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Möbius strips. Plenty written on them, but I made a few for play and I have some questions and observations that I haven't seen mentioned. Here's the first challenge.

 

By experiment, determine the smallest ratio of length-to-width that a piece of paper may have (that is to say, what is the widest possible strip of paper) and still have the capacity for forming into a Möbius strip. :doh: :lol:

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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.

Here's a scan of my 16 3/4" X 3" Möbius strip partially flattened and forming a regular hexagon.

Ugly ‽ Clunky‽ :eek: I get the sense you are prodding this ol' turtle‽   Very well; I'll assume the role of umbrage taker. :hihi: Möbius strips of all proportions are not kinky, only half-twisted, a

I love these things - I once made a bet with a friend, I said "how much do you want to bet I can cut this peice of looped paper in half but still keep it in a loop?" :hammer:

 

:doh: That's always a good one! :rose: Now since you have such a good rapport with the little devils, care to offer an answer to my question? Did I express the question well enough?

 

I now have an exact answer, but I'm interested to see responses from you-all dear readers. I must say that I only came to my answer by actually making a collection of Möbius strips of varying ratios of length-to-width, and I would be surprised to see any of you arrive at the exact answer by other means. Fascinating. :) :lol:

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So, Im going to have to talk this out..

 

-for a modius strip to be sucessful one corner of a rectange must reach the opposite diagonal corner and likewise for the other pair.

 

-For a rectagle of length L and width W, made into a strip one complete perimeter path will equal L

 

-If that rectangle is made into a mobuis strip, starting at one point on the edge of the strip and following the edge around untill you get back to the starting point, the path will be of distance 2L. While traveling L around the stip edge will place you at W displacement from your starting point.

 

more to come.. :beer-fresh:

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So, Im going to have to talk this out..

 

-for a modius strip to be sucessful one corner of a rectange must reach the opposite diagonal corner and likewise for the other pair.

 

-For a rectagle of length L and width W, made into a strip one complete perimeter path will equal L

 

-If that rectangle is made into a mobuis strip, starting at one point on the edge of the strip and following the edge around untill you get back to the starting point, the path will be of distance 2L. While traveling L around the stip edge will place you at W displacement from your starting point.

 

more to come.. :beer-fresh:

Im not sure if this constitutes a full mathematical proof but it does logically work.

 

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.

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Im not sure if this constitutes a full mathematical proof but it does logically work.

 

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.

 

Perhaps, but I want a value. What is the smallest ratio of the long side L to the width W, or in effect the largest possible W for a given L

 

Here's my little family of Möbius strips, each made from 17" long paper with 1/4" overlap on the glue joint (I recommend glue over tape for these) giving an effective length for L of 16/3/4". The widths W varies in increments of 1" for the 8 Möbius strips in a row. You can work out some of the ratios before I post them if you like.

 

(I didn't realize when I was making them that Möbius strips are chiral, so I made both lefts & rights. Can you see the differenc? ):) :beer-fresh:

 

Click image to open full-size view in new window >>

 

 

Dont neglect hexaflexagons: Hexaflexagon Toolkit

 

That may give a clue, but a hexafleagon is 2 times more than what we really have in the Möbius strip of limit that I am asking the value for. See if my photo helps. :)

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By experiment, determine the smallest ratio of length-to-width that a piece of paper may have (that is to say, what is the widest possible strip of paper) and still have the capacity for forming into a Möbius strip. :cup: :turtle:
No physical construction experiments for me, sir Turtle – I’m a numbers person, not a scissors and paper person (or at least not a very precise scissors, paper, and ruler person ;))

 

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. Giving yourself a little “room to turn the corner”, this makes a multi-thickness hexagonal thing. Cutting the corner-turning room to zero, its shape goes to a triangle. It’s length, then is [math]\frac{3}{\sqrt{3}} \dot= 1.73[/math] time its width, not including any overlap for the ends. Laid out flat with its creases marked, it looks like 2 equilateral triangles laid edge-to-edge with 2 30/60/90° triangle on each end to make a rectangle.

 

Short, wide Mobius strips seem kinda ugly and clunky. I wonder if any objective “looks like a pretty Mobius strip” criteria can be stated?

 

All Mobius strips made of twisted rectangles, even long, thin ones, are slightly kinky. I wonder it one can make a kink-free strip out of a flat, non-rectangular shape?

