Möbius Showcase!

I have been interested in topology for quite a while, but today was the first time I went beyond simple Möbius strips and did some really cool mathematical experiments! I made quite a lot of contraptions using only a pair of scissors, glue, and strips of paper. Presenting to you, the Möbius Showcase! -

1. Simple Möbius Strip

Specifically, with 1 half-twist

2. Cutting a Möbius Strip in Half

Surprisingly, this gives a new Möbius strip which is twice as long and has four half-twists/two twists. The photograph is a bit unclear.

3. Cutting a two-twist Möbius strip in half

Another unexpected result: we now have two strips going through each other.

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4. Cutting a three-twist Möbius strip

We get a single loop with - get this - a knot!

5. Cutting a normal loop in half

Pretty much what you'd expect

6. Cutting two normal loops which are at right angles in half

Strangely, what we get is: a flat square (which makes for a nice photo frame).

7. Cutting a Möbius and a normal which are at right angles into half

This gives us - yet again - a flat square.

8. Cutting two Möbiuses (Möbii?) which are at right angles into half - opposite chirality

I saved the best for the last. Our result is *drumroll* two linked hearts!

In One Photograph

If you are wondering about equivalents to the Möbius strip, it is part of a group called non-orientable surfaces. Its next-dimensional equivalent is the Klein bottle (named after Felix Klein), but sadly, they are not delivered to three spatial dimensions. You can create some, but they will always have to intersect with themselves, since the 4D 'twist' is in another imperceptible dimension.

It is kind of like the shadow of a Möbius strip. 

You will notice that at one point in the upper left corner it is reduced to just a single line - resembling the intersection of a Klein bottle.

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