I have been recently intrigued by the 2011 IMO Problem 2 which describes a windmill process exemplified in this video by 3Blue1Brown. The question is fundamentally about invariance in that the number of blue points and the number of red points remain constant throughout the process (you may read the solution to the problem presented here to understand how the points have been coloured).
Moreover, the question, or rather the answer is beautiful in that it demonstrates invariance within a complex and chaotic unfolding. Besides, it invokes the notions of infinity and randomness that in my opinion adds to its beauty.
I'd like to define this problem or rather process as an example of a broader class of processes under the name "beautiful invariance". More precisely, this class of processes can be characterised by "invariance within complex/chaotic unfolding". Feel free to add your own properties/definitions that you believe fit the concept of "beautiful invariance".
The question I seek is to find more examples of problems/processes that could be classified under this class and any supporting documentation would also be great.
Further, examples of any real-world implementation of these processes (for example, a system that follows a process similar to the windmill process) would also be much appreciated.
Edit: A Response to Mark's comment
I understand that this post may come off as opinion-based but I've tried to specifically define what I mean by beautiful in the given context i.e. "invariance within complex/chaotic unfolding". As for the notion of randomness in the problem, I related it to the fact that the points in the space were randomly chosen. I don't wish for it to be viewed in any other abstract way. As for the use of the term "chaos", it was probably a bad choice as I didn't foresee the conflicts it may generate with traditional definitions of chaos. In layman terms, when I say "chaos", I simply mean "a lot of stuff going on". With reference to the problem, the "lot of stuff going on" is the constant changing of the pivot of the line. I hope this generates a very specific direction to the question that is not based on opinions but facts. To reiterate, I am seeking examples, documentation and real-world implementation of problems/processes that follow the above definition of beautiful. You may still argue that my question is open-ended especially because I said
Feel free to add your own properties/definitions that you believe fit the concept of "beautiful invariance".
I said this because I didn't want to restrict myself to only the properties of the problem I observed as the wealth of mathematical knowledge in this community surely has more to provide. To further strengthen my case, I must mention that many highly upvoted posts, including the most highly upvoted post on math.se use rather vague notions of beauty. If this line of questioning is unacceptable on this site, however, I understand and I would request than to ignore the "feel free to add..." part.
I am open to any other comments on how I can improve this post.