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No physical construction experiments for me, sir Turtle – I’m a numbers person, not a scissors and paper person (or at least not a very precise scissors, paper, and ruler person ;))

 

No worries; that's what you-all keep me around for. :hihi:

 

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. Giving yourself a little “room to turn the corner”, this makes a multi-thickness hexagonal thing. Cutting the corner-turning room to zero, its shape goes to a triangle. It’s length, then is [math]\frac{3}{\sqrt{3}} \dot= 1.73[/math] time its width, not including any overlap for the ends. Laid out flat with its creases marked, it looks like 2 equilateral triangles laid edge-to-edge with 2 30/60/90° triangle on each end to make a rectangle.

 

Short, wide Mobius strips seem kinda ugly and clunky. I wonder if any objective “looks like a pretty Mobius strip” criteria can be stated?

 

All Mobius strips made of twisted rectangles, even long, thin ones, are slightly kinky. I wonder it one can make a kink-free strip out of a flat, non-rectangular shape?

 

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

 

Now I don't know about other shapes yet, but I have a bit more on the Möbius strip. As one approaches the length/width limit of [math]{\sqrt{3}[/math], the Möbius strip gets flatter & flatter, but it is possible to flatten down the other springy strips as well. Doing this to a particular Möbius ratio gives a regular hexagon, and I suspect the width then is 3* [math]{\sqrt{3}[/math]. (For my 16 /3/4" length, if I cut a strip ~3 1/4" wide & make a Möbius strip of it, that strip will flatten into a regular hexagon.)

 

Now for my wild speculation>>> :hihi: The Wicki article I linked to says this at the bottom:

...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)...

 

Then we have a recent report here on a strange hexagon at Saturn's North pole >> http://hypography.com/forums/space/10994-hexagon-saturn.html?highlight=Saturn+hexagon

 

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

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Now I don't know about other shapes yet, but I have a bit more on the Möbius strip. As one approaches the length/width limit of [math]{\sqrt{3}[/math], the Möbius strip gets flatter & flatter, but it is possible to flatten down the other springy strips as well. Doing this to a particular Möbius ratio gives a regular hexagon, and I suspect the width then is 3* [math]{\sqrt{3}[/math]. (For my 16 /3/4" length, if I cut a strip ~3 1/4" wide & make a Möbius strip of it, that strip will flatten into a regular hexagon.)

 

Now for my wild speculation>>> :hihi: The Wicki article I linked to says this at the bottom:

 

Then we have a recent report here on a strange hexagon at Saturn's North pole >> http://hypography.com/forums/space/10994-hexagon-saturn.html?highlight=Saturn+hexagon

 

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

 

Here's a scan of my 16 3/4" X 3" Möbius strip partially flattened and forming a regular hexagon.

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Short, wide Mobius strips seem kinda ugly and clunky. I wonder if any objective “looks like a pretty Mobius strip” criteria can be stated?

 

All Mobius strips made of twisted rectangles, even long, thin ones, are slightly kinky. I wonder it one can make a kink-free strip out of a flat, non-rectangular shape?

 

Ugly Clunky :eek: I get the sense you are prodding this ol' turtle :jab:

 

Very well; I'll assume the role of umbrage taker. :hihi: Möbius strips of all proportions are not kinky, only half-twisted, and objectively simply beautiful. Moreover, as a family where the length is held constant while the width varies, the beauty is compounded at the [math]{sqrt 3}[/math] for each additional member. Tooooo sexxxxy!! :hihi:

 

The kink-free business from a flat non-rectangular shape sounds like a subject for a thread on topology in general. A Klein bottle perhaps? At any rate, not a Möbius strip, so back to them.

 

Having covered the surface itself a bit, I have in mind to poke about in the hole a Möbius strip bounds. What shapes can the hole assume? Can you pass one Möbius strip through the hole of one equally proportioned? For the same question, does it matter if the twin is of opposite hand? What are the limits for passing Möbius loops of a single family through one another? Can they get stuck together? :cup: :turtle:

 

Then we have a recent report here on a strange hexagon at Saturn's North pole >> http://hypography.com/forums/space/1...Saturn+hexagon

 

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

 

Better call NASA, Turtle accidentally did their job again..

 

:hihi: Only one caveat; I did it on purpose. It's what we generalists do. :) To quote Roger Thelonious George, "chaos favors the prepared imagination." :cup: :turtle:

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Now for my wild speculation>>> :hihi: The Wicki article I linked to says this at the bottom:
...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)...

