I've been playing around with more Pisano circle stuff, and I've realized that there's much more to be done with this stuff.
The entire time we've been looking at writing the remainders along the outside of the circle and making a bunch of connections from remainder to remainder in the period.
But what about writing the entire period around the edge of the circle? Then what you can do is connect every single point with the same remainder value.
I tried this out with Fib (mod 10), which has a period length of 60. So that means all the 1s were connected to each other, all the 2s, all the 3s, etc.
The result was a pattern that was not vertically symmetrical like before, but instead it has
rotational symmetry.

- f60.jpg (752.08 KiB) Viewed 4456 times
Moreover I noticed that the amount of remainders in the period repeats:
# of 0s = 4
# os 1s = 8
# of 2s = 4
# of 3s = 8
4=4
5=8
6=4
7=8
8=4
9=8.
I also drew out the new period circles for mods 1 through 6. Below is the original circles from the last video and underneath is the new set of circles (with Roman numerals to differentiate) and listed under those is how many times each number appears on the circle.

- pis compa.png (978.89 KiB) Viewed 4456 times
So there's an expansion to this exploration
