# Möbius strips

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### #18 CraigD

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Posted 25 May 2007 - 10:09 AM

Then we have a recent report here on a strange hexagon at Saturn's North pole >> [ ”Hexagon on Saturn” and ”Cassini Images Bizarre Hexagon on Saturn” ]

My speculation is that we see a hexagon, but it's really a Möbius strip of charged particles.

I can’t see how an image produced by swirling gas can be equated to a Mobius strip, other than by a superficial resemblance.

The Saturn hexagon appears to be a cloud and gas formation. It’s not a true object in the sense that that an ordinary Mobius strip is. While the latter is made of a thin rectangular solid comprised of a bunch of atoms bonded into molecules, stuck together into cells, which are stuck together into fibers, which are stuck together into the thin sheet, the Saturn “hexagon” is a bunch of atoms and molecules in gas form, some likely clumped into dust/smoke particles, but not a solid whole. Matter is constantly entering and leaving the formation, and moving within it. It’s no more really a “band” than are the various bands of Saturn’s rings, or a procession of cars on a busy highway is a “snake”.

Another problem with drawing an equivalence between a Mobius strip and Saturn’s hexagon is that the latter’s not shaped much like a strip of paper. Though overall shaped like a thin band with 6 somewhat strait segments and 6 sharp angles, the segments vary a lot in thikness, from about 100 to 1000 km in width. Observations have revealed them to extend about 100 (but not as much as 1000) km from their tops to their bottoms. Their shape, then, appears to be roughly that of a flattened cylindrical tube, with no sharp edges to allow them to be divided into a top and bottom face for the Mobius construction to make a single surface. Scrutinizing the JPL images reveals no sign of the (single) distinct edge that defines a Mobius strip. In short, it’s hard to make a decent Mobius strip out of a garden hose – especially one that’s made out of wind and clouds.

The Mobius strip analogy might be more applicable if the wind direction within the Saturn formation showed a distinct pattern of alternating up and down at adjacent sharp corner (under high magnification, the corners don’t appear all that sharp, with radiuses of at least a couple of tens of kilometers, but still, clearly, a lot of air is changing direction more abruptly than usual). This would indicate that the formation was in a sense “folded” rather than curved. Visual examination of the images doesn’t appear to suggest this, and, AFAIK, Cassini’s instruments capable of imaging this don’t include ones capable of directly measuring wind velocity (eg: doppler radar). It’s imaging radar is, I believe, intended for surface mapping, not velocity measurement – though it’s unwise to underestimate the flexibility with which mission staff – some very smart folks - can use their instruments.

What’s forming Saturn’s strange “hexagon” is a fascinating mystery, on a par, I think, with that of Jupiter’s Great Red Spot. We’ve got to be careful to avoid finding patterns that aren’t really there, however – the human brain seems optimized to finding such patterns, even when it must be excessively inventive to do so.

### #19 Turtle

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Posted 25 May 2007 - 01:00 PM

I can’t see how an image produced by swirling gas can be equated to a Mobius strip, other than by a superficial resemblance.

...What’s forming Saturn’s strange “hexagon” is a fascinating mystery, on a par, I think, with that of Jupiter’s Great Red Spot. We’ve got to be careful to avoid finding patterns that aren’t really there, however – the human brain seems optimized to finding such patterns, even when it must be excessively inventive to do so.

That you (or 'they') can't see how, does not affirm that my conjecture is false. We've got to be careful not to discount explanations simply because they don't come from a 'specialist'.

Imagine if you will, in your best gedanken style, a rain storm at night and you turn on a spotlight which you slowly rotate through the sky. The apparent motion of the beam has nothing to do with the motion of the rain drops, which continue to fall whether they're in the light or not. I see the hexagon on Saturn in a similar light, in that the magnetic field is the light and charged particles in the atmosphere are the drops. They only 'glow' when they're in the path.

Anyway, discounting my conjecture seems to require more than casual objections.

PS Let's not forget that we have a source stating that charged particles doform Möbius bands in Earth's atmosphere. I can't find the actual article, but here is the reference Wicki gave:

Charged particles, which were caught in the magnetic field of the earth, can move on a Möbius band (IEEE Transactions on plasma Science, volume. 30, No. 1, February 2002)

PPS I just sent an e-mail to the webmaster at SpaceWeather.com with my conjecture and a link to this thread. We might as well get this over with.

