Möbius strips

[Acme]

If it was a mobius strip, wouldn't if have to be squashed for you to go across? So at the ends you couldn't move across.

Necromancy! I searched the forum for 'Möbius' and this thread was the only one returned. And wonder of wonders you have mentioned just the concept I had in mind, i.e. squashing. Before I launch into squashed space I would just point out that above you say 'at the ends' and a Möbius band has no ends.

 

So a few years back I got to wondering if there was a limit of L to W beyond which I could not join W's to get a Möbius band. I made paper bands as well as sewed some of canvas. Anyway, yes Möbius bands beyond certain ratios must be squishified. At some ratios the band squishes into a hexagon. After seeing it [the hexagon] and having recently read about the mysterious hexagon at Saturn's N pole, I wondered if it might have its origin in a squashed Möbius band of either an electric field or winds.

 

Just before replying here I did a search and found the area of squished Möbius bands has been explored and while some facts have been forthcoming, others remain unsettled which may have a bearing on the space geometry discussed here.

 

Fattest rectangular Möbius strip in 3-space @ Wikipedia

If a smooth Möbius strip in 3-space is a rectangular onethat is, created from identifying two opposite sides of a geometrical rectanglethen it is known to be possible if the aspect ratio of the rectangle is greater than the square root of 3. (Note that it is the shorter sides of the rectangle that are identified to obtain the Möbius strip.) For an aspect ratio less than or equal to the square root of 3, however, a smooth embedding of a rectangular Möbius strip into 3-space may be impossible.

 

As the aspect ratio approaches the limiting ratio of \sqrt{3} from above, any such rectangular Möbius strip in 3-space seems to approach a shape that in the limit can be thought of as a strip of three equilateral triangles, folded on top of one another so that they occupy just one equilateral triangle in 3-space.

 

If the Möbius strip in 3-space is only once continuously differentiable (in symbols: C1), however, then the theorem of Nash-Kuiper shows that there is no lower bound.

 

A method of making a Möbius strip from a rectangular strip too wide to be simply twisted and joined (e.g., a rectangle only 1 unit long and 1 unit wide) is to first fold the wide direction back and forth using an even number of foldsan accordion foldso that the folded strip becomes narrow enough that it can be twisted and joined, much as a single long-enough strip can be joined.[5] With two folds, for example, a 1×1 strip would become a 1x⅓ folded strip whose cross section is in the shape of an N and which would remain an N after a half-twist. This folded strip, three times as long as it is wide, would be long enough to then join at the ends. This method works in principle but becomes impractical after sufficiently many folds, if paper is used. Using normal paper, this construction can be folded flat, with all the layers of the paper in a single plane. But mathematically, it is not clear whether this is possible without stretching the surface of the rectangle.[6] ...

Möbius Hexagon

 

Möbius Band Ratio 10/8

 

Möbius Band Ratio 10/8 Unfolded

Edited June 22, 2014 by Acme

[Mordred]

WHAT IF ????

 

What if science has gotten it all wrong ? What if science and the principles it is based on are all wrong ? What if there is another possibility that the science community has not grasped ? A collection of good questions, but how do we answer them ? Simple, I just tell you what I think, it will probably not make alot of sense because it comes from my mind and not yours, or it may make alot of sense because you think in a similar fashion to me.

 

you can see how the gravity of our Earth would cause an expansion of space on the other side of the loop, but because the loop is mobius we would see the expansion of space occurring at a point some distance from us rather than under us.

This not only could explain the expansion of the universe but it also would allow for the exceptance of worm-holes, these would be then a slight tear in the fabric of space time allowing us to travel to the other side of the loop, it would appear that we have traveled a huge distance but in reality we have only traveled a short distance but the downside to this would be that worm-holes would only have a limited use.

 

Another little question bothers me, what if the basis for our mathematical formulae are all wrong? what if instead of two equations having to yield the same result to prove a theory we have the same value yielded but as a positive in the first instance and a negative in the second instance. This would allow us to apply maths as it is in nature, opposites attract.