 

I tried to find this article, but it appears the IEEE archive is for paying folk only. :turtle:

 

A device called a Möbius resistor is an electronic circuit element which has the property of cancelling its own inductive reactance. Nikola Tesla patented similar technology in the early 1900s: U.S. Patent 512,340 (patent link) "Coil for Electro Magnets" was intended for use with his system of global transmission of electricity without wires.

 

This is an interesting design, but I don't quite yet see why it is Möbius. For some reason I can't get to the drawing again!???

 

I think I'll make some nested strips. :hihi: :hihi: :turtle:

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Then we have a recent report here on a strange hexagon at Saturn's North pole >> http://hypography.com/forums/space/10994-hexagon-saturn.html?highlight=Saturn+hexagon

 

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

 

Quite an interesting proposition Turtle. The only problem I see right away is that the angle from the JPL site photo is not from directly above the north pole. I suppose it doesn't matter if the Mobius belt is in an extremely compressed state (ie occupies a thin layer of atmosphere). The JPL site also states that the hexagon was far lower in the atmosphere than they expected it to be: "The new images taken in thermal-infrared light show the hexagon extends much deeper down into the atmosphere than previously expected, some 100 kilometers (60 miles) below the cloud tops. A system of clouds lies within the hexagon. The clouds appear to be whipping around the hexagon like cars on a racetrack." - JPL. I'm not an astrophysicist, but it seems to me, logically, that a Mobius belt would form high in the atmosphere. Even if you consider the circular motion about the pole and a potential magnetic vortex, the Mobius belt would still have to be isolated in a relatively thin atmospheric layer. I'll admit to not knowing much about magnetic fields, so perhaps I'm overlooking something basic here.

 

How can the "Flattened-Mobius-Hexagon Speculation" explain the photo (and video)?

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When you cut a mobius in half (long ways), you get a sort of double mobius happening - whats more interesting is if you then cut that in half you get 2 double mobius' but linked together!

 

Aye maty; but we'll not be puttin' me family to the shears I'll be tellin' yer. :pirate:

 

Then we have a recent report here on a strange hexagon at Saturn's North pole >> http://hypography.com/forums/space/1...Saturn+hexagon

 

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

 

Quite an interesting proposition Turtle. The only problem I see right away is that the angle from the JPL site photo is not from directly above the north pole. I suppose it doesn't matter if the Mobius belt is in an extremely compressed state (ie occupies a thin layer of atmosphere). The JPL site also states that the hexagon was far lower in the atmosphere than they expected it to be: "The new images taken in thermal-infrared light show the hexagon extends much deeper down into the atmosphere than previously expected, some 100 kilometers (60 miles) below the cloud tops. A system of clouds lies within the hexagon. The clouds appear to be whipping around the hexagon like cars on a racetrack." - JPL. I'm not an astrophysicist, but it seems to me, logically, that a Mobius belt would form high in the atmosphere. Even if you consider the circular motion about the pole and a potential magnetic vortex, the Mobius belt would still have to be isolated in a relatively thin atmospheric layer. I'll admit to not knowing much about magnetic fields, so perhaps I'm overlooking something basic here.

 

How can the "Flattened-Mobius-Hexagon Speculation" explain the photo (and video)?

 

To the pink, I reason that 'they' were surprised because they are not looking for a electro-magnetic Mobius band, but rather thinking in terms of clouds & fluid dynamics.

 

To the bold & the rest, the Saturn Mobius band is thin indeed, but not paper thin according to my new calculations. Lacking a micrometer, I measured a stack of 38 sheets of my 32 lb paper at 5/16" and calculated a mean thickness of .016447368". Then, I calculated the maximum diameter of the 3[math]{sqrt 3}[/math] wide x 16/3/4 long hexagon forming Mobius strip when it is completely flattened at 7.4". Next, I took the ratio of 2 thickness' to diameter and found it .0022222617.

 

Now taking the ratio of thickness to diameter from Saturn's hexagon Mobius band we have 100km thick by 25,000km diameter, for a ratio of .004. This is twice that of the paper.

 

Then again, as you point out about the perspective, we can't tell if the hexagon is curved or flat. :( I don't know enough about magnetic fields either to say exactly how this works, and sorry to say it seems we gotta pay for that knowledge. :( :pirate: I say, we storm the archive & take some booty. Ar ye wi' me mates!!?? :turtle:

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