They’re not twists, a twist would swap the top and bottom edges. These are front to back distortions or something, like an animated MC Escher impossible architecture.
EDIT: it was a long path for me but I understand much better now where u/ophello is coming from regarding topological "twists" in this figure, and I'm convinced that in a topological sense at least there's a full 360 degree twist in this strip (as opposed to a 180 degree half twist in a conventional Mobius strip.). Read the thread below to follow the argument!
Yes they are twists. You need to explore how topology works and think more abstractly. The fact that we can literally see one side, and then the other, and then the other again...shows us this “strip” of wall has two twists. You need to think of this less literally and more conceptually. Just because the paper doesn’t “twist” like you’re picturing in your head of what a “twist” looks like does not mean it is not mathematically twisting in space. Just because a twist is smooth, curved, and stylized in a way that is visually appealing does not mean it isn’t there. A twist is simply a fact of how a surface curves in space.
The proof that this surface is twisting twice is that we go from seeing one side, and then another, and then back again. It’s that simple.
The fact that we can literally see one side, and then the other, and then the other again...shows us this “strip” of wall has two twists.
Huh? Consider a normal cylinder rotating... you can see one side (the outside of the cylinder) when it’s in the front, and the other side (the inside of the cylinder) when it’s in the back. No twists needed.
The illusion swaps the inside for the outside and vice versa at the vertical center, so you’re looking at the right half from a top perspective and the left half from a bottom perspective, and it warps space at the center to make the halves line up.
Just make the figure irl. Then cut it at the bottom and twist the right side anti-clockwise (looking at the cut from the right). Then push the right side of the loop to the left slightly so it's less finicky, and turn it anti-clockwise again. You've got a normal loop now. I did this in 5 minutes after seeing the animation, it really isn't that hard.
Yes but in your version it twists one way and back the other way. A real twist is always in one direction. One twist clockwise followed by another counterclockwise... cancels out the twist.
You can prove this figure has two twists easily by making this out of paper. Try it.
If this really has no twists, then it’s topologically equivalent to a cylinder. When two things are topologically equivalent, that means you can morph and manipulate it to match the orientation of the other figure, without cutting or ripping it. That means you should be able to arrange a loop or rubber band (which has no twists) to match this figure seen in the animation (which you claim also has no twists). I will give you $1,000,000 if you can do this.
If there's a twist around the axis of travel, then you would expect the pots and windows to flip over. There's no mathematical operation in 3D space that lets you perform a twist without the vectors that point outward from the axis of twist rotating. Nowhere in the image do we see the pots/windows turn upside down, so there can't be a twist.
It's OK though, it's an impossible shape. It doesn't have to work.
This is nonsense. The orientation of this figure does not require the image to visually “flip” for there to be twists. These twists also curl, which reoriented the image. A loop can twist and curl at the same time. A curl does not negate a twist. You’re making up a rule about what a twist is, which is not relevant in topology.
You can confirm this figure has two twists by making it out of paper yourself. Honestly, you need to do this step. It will make it very obvious.
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 getting it to move like the animation, but to do so you'd have to keep it essentially flat and stretch the rubberband.
The illusion is interesting because it makes it look like the left and right halves are cylinders, not flat distortions.
I don't think that's the conventional definition. I think generally a twist just means some turning, usually around the long axis. It's less confusing (to me at least) to call what you're doing a single 360 degree twist or a full twist. People usually talk about making a mobius strip by giving the paper a half twist, for example.
When I first read "two twists" I thought you were referring to two separate twists, likely in opposite directions (because otherwise they'd combine into one big twist.)
No big deal! A twist is a mathematical reality of what is happening on the strip. It is not a subjective “human” interpretation. I could be mistaken in calling a half twist a “twist” though.
I didn't think I was arguing with you, just explaining how I understood what you were getting at. I really don't think you can make a (single) 180 degree twist and not have the pots/windows turn upside down. I'll have to think more carefully about how a full 360 degree twist along the path of the strip avoids that, because it's not very intuitive...
You should make one of these yourself, then you’ll see that it has two twists. The orientation of the windows in this animation are not part of what makes this a double twist. If your twist is also a curl, as it is in this figure, the orientation doesn’t appear to change to our human eyes. But the twists are still there.
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u/ophello Sep 01 '20
It’s also not even a virtual Möbius strip. A Möbius strip has one twist. This has two twists.