Twisted paper

[Keiji]

Well, I did some "practical research" (ie, cutting up bits of paper ) into how different paper twists would end up after you cut them in half. I have, so far, found results for the first 5 twists:

The 0-twist splits into two seperate 0-twists (rather obviously).
The 1-twist, or Möbius strip, becomes a single 2-twist, twice as long as the original strip.
The 2-twist splits into two linked together 2-twists.
The 3-twist is strange; it becomes a single 4-twist with a knot in it. When I cut open the knot and reassembled it without removing any twists, I counted 8 twists.
The 4-twist splits into two linked together 4-twists.

[pat]

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?

[Keiji]

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).

Yes, that's an interesting thought. So, the 3-twist would have one edge (since it is a Möbius strip with another 2 twists, of course) which would have 6 twists in it. These twists of course would twine hence the knot.[/quote]

But, it will get you as far as which configurations are linked rings, which are just one ring, which are knotted, etc.

Probably, but you wouldn't be able to easily work out exactly what kinds of knots are created. The knot I drew for the result of a 3-twist up there I drew as accurately as possible by taping the knot onto the side of my monitor.

Nice pictures, BTW. What did you use to draw them?

Serif DrawPlus 5. Hence the lines between parts of the strips.

[PWrong]

I did the next two strips, and I've found an even/odd pattern.

The 5-twist becomes a knot like the 3-twist, except that where the 3-twist becomes a trefoil knot, (3 crossings), the 5-twist becomes a more complicated knot with 5 crossings. The knot from the 3-twist crosses 3 times, so there's one pattern.

After cutting open the 5-knot, I count 12 twists.

So I have two conjectures for the result of cutting an n-twist, where n is odd.

1. The result will be a knot that crosses over n times.

2. The band will have 2(n+1) twists.

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.[/i]

[Keiji]

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.

I shall try that myself with a longer strip.

EDIT: It is indeed a 6-twist. I shall now try a 8-twist to see if there is a pattern somewhere...

EDIT2: The 8-twist becomes two 8-twists. I think it would be a sensible idea to retest the 4-twist to see if I made a mistake previously...

EDIT3: I was right. The result of a 4-twist is two linked 4-twists. I take it I'm going to have to update that first post...

[quickfur]

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>

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. :-)

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.)

[pat]

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?)

A Klein bottle is a 2-D surface. Realmar paper would be a 3-D surface. Also, a Klein bottle needs to be glued on two edges. It sounded like you were only gluing one edge of your realmar paper.

It's easier to get a handle on these things if you leave them flat and visualize the way the edges fit together. For example, your realmar paper is likely a rectangular prism. If you make it so that something that leaves the left face comes back on in the same spot on the right face, you've got basically a tube. If you make it so that something that leaves the left face comes back on the right face but at a spot that's horizontally mirrored from where it left, then you've got something akin to a Moebus tube. Also, if you make it so that something that leaves the left face comes back on the right face but the whole face is 180-degrees rotated, then you've also got something akin to a Moebius tube. Many other variations are possible depending on where you decide to tape things together and what sort of twisting goes on before the taping.

[Keiji]

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. :-)

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?

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.)

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. Aa lathe cut in the direction perpendicular to the twist.

[quickfur]

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?

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.

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.

What's a lathe?

[Keiji]

A lathe in 3D would be a surface of revolution of a planar object. A lathe in 4D would be the surface of revolution of a realmar object. :wink:

And I will try cutting it into quarters. I expect it will give 2 linked twists.

EDIT: I was right. 2 linked 4-twists.

[PWrong]

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?

Now, here's a ridiculously complicated question. What about iterated cutting? If you cut a 1-twist, you get a 2-twist. Cut this 2-twist, and you get two 2-twists linked together. Now cut one of these, and you probably get three 2-twists, and so on. That's a pretty simple one. I'm going to try cutting a 3-twist, then cutting the result of that. It should get pretty messy.

[/b]

[PWrong]

Hey, I was right, it's the messiest one yet. Just remind everyone, the 3-twist becomes an 8-twist with a trefoil knot.

Now, cutting this 8-twist results in two linked strips with a huge number of twists. Each strip has a trefoil knot in itself, and is linked with the other strip with several crossings. It's confusing, because the crossings of the trefoil knot are placed in between the crossings between the strips. :?

[quickfur]

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?

It matters because it's a Moebius strip, and as such, how close you are to the edge determines how many times you cut through any given segment of the original strip. It's this latter (the number of pieces a segment is split into) that causes weird things to happen.

[PWrong]

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?

[Keiji]

Yes, you are.

[quickfur]

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?

The difference is that because it has a twist, you'll be cutting on the other side of the strip when you got back to where you started, so you'd have to keep going. That will bring you back on the right side, but now another 3rd/5th away from your starting point. So you'll have to keep going until you finally get all the way back, by which time you've split the entire strip into 3rds or 5ths. And yes, the results are very different. :-)

[Keiji]

Massive four-year bump!

I added info on paper cutting to HDDB: http://teamikaria.com/wiki/Paper_cutting

I'm particularly interested in the results of iterated cutting now, since straightforward cutting is boring by now

I also might add info on cutting stuff into thirds or more later.

Notice I redid all the images in Inkscape, to remove the lines between sections of the strips, and to make the trefoil knot look a /lot/ nicer. I must find out what the 5-crossing knot looks like though so I can make a picture of it.