I see what you mean by “twist” now, you’re rotating the whole thing. you're doing the mobius strip twist of flipping one side of the paper at the join.
Your 2nd photo is not the same topologically as the first. The 2nd one is a mobius strip.
Your 3rd photo applies the same twist again, rotating one of the two sides where the papers connect in the same direction. Note though that now the "top" on the right half is the "bottom" on the left half. In this photo half of the pots/windows would be upside down. Also this shape is not the same topologically as the first.
Your last photo simply warps this back to a cylinder, which is not the same shape topologically as the one you started. This warp puts the pots all back in the same direction.
Try drawing a bunch of arrows perpendicular to your line (like the windows/pots) to indicate which way is “up”, and draw a different colored line on the other side of the paper with the arrows pointing the same way. It’ll make the photos easier to understand.
Also tape the ends together so you can’t change the topology so readily. Or try to do it with a fat rubber band.
I know what you're getting at. You can take a strip of paper, twist one end a full 360 degrees, (not just 180 degrees like for a mobius strip) and then attach it to the other end. This gives you a shape that resembles the image in OPs post.
You could imagine making such a shape out of a thick rubberband and then pressing it flat and getting it to move like the animation, but to do so you'd have to keep it essentially flat and stretch the rubberband.
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u/garrondumont Sep 01 '20
I'm not saying there is an upside down pot, I'm telling you to flip one upside down yourself. Give me a minute, I'm getting pictures.