### #20 Qfwfq

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Posted 30 May 2007 - 07:56 AM

Whether by paper or by gedanken, you have given the exact value I asked for, and it agrees with my result; ${\sqrt{3}$ Nicely done!

And yet I can make one with length equal to width plus overlap, that is a perfect square if you don't count the overlap. It's the extreme limit case, of course.
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### #21 CraigD

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Posted 30 May 2007 - 10:13 AM

And yet I can make one with length equal to width plus overlap, that is a perfect square if you don't count the overlap. It's the extreme limit case, of course.

At first glance, I though Qfwfq err in this claim. At second glance, I reexamined my own assumption,

As best I’m able to figure, the shortest possible Mobius strip is made by creasing the strip 3 times at a 60° angle, doubling it back around on itself while flipping it upside down.

An realized the error was mine. It’s possible to make a Mobius strip using a perfectly square (ignoring any overlap used for practical reasons to stick a physical model together) piece of paper, or any rectangular piece of paper with a length/width ratio $\lt \sqrt3$, as follows:
Assuming corner labeled
1 2
3 4
, fold 4 onto 1, press flat, giving
1/4 2
3
. Then fold 2 onto 3, giving
1/4
3/2
. Join edge 2-4 to edge 3-1 (this is the “third flip” required for opposite sides of the paper to become one). Done.

It’s not very pretty, but counting its edges, it has 1, and sides, 1. It’s a Mobius strip. To traverse it, an and must force its way through a tight passage where the paper is pressed flat against itself.

With apologies to turtle for having my crude fold and tape work share a page with his beautiful models and photons, here’s a picture of one made from a standard 8”x11” (7.5” x 10.5”) piece of planning pad paper (square ruled on one side, line ruled on the other):

Qfwfq, you a paragon of seeing though unwarranted assumptions, and an all-round smart and well-spoken guy!

### #22 Turtle

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Posted 30 May 2007 - 01:42 PM

And yet I can make one with length equal to width plus overlap, that is a perfect square if you don't count the overlap. It's the extreme limit case, of course.

More to the point now, did you make it? That is to say, I'm curious what brought you to your conclusion.

Qfwfq, you a paragon of seeing though unwarranted assumptions, and an all-round smart and well-spoken guy!

If not timely.

At first glance, I though Qfwfq err in this claim. At second glance, I reexamined my own assumption,

An realized the error was mine. It’s possible to make a Mobius strip using a perfectly square (ignoring any overlap used for practical reasons to stick a physical model together) piece of paper, or any rectangular piece of paper with a length/width ratio , as follows:
Assuming corner labeled
1 2
3 4
, fold 4 onto 1, press flat, giving
1/4 2
3
. Then fold 2 onto 3, giving
1/4
3/2
. Join edge 2-4 to edge 3-1 (this is the “third flip” required for opposite sides of the paper to become one). Done.

It’s not very pretty, but counting its edges, it has 1, and sides, 1. It’s a Mobius strip. To traverse it, an and must force its way through a tight passage where the paper is pressed flat against itself.

With apologies to turtle for having my crude fold and tape work share a page with his beautiful models and photons, here’s a picture of one made from a standard 8”x11” (7.5” x 10.5”) piece of planning pad paper (square ruled on one side, line ruled on the other):

My own reexamination went your usual gedanken route however, and while not posting the result in a timely manner, I realized how easy it would be to sew a mobius strip from a sqaure of fabric. Tan mieux the topic is in Engineering & Applied Science.

### #23 Turtle

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Posted 30 May 2007 - 05:23 PM

...I realized how easy it would be to sew a mobius strip from a sqaure of fabric. Tan mieux the topic is in Engineering & Applied Science.

What can I say; I meant 'relatively easy.'
YouTube - sewing MÃ¶bius canvas band

### #24 Jay-qu

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Posted 30 May 2007 - 05:25 PM

urrr isnt that what I said about 16 posts ago ?
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### #25 Turtle

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Posted 30 May 2007 - 05:37 PM

urrr isnt that what I said about 16 posts ago ?

