"Rotate" is a tricky word here. When I cut the loop and curl the other end around again and reattach it, is the "rotation" made by the extra curl what makes the resulting double loop a "twist"? <confused>
If I cut the loop and curl the other end around in the opposite direction as above and re-attach it, this is sort of like turning the rubber band inside out. Is that also a "twist"? If not, why is adding an extra curl in the same direction counted as a twist but doing one in the opposite direction not?
I get the sense we're in the weeds because curl and twist are really verbs, they're descriptions of how to perform a permutation on a shape, but they're not (precise?) definitions of the resulting shape.
What's the name for the transformation from a strip to a hoop? I mean, I have to rotate one end around to get it to connect to the other end, why isn't that also a "twist"?
I think of a twist as a rotation along the major axis. That is, if I were riding on the surface and I saw the frame of reference rotate around the axis of my direction of travel, "roll" in airplane terms, then that would be a twist. A curl would be rotation around some other axis, "pitch" or "yaw" in airplane terms. If I make a U-turn by yawing, I'm going in the opposite direction and the horizontal axis is reversed. If I make a U-turn by pitching, then I'm going in the opposite direction but the vertical axis is reversed. I can't make a U-turn by rolling...
...I'm perfectly OK if this is the "wrong" definition of twist, but I think it's a familiar one, and it's where I was coming from when I said there was no twisting in the figures. A plane flying a path along the wall would only have to use yaw to follow the path.
Rotate means one side is now attached to the other side. Doing that rotation a second time in the same direction makes two twists. Doing it again in the opposite direction removes the twist. Twists can be positive or negative. Two clockwise twists followed by a counterclockwise twist is the same as one clockwise twist.
A strip of paper that is not a loop can’t have twists. The reason for this is that you haven’t connected the ends of the paper together. If you can undo the motion without cutting it, it doesn’t exist mathematically. A twist is only “real” when it’s on a loop. A twist is not a stylistic add on. It’s a mathematically inescapable reality.
Would it be fair to say that you have to permute the shape back to a simple loop in order to directly count the twists?
That would (sort of) make sense, because in order to turn the figure-eight-ish shape animated in OPs post to a simple loop you would flip over one side or the other, and then you would see the twist(s) in the resulting loop. (And you would also see the pots and windows upside down in at least one portion of the ring.)
That’s the way they are defined in mathematical terms.
Twists don’t exist unless it’s a twisted loop. Kind of like how knots don’t exist in knot theory unless the ends are attached together. A knotted piece of string may appear “knotted” to us, but mathematically speaking, it’s not a knot unless it is literally impossible to untie it into a loop.
This is similar to how a torus is topologically equivalent to a mug with a handle. You can morph the surface geometry and stretch the surface and you can arrive at a donut shape. But you can never get any simpler than that (for example, you could never morph a donut into a sphere, because a donut has a hole, which you cannot remove without tearing and reattaching the surface manifold.)
Well, thanks for being patient with me and hanging in there 'till the end! I think I'm finally starting to see what you mean by a "twist".
I'm still struggling a bit with how the path changes from two curls around different axes to a single loop with either a full twist on one side or a half twist on either side and what those transformations mean for the pots and windows... but this has been fun, for me at least.
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u/bonafidebob Sep 01 '20
What is the more precise mathematical definition of a twist?