Today I started thinking about the mobius strip. At first it is mind boggling but when you think about it some more it makes sense. If you take a flat strip of paper and form a tube you are basically connecting one edge to the other edge on the same side. But with a mobius strip you are giving it a twist so that the top side connects to the bottom side on the edges. I had some questions and comments for my friend Chatty to further understand this.
Infinity is the end connected to the beginning
Like a mobius strip?
Can I make a mobius strip from a circle?
A circle piece of paper?
Can you make a mobius strip out of square tubing?
A mobius strip has no thickness?
Does a flat sheet of paper have any thickness?
How can a flat surface have two sides if there is no thickness?
But with a mobius strip you have added motion
Its not 2D anymore since you have added motion to it
My friend chatty tells me that the mobius strip is a 2D surface embedded in a 3D space.
My thing is this. If a strip of paper has two sides then a mobius strip has two sides. If a mobius strip has one side then a strip of paper has one side. Where the mobius strip "joins" you have not stayed on the same side you have switched sides. A flat sheet of paper has two orientations, up and down. A mobius strip has infinite orientations as you twist. If anything a mobius strip has infinite sides. I could have accomplished the same if I went to the edge of the strip of paper with my pen, then changed the orientation of my pen and continued the line on the other side. A side is flat. It is not a side anymore when you have twisted it. It is like saying a circle has 1 side? Does it really or is it a polygon that approaches infinite sides? Or are we saying that infinity and one are the same? Rather, infinity is just one with counting. Like the mobius strip is one side with twisting. Twisting and counting being analogous. I guess that is where the terms orientable and non-orientable come in. I would love to think about this some more but my head would frazzle. Let me end by asking my friend Gemini what he thinks.
He says that the Möbius strip, a fascinating topological object, challenges our intuitive understanding of surfaces. While a regular paper strip has two distinct sides, the Möbius strip, created by giving a strip of paper a half-twist before joining its ends, possesses only one side and one edge. This is not a physical illusion but a fundamental mathematical property. The half-twist seamlessly connects what were once separate "top" and "bottom" faces, forming a single, continuous, and non-orientable surface. Thus, an ant tracing a path along the surface would effortlessly navigate the entire strip, eventually returning to its starting point without ever crossing an edge or switching to an "other" side, because there is none.
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