Errr...maybe you meant to say that [a square sheet can be Mobius], but you lost my confidence when you said

...Consider the mobius case where W > L, it would then be impossible to traverse a distace of L and end up a displacement of W from you starting point.

In standard parlance, if there is a difference between L (length) and W (width), the L (length) is always assigned the larger dimension. So in effect, you just switch the assignment of variables.

So, just how 'narrow' can a Mobius band be anyway?

### #26 InfiniteNow

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Posted 30 May 2007 - 06:21 PM

What can I say; I meant 'relatively easy.'
YouTube - sewing MÃ¶bius canvas band

For some reason, your video clip reminded me of the movie Pi (full length film available below).

### #27 Turtle

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Posted 30 May 2007 - 07:11 PM

For some reason, your video clip reminded me of the movie Pi (full length film available below).

I'm familiar with that film. I have it playing while I post.

So I set to twiddling my 1 / 1 canvas Möbius band, and I decided to jam my hand in and see if I could get it on my wrist as a bracelet. My hand is a little too big, but low & behold in the process I jammed that little pecker right into Craig's described form. Well, a cone at first, long & tapering with a sort of tail, but when I flattened it there was Craig's directions and in the form of a 45/45/90 triangle no less! No small coincidence the cone form makes a great dunce hat.

So, a 1/1 Möbius band flattens into a 45/45/90 triangle; a ${sqrt 3}$/1 Möbius band flattens to a 60/60/60 triangle; and a 3${sqrt 3}$ Möbius band flattens to a regular hexagon.

Well, it's a near record temp here in a hundred years just now (90ºF), so off to chill & twiddle my bits.

### #28 Qfwfq

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Posted 31 May 2007 - 02:58 AM

Qfwfq, you a paragon of seeing though unwarranted assumptions, and an all-round smart and well-spoken guy!

Ooooh, you're embarassin' me!

More to the point now, did you make it? That is to say, I'm curious what brought you to your conclusion.

Just thought it over, did the paper folding when I got back home although I had no doubt of the possibility, just to see to what point it can be unfolded. Obviously not much, being the extreme case it's hardly recognizable but can be better understood in topological terms by just considering how the joined edges go through the two folds.

In standard parlance, if there is a difference between L (length) and W (width), the L (length) is always assigned the larger dimension. So in effect, you just switch the assignment of variables.

So, just how 'narrow' can a Mobius band be anyway?

I must speak in JQ's defense, he was just recycling the ordinary terms. In the case in point, by length of the rectangle he meant the distance between the edges to be joined. In order to resolve the conflict with standard parlance, call this banana. By its width he meant the distance between the other two edges, to the same purpose call this apple.

This said, JQ correctly argues that banana can't be less than apple. The reason is obvious enough. This is a necessary requisite for Möbius, whereas a segment of cylindrical surface can be made joining edges of a rectangle without this constraint between banana and apple. All that remained for JQ to demonstrate was that "banana not less than apple" is sufficient for Möbius.

### #29 Jay-qu

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Posted 31 May 2007 - 03:38 AM

Yeah.. I was only saying that you cant go further than the square case, but failed to proove that the square case is possible

### #30 Turtle

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Posted 31 May 2007 - 01:58 PM

... I'm curious what brought you to your conclusion.

Just thought it over, did the paper folding when I got back home although I had no doubt of the possibility, just to see to what point it can be unfolded. Obviously not much, being the extreme case it's hardly recognizable but can be better understood in topological terms by just considering how the joined edges go through the two folds.

Ahhhh...that is very satisfying to hear. That is to say (at the risk of embarrassing you further) I figure I got a tiger by the toe when something I say gets you to thinking.

I must speak in JQ's defense, he was just recycling the ordinary terms. In the case in point, by length of the rectangle he meant the distance between the edges to be joined. In order to resolve the conflict with standard parlance, call this banana. By its width he meant the distance between the other two edges, to the same purpose call this apple.

This said, JQ correctly argues that banana can't be less than apple. The reason is obvious enough. This is a necessary requisite for Möbius, whereas a segment of cylindrical surface can be made joining edges of a rectangle without this constraint between banana and apple. All that remained for JQ to demonstrate was that "banana not less than apple" is sufficient for Möbius.

