Mobius Strip Transformation Symmetry - Topology

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?

Category:

Science & Technology

Tags:

• Mobius
• strip
• transformation
• symmetry
• topology
• new
• discovery

[kitefrog]

There's a book called topology that talks about the twists after = 2N +2. Where N is the number of twists before. ~I forget the title.

[kitefrog]

yes, thanks zyxon, I was premature and excited.

[kitefrog]

I'm bipolar, in a manic stage I'm creative with big ideas. This work came out of my last mania that last 6 weeks. I could spend 9 to 12 hours straight making paper models and studying the resulting graph theory. Unfortunately, it remains and open problem in math to count how many resulting eulerian ciruits that result in the graphs created from the paper models. VEry stranfe stuff. I'm not working on it now, just studing graph theory

[starlite528]

would be nice if the video was oriented properly

[thedelta88]

very cool. do u just do that in your free time?

[relike868p]

Which paper do you find that 13 flips give 28 twists after cutting in half?

(I mean the M = 2n+2 formula, which appears on Wikipedia, but there is no source for that)

[Izzy827M]

Exellent. You have some great ideas- I would never look for the center of gravity for the pattern.

[kiakaku]

i don't know what do he want to do ????????????

[glowingdarkmatter25]

i love math, but it hates me. This is bizarre, but cool.

[TheAlansonTobias]

good job nice way to prepare information :)

[thetommantom]

I had to do this, I just found this video and started watching it now, 819

[melman2004]

you should write a 3 dimensional surface animation program to clearly see the intersections. it seems like your comparing your home made origami to a field equation.

[mike6789k]

hahaha, thats what he said!

[OwnedOver9000]

Amazing!!!!