 

Thanks for reading the ramblings of a madman

 

 

In terms of expansion localized gravitational influences is negligible. Expansion is a global influence so your Earth example above wouldn't really work. To explain that further you need to realize that our measurements of expansion is uniform (defined by the terms homogeneous and isotropic)

Homogeneous- no preferred location

isotropic -no preferred direction.

now think of those terms in regards of expansion being the influence from every gravitational body. wouldn't be too uniform.

The FLRW metric is an equivalence to the Einstein field equations, however both include correlations to the ideal gas laws. The universe dynamics can be described as an ideal gas or perfect fluid. (provides a very good approximation)

Essentially radiation(relativistic,non relativistic),matter (baryonic,nonbaryonic-dark matter), the cosmological constant (dark energy possibly) each have positive energy-densities however they contribute to the pressure in different relations defined by their equations of state.

http://en.wikipedia.org/wiki/Equation_of_state_%28cosmology%29

here is an article that covers the fluid equations and how its derived from the EFE

http://www.helsinki.fi/~hkurkisu/cosmology/Cosmo4.pdf

the resulting rate of expansion is defined by the acceleration equation (see link)

Universe geometry is also defined by the above relations as compared to the calculated critical density. If the total actual density equals the critical density then the universe is flat.

see the following for how universe geometry affects light paths and distance measurements

 

http://cosmology101.wikidot.com/universe-geometry

page 2 covers the distance measures

http://cosmology101.wikidot.com/geometry-flrw-metric/

 

the links in my signature has more articles on these aspects as well as others.

one of my favorites covering above in regards to GR is by Mathius Blau, though its extremely long and technical

http://www.blau.itp.unibe.ch/newlecturesGR.pdf

(its an excellent reference particularly for the relativity forum lol)

section F covers cosmology

Edited June 22, 2014 by Mordred

[Acme]

 

yada, yada, yada...

In terms of expansion localized gravitational influences is negligible. Expansion is a global influence so your Earth example above wouldn't really work. snip...

 

Just a thought to note that skartag is long gone, i.e. no longer a member for whatever reason. I only necromanced the thread to comment on the invocation of Möbius bands. Perhaps I should have started a new Möbius thread or maybe the staff will see fit to split off a new one from this.

[Mordred]

oh lol didn't notice that

Edited June 22, 2014 by Mordred

[imatfaal]

Acme - as this threads is so old I am not sure it can be hijacked - here is another variant on the standard mobius. A prismatic ring - ie a mobius with thickness and an interior and exterior. I always like it a lot - took me quite a while to work out what the set looked like - but as soon as I had it mathematically sorted on paper it wound up together very neatly.

 

 

 

 

[Acme]

Acme - as this threads is so old I am not sure it can be hijacked - here is another variant on the standard mobius. A prismatic ring - ie a mobius with thickness and an interior and exterior. I always like it a lot - took me quite a while to work out what the set looked like - but as soon as I had it mathematically sorted on paper it wound up together very neatly.

Prismatic Ring.jpg

Simply stunning! Perhaps ajb will stumble over it and opine.

 

The mutually exclusivity of interiority/exteriority reminds me of a flat-folding honeycomb structure I ran across in (I think) The Penguin Book of Strange and Curious Geometry. While the drawings were sufficient to get across the separation, I had to build it to grokk the flat-folding.

 

Might as well post my last Möbius pic.

[hypervalent_iodine]

!

Moderator Note

Just FTR, these posts are probably better suited to their own thread and so I have moved them accordingly.

[Acme]

Thanks for the move HI. While I did make one speculation about the hexagon on Saturn, Möbius bands and their study are anything but speculative and this thread properly belong in the math section.

 

Just so, to refresh my memory and get all hands-onny I performed the ol' cut the Möbius band in half exercise last night. The cut is of course parallel to the edge. The result -to me at least- is anything but intuitive.

