What is a Möbius strip and what does it portray?

In: Physics

1. take a sheet of paper

2. cut out a strip along one of the longer sides. make it not too narrow, but not too wide. 3-4cm will suffice

3. turn one of the (short) edges around, and, without letting it go, glue the opposite end to it

4. take a pencil and start drawing a line along the long edge

5. you’ll find yourself going back to the starting point despite not crossing any edges

[some](https://lh3.googleusercontent.com/proxy/ImoL-fWsLmWpctchwzXA65ROKFcwbxeiliQt72Jicvi0vyo0g9HFrYNuQbdV8Vlp-cujcUr_rAbozRro6FYg8A) [pictures](http://www.indepthinfo.com/mathematics/images/mobius-strip.jpg) [to](https://darylchow.com/frontiers/wp-content/uploads/2020/07/images-6.jpeg) help illustrate what’s going on

and I wouldn’t say it ‘portrays’ anything apart from being quite an interesting geometrical thingy in itself.

A Mobius strip is an (idealized) 2D surface with only one side.

Take a normal strip of paper, and ignore the thickness (that’s the idealized part). It has two sides. Even if you connect the ends to make a ring, it still has two sides (and “inside” and an “outside”). Any normal 2D surface is like this, regardless of shape.

Except try twisting one end of the strip 180 degrees before you attach the ends. Now you’ve got a ring with a half-twist in it. And if you trace your finger along the surface you’ll find that you cover the *entire* surface before you get back where you started…it only has one side. This is weird. It’s a 2D object in 3D space that only has one side. It makes topologists (mathmaticians who study surfaces and such things) excited.

The 3D version is called a Klein Bottle. It’s equally weird. You can buy them here: [https://www.kleinbottle.com/](https://www.kleinbottle.com/)

If you take a flat strip of paper, twist it once and attach the two short ends to each other you have a mobius strip.

In mathematical terms, it is a topological construct with only 1 surface (1D) but taking up 3D space.

It is used to illustrate the idea of how things in lower dimensions might exist in higher dimensions.

Mobius loops are the basic prototype of non orientable surfaces. That would be a lot of math to define, so I’ll just give an example. An orientable surface would be a sphere. Stand on the equator and point straight out. Walk a full circle along it. You’ll be at the same point in space as you started and pointing in the same direction. Do the same thing on a mobius loop. You’ll end up pointing in the opposite direction (toward the center) even though you’ve returned to the same point in 3d space. That makes the mobius loop NON orientable. This gets a bit more complicated in different kinds of geometries (like what constitutes a full rotation for example). Physical systems confined to non orientable surfaces (mathematically speaking) admit other mathematical properties. It’s a broad topic. The mobius loop is the simplest example of these kinds of surfaces.

Take a ribbon of paper. Twist one end 180 degrees. Tape both ends together. This is a Mobius strip. It has some fairly interesting properties. One property is that it only has one “side.” If you start drawing a line down the middle of the paper, that line will eventually cover both “sides” of the original paper without you ever living the pen, eventually connecting back from where you started.

If you cut a Mobious strip in half (down the length of the ribbon) you’ll end up with a single, longer but thinner, strip with two twists in it.