Yeah.. I was only saying that you cant go further than the square case, but failed to proove that the square case is possible.

In my own defense, I have to 'hear' things just the right way to understand them; but you guys knew that. You know as well that I am frequently not above poking a little fun.

Now what I want to know/hear is whether or not we have generated new descriptions of Möbius bands in regard to flattening particular length/width strips, or if somewhere there is a book or pages that lay out what we have developed? If this is new, we ought to make some terms to describe the family of varying ratios.

To answer my own question about how narrow a Möbius band can be (I was teasing Jay a bit with this one too:lol: ), the limit approaches banana = apple where apple is the thickness of the band. But then theoretically, apple has no thickness.

If I make a stiff paper 3${sqrt 3}$ Möbius strip and leave it 'open', and then glue thin copper wire onto it round-n- round til it's covered and them attach a battery to the coil, how will the resulting magnetic field differ from a simply wound round coil of the same gauge & length of wire?

### #31 Turtle

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Posted 02 June 2007 - 02:11 PM

...I must speak in JQ's defense, he was just recycling the ordinary terms. In the case in point, by length of the rectangle he meant the distance between the edges to be joined. In order to resolve the conflict with standard parlance, call this banana. By its width he meant the distance between the other two edges, to the same purpose call this apple.

This said, JQ correctly argues that banana can't be less than apple. The reason is obvious enough. This is a necessary requisite for Möbius, whereas a segment of cylindrical surface can be made joining edges of a rectangle without this constraint between banana and apple. All that remained for JQ to demonstrate was that "banana not less than apple" is sufficient for Möbius.

I thought so too; however, using another canvas square 9"x9", I marked off a 7" edge and then puckered and pinned said band banana Möbiusly (Möbius bananally?), i.e. I joined the longest edges. Seems we all were mistaken about the limiting case.

### #32 Turtle

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Posted 02 June 2007 - 03:58 PM

I thought so too; however, using another canvas square 9"x9", I marked off a 7" edge and then puckered and pinned said band banana Möbiusly (Möbius bananally?), i.e. I joined the longest edges. Seems we all were mistaken about the limiting case.

I went ahead and cut a 8"x10" canvas rectangle and sewed the long 10" edges together. I'll spare you the movie this time, but I sewed the edges from the outside rather than the inside and it went more quickly. Then proceeding as before, I starting poking my fingers into it until I forced it flat. Here's the scan of one side, and the other side has the seam down the middle. The seam continues inside, where there are several unseen folds.

Interesting to note that this flattened band is 6 layers thick at it's thickest, the square is 4 layers thick, the triangle 3 layers thick, and the hexagon 2 layers thick at their thickest(s:hyper: )

### #33 Turtle

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Posted 02 June 2007 - 07:13 PM

I went ahead and cut a 8"x10" canvas rectangle and sewed the long 10" edges together. I'll spare you the movie this time, but I sewed the edges from the outside rather than the inside and it went more quickly. Then proceeding as before, I starting poking my fingers into it until I forced it flat. Here's the scan of one side, and the other side has the seam down the middle. The seam continues inside, where there are several unseen folds.

Interesting to note that this flattened band is 6 layers thick at it's thickest, the square is 4 layers thick, the triangle 3 layers thick, and the hexagon 2 layers thick at their thickest(s:hyper: )

Keeping in mind that sometimes a pen is just a pen, I poked mine deep into as many folds as I could reach and drew lines on they edges as straight & dark as merited the circumstance. Likewise with their outside edges. Then I cut the seam threads and laid 'er flat. Then I scanned it. I added red lines with arrows on the sides seamed together. Here it is attached: (It, is what they call a 'planar net'.)>>

I suppose giving an order to folding is de rigueur, but I'm dee kneep in IPA so that will just have to wait.

### #34 InfiniteNow

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Posted 02 June 2007 - 08:18 PM

[attachment=886:2/6/9/4/1486.attach]

It reminds me a bit of one of the "petals" of the vesica piscis flower:

[attachment=887:2/6/9/4/1487.attach]

I suppose giving an order to folding is de rigueur, but I'm dee kneep in IPA so that will just have to wait.

Bhat Wrand?