We will develop rules for writing down what we see so as to be most convenient and likely correct.
concepts
A starting place is chosen to be the chess point most completely touched during the first few segment frames.
Place names follow chess points.
The final journey segment ends with the final chess point.
A journey ends when the a character disappears or transforms into a distinctly different character.
A cycle can be open or closed. A closed cycle would have a character return to the same spot after possibly going through multiple transformations.
Where multiple closed cycles touch we can ask how long it takes for the same characters to meet again.
checks
The character, cycle count equivalence suggests that as many groups will be visible at once as cycles required to complete the journey.
questions
Here is a question from number theory. If I have n gears, each with T(n) teeth, all collectively engaged and still freely turning, how many teeth will pass one point before every gear returns to its starting position? Assume all T(n) are different and relatively prime.
(5*6*7*9+1).times {|i| p [i%5,i%6,i%7,i%9]}
The answer appears to be the product of unique prime factors. This ruby program models four gears. One factor, 3, is shared between 6 and 9 so the start position repeats three times.