 

I'll write up my experiments and put the results in spoilers so folks can test their intuition, and then I'll pose a new question. Please excuse and/or correct any improper terminology I employ. Where I say band it should be understood that I refer to a closed loop.

 

Möbius band [1/2 twist] cut in half yields

1 band with 2 twists.

 

 

Band with 1 twist cut in half yields

2 intertwined [knotted?] bands each with 2 twists

 

 

Band with 1 1/2 twists cut in half yields

1 band with an overhand knot and 4 twists

 

 

Band with 2 twists cut in half yields

2 bands interlaced, each band with 1 twist

 

 

So the question is, if imatfaal were to cut his prismatic ring in half, what would it yield? I was thinking to reproduce a printout of his planar net and build the structure and then cut it in half, but in the mean time I have convinced myself of the result. Here's my guess in spoilaform.

I think the prismatic ring is equivalent to a 1 1/2 twist band and so cut in half would yield 1 band with an overhand knot and 4 twists. ??

 

 

imatfaal's prismatic ring:

Edited June 24, 2014 by Acme

[imatfaal]

I might have to dig out the sets to try it.

 

But to get rules straight - where do I cut? A single cut will just undo the winding

[Acme]

I might have to dig out the sets to try it.

 

But to get rules straight - where do I cut? A single cut will just undo the winding

I prepared a video. [and damned if I can figure out how to get the player to display! ]

 

Video of loop cuttings: >> https://flic.kr/p/o6QN96

 

Edit: video preview

Edited June 25, 2014 by Acme

[Acme]

I might have to dig out the sets to try it.

 

But to get rules straight - where do I cut? A single cut will just undo the winding

Anything yet imat? I did some reading and as your structure has an inside and outside surface and seemingly 1 boundary I'm not sure it is properly called 'Möbius'. I'm sure ajb could expound on the correct topological nomenclature.

 

I also found the general rules for the results of making the cut on bands of different number of twists.

 

Mobius band @ Wiki

 

...Other analogous strips can be obtained by similarly joining strips with two or more half-twists in them instead of one. For example, a strip with three half-twists, when divided lengthwise, becomes a strip tied in a trefoil knot. (If this knot is unravelled, the strip is made with eight half-twists in addition to an overhand knot.) A strip with N half-twists, when bisected, becomes a strip with N + 1 full twists. Giving it extra twists and reconnecting the ends produces figures called paradromic rings.

 

A strip with an odd-number of half-twists, such as the Möbius strip, will have only one surface and one boundary. A strip twisted an even number of times will have two surfaces and two boundaries.

 

If a strip with an odd number of half-twists is cut in half along its length, it will result in a single, longer strip, with twice as many half-twists as in the original plus two more. Alternatively, if a strip with an even number of half-twists is cut in half along its length, it will result in two linked strips, each with the same number of twists as the original.

...

[imatfaal]

Acme - bit too tied up to go seeking my sets at present (and I am sure I remember that it needed to be very accurate - no margin for cludging). And you are correct it is not strictly a mobius; it is a prismatic ring.

====

 

I think youtube is the only player to get embedded, maybe vimeo

 

====

 

I can still remember my brother telling me (as a v young child) how difficult it is to visualise what happens when you slice mobius strips - and then demonstrating - he went on to teach this sort of stuff so I guess it was probably him learning about teaching more than me learning maths!

[tar]

 

Imatfaal,

 

The prism moibus is neat. I realized in recreating it in clay that I have done it before, with strips of clay triangular in cross section, just twisting the one end one side before joining. Just did it with a cross section of a square (four sided flexible beam) as well.

 

In terms of slicing the triangular one in half, I would imagine cutting from the apex to the base would be appropriate but since there is that one side twist built into the figure, there seems to me would be an issue when you got back around to where you started, because the cross section cut you are executing would not line up with the one you began, as it would be 120 degrees off. What to do then?

