pat wrote:I am not sure if this helps, but one can think just about the two edges of the initial strip. The resulting strips after the cut bear the same relation as the initial edges did (except that the strips can be twisted).
But, it will get you as far as which configurations are linked rings, which are just one ring, which are knotted, etc.
Nice pictures, BTW. What did you use to draw them?
PWrong wrote:Now the 6-twist becomes two 6-twists linked together, except the link is more complicated than the 4 twist. It's annoying; they should be 4 twists. Maybe I counted something wrong.
bobxp wrote:Well, I did some "practical research" (ie, cutting up bits of paper :wink: ) into how different paper twists would end up after you cut them in half. To clarify exactly what I mean by "cutting", I mean cutting along the middle of a strip like this:
<snip>
quickfur wrote:Now, the real question is, what happens if we twisted realmar paper through 4D and glued it into a Mobius tube (is that by any chance related to the Klein bottle btw?)
quickfur wrote:For even more fascinating results, try cutting the paper at 2/3 the width instead of 1/2. For the Moebius strip, you will eventually end up cutting the original paper into thirds, but the resulting shape will be radically different. Try it and see. :-)
quickfur wrote:Now, the real question is, what happens if we twisted realmar paper through 4D and glued it into a Mobius tube (is that by any chance related to the Klein bottle btw?), and then cut through it with flumar scissors. (Tries to imagine the resulting flumar twists... owie, my brain hurts.)
bobxp wrote:OMG. A Möbius strip linked to a 4-twist? I can understand why you get a Möbius strip and the link, but why a 4-twist?
Ehh. Weird. I'll imagine it an extension of a 3D Möbius strip. Now there are four possibilities:
1. An extrusion cut in the direction of the twist, which should yield an extrusion of the 2-twist.
2. An extrusion cut in the direction perpendicular to the twist.
3. A lathe cut in the direction of the twist.
4. A lathe cut in the direction perpendicular to the twist.
quickfur wrote:Hehe, beats me. Moebius strips are very strange. You could also try cutting the original strip at 1/4 the width (which does not do what you might think it does at first...) or 1/5 for really strange results, if your original is thick enough.
PWrong wrote:quickfur wrote:Hehe, beats me. Moebius strips are very strange. You could also try cutting the original strip at 1/4 the width (which does not do what you might think it does at first...) or 1/5 for really strange results, if your original is thick enough.
Why would it be any different from two thirds? You start near the edge, then get back to where you started except on (what appears to be) the other edge. Why does it matter how close you are to the edge?
PWrong wrote:So are you actually cutting it twice? Take an ordinary 1-twist. You start a quarter from the edge and cut around once. Then you're a quarter away from the other edge. You continue cutting until you get back where you started. Surely it would be the same for a third or a fifth?
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