How to Play with a Mobius Strip & topology tricks

This was a three flip mobius strip with one surface that yields a loop with 8 twists. It takes 6 lines to flatten it which leads to 7 zones, one less than the number of resulting twists.

I tend to flatten mobius strips as fast as possible and the solution that I get. My rule broke down for solutions beyond 13 flips in mobius strips, so I decided to go back and start over.

This time, I wanted to understand how 6 intersections can flatten 8 twists. I found this answer! When the strip crosses itself, this kills 2 twists. When you solve mobius strips with 3 twists that have been cut down the middle you always get a solution where the strips crosses itself twice. Once inside, and once outside the total boundary.

The two self crosses therefore, kill 4 twists, there are four interestions left and four twists to kill. different sections of the strip that intersect therefore destroys one twists. Easy.

I examined the mechanics of this easy relationship and decided to explore it for symmetries, afterall symmetry exists everywhere in the universe from nature, maths, the cosmos, etc.

I found a nice example of symmety, but I'm not sure if it will stand up and how the paper model will apply to the linear algebra that I'm working on. The paper model had a center of gravity that was within one CM of the intersection that was the most symmetric! WOW

Was this luck? I don't know.

I now feel that I have to find all the solutions for the 3 flip mobius strip and see if this pattern really exists.

The mystery depends. I also want to know if there are any conditions that can be put on the solutions beyond 13 flips that would yield results consistent with the patten that I found from 1 to 9 flips?