 

Regards, TAR

You would have to keep cutting, go around again, find you where 240 off, go around again and "maybe" find at the end of your third round that the cuts would line up. Have no physical way to hope to execute that in clay, nor have any clue as to what you would wind up with, intuitive or not.

Acme,

 

Putting the strip and the prism in the same terms, I would say that the moibus strip is equivalent to the four sided beam, rotated two sides before joining. What you are calling a half twist. A full twist would be to turn the beam four sides, joining the original side with itself, or a complete 360 rotation. What you call a half twist would be a 180 degree or two side rotation.

 

In Imatfaal's prism there is just a one side twist, or a 120 degree twist.

 

If the two are to be considered in the same terms, his figure has executed a 1/3 twist, so I don't think the results of cutting it in half would be the same as cutting a figure in half that has undergone 1 and a half twists, as you project. I say this because 120 and 540 are not equivalent and the results of cutting in half, twists of these two different degrees are likely different.

 

Regards TAR

[Acme]

Acme,

 

Putting the strip and the prism in the same terms, I would say that the moibus strip is equivalent to the four sided beam, rotated two sides before joining. What you are calling a half twist. A full twist would be to turn the beam four sides, joining the original side with itself, or a complete 360 rotation. What you call a half twist would be a 180 degree or two side rotation.

 

In Imatfaal's prism there is just a one side twist, or a 120 degree twist.

 

If the two are to be considered in the same terms, his figure has executed a 1/3 twist, so I don't think the results of cutting it in half would be the same as cutting a figure in half that has undergone 1 and a half twists, as you project. I say this because 120 and 540 are not equivalent and the results of cutting in half, twists of these two different degrees are likely different.

 

Regards TAR

The 'planar map' for imatfaal's ring is in post #5. My printer is put away just now but my plan is to print out a copy and try putting it together so I can cut it apart & see the result. For things like this my minds eye is not to be trusted.

 

As to your clay constructions, I'd say they are substantially different than the paper constructions like imatfaal's ring as they have no interior, i.e. they are not hollow. ajb is our topology expert so maybe he will drop in to correct us or you could ask him directly if you like.

[Janus]

Acme - as this threads is so old I am not sure it can be hijacked - here is another variant on the standard mobius. A prismatic ring - ie a mobius with thickness and an interior and exterior. I always like it a lot - took me quite a while to work out what the set looked like - but as soon as I had it mathematically sorted on paper it wound up together very neatly.

Prismatic Ring.jpg

Once I saw this, I had to recreate one with POV-Ray.

 

Here's an animated verison with a "ball bearing" rolling along the surface. The reflection lets you see the bearing when it's on the underside.

 

[tar]

Acme,

 

So when you say cut in half you are wanting to put a rend in the "surface"?

 

Oh, I was trying to imagine the thing as a solid, and split the crossection in half.

 

In the simple moibus strip, you have to physically cut the paper or canvas, substantially cutting two sides at once, the up side and the downside. This made me consider the two sided strip as analogous to a beam, just very thin in one crossectional dimension.

 

So you must define what you mean by cutting in half, so that we can attempt to visualize it.

 

Are we cutting just the surface of Imatfaal's prism, as if it is paper, along the line of Janus' ball bearing, or are we to cut clean through the cross-sectional triangle?

 

Regards, TAR

[Acme]

Acme,

 

So when you say cut in half you are wanting to put a rend in the "surface"?

 

Oh, I was trying to imagine the thing as a solid, and split the crossection in half.

 

In the simple moibus strip, you have to physically cut the paper or canvas, substantially cutting two sides at once, the up side and the downside. This made me consider the two sided strip as analogous to a beam, just very thin in one crossectional dimension.

 

So you must define what you mean by cutting in half, so that we can attempt to visualize it.

