $k$ and $m$. After further research, this isn't terribly surprising due to the fact that the $f_n \operatorname{mod} m$ is periodic with period equal to the Pisano periods. This explains why patterns repeat themselves in the images, but I do not see any reason that the pattern should come precisely full circle. Is it because $k | 360n$ for some $n$?
I was wondering if anyone has seen these types of shapes before and if so, in what context? My coworker pointed out that they're quite similar to spirograph images. What I find most interesting is that some of the values for $k,m$ diverge. This is not consistent at all with the Pisano Period thought.
Here are some observations I've made:
for $m=2$ and $k$ the exterior angles of regular polygons: $$\begin{array}{ccc}k&\text{Image}&\text{Period}\\120&3 \text{ tris}&9\\90&\text{rect}&6\\72&2 \text{ quads} &10\\ 60&\text{truncated tri}&6\\45&\text{truncated square}&8\\ 40&2 \text{ truncated squares}&16\\36&\text{truncated pentagon}&10\\ 32.\bar{72}&2 \text{ truncated pents}&20\end{array}$$ And then this pattern of $(n+1)$-gon, two $(n+1)$-gons continues. Also, all images I've seen of the form $m=3n, k=20(n+1)$ have diverged. So, 2 questions:
Do the patterns return to their origin because $k|360n$ for some $n$?
And
What causes the pattern to diverge?
Thanks for your help!