 

Are we cutting just the surface of Imatfaal's prism, as if it is paper, along the line of Janus' ball bearing, or are we to cut clean through the cross-sectional triangle?

 

Regards, TAR

Watch this video that I posted in post #10; it illustrates the cuts : >> https://www.flickr.com/photos/114331103@N07/14505951965/

My post #11 has a description of the cuts from the Wiki article on Möbius strips.

 

I got out my printer and printed up several copies of imatfaal's strip. I have 1 cut out & patched and today I will try and tape the little bugger into the prismatic ring. Good times.

 

Janus: Trey kewl POV! Can you model my proposed cutting of the prismatic ring?

[Janus]

Janus: Trey kewl POV! Can you model my proposed cutting of the prismatic ring?

Here's three "views" of the cut.

 

No.1 in left top is with a "cut" down the center of the Side. The sides on either side of the cut are colored red and blue. The sides on are joined at the vertex.

 

If you break the join at the vertex, you can separate the figure into two separate strips, Red and Blue, as shown on the right.

 

If you widen the "cut", but maintain the vertex, You see that the vertex in one continuous loop even after the cut is made.

 

 

Now assume that we add a third side, one that joins the cut edges. So that we once more have a Prism with three sides (red, white and blue.) You then have a red-blue vertex, a red-white vertex and the blue-white vertex, with each vertex forming a continuous but separate loop. (whereas in the original "pre-cut" figure we had just one continuous vertex.

 

If you now break this prism loop at some point, give it a 120 degree twist and rejoin it together, you once again will have one continuous vertex and one continuous edge.

 

Rinse and repeat. (split the new prism along the side, join the edges with a new side, break this prism at some point, twist and rejoin.)

 

Keeping going and you can create an almost fractal like figure with just one continuous side and one continuous vertex.

[Acme]

Here's three "views" of the cut.

No.1 in left top is with a "cut" down the center of the Side. The sides on either side of the cut are colored red and blue. The sides on are joined at the vertex.

If you break the join at the vertex, you can separate the figure into two separate strips, Red and Blue, as shown on the right.

 

If you widen the "cut", but maintain the vertex, You see that the vertex in one continuous loop even after the cut is made.

 

 

Now assume that we add a third side, one that joins the cut edges. So that we once more have a Prism with three sides (red, white and blue.) You then have a red-blue vertex, a red-white vertex and the blue-white vertex, with each vertex forming a continuous but separate loop. (whereas in the original "pre-cut" figure we had just one continuous vertex.

 

If you now break this prism loop at some point, give it a 120 degree twist and rejoin it together, you once again will have one continuous vertex and one continuous edge.

 

Rinse and repeat. (split the new prism along the side, join the edges with a new side, break this prism at some point, twist and rejoin.)

 

Keeping going and you can create an almost fractal like figure with just one continuous side and one continuous vertex.

Ack!! I can't quite get my head around it. Two cups of coffee and I still can't assemble my ring.

 

So on just the first cut, -the one that I wanted- is the result like any of the cuts I did of 'simple' loops in my vid? (For some reason my cut of the double-twist got cut out of the Flickr version of my vid; perhaps a time limit.) Anyway, I guessed that cutting imatfaal's ring as in your #1 would be equivalent to my 1 1/2 twist cut [with the vertex flattend and becoming a surface]; was I right? Help me Obi Wans!

 

Edit: Erhm...given your second cut along the vertex, am I supposed to be using 2 of imatfaal's planar strips to build the prismatic ring? Doble ack ack!! :doh:

Edited July 5, 2014 by Acme

[imatfaal]

I love the animation!

 

Acme - that is the entire set. There is a colour code; ie to tell me where to attach edge to edge. I will have to try and remember it.

===

Acme

 

Sorry - you do realise that it is not a simple set - that is a crossover the strip passes over itself; you cannot make it from a single sheet of paper. I will dig out the actual jpg - or redo the calculations (frankly it was pretty vile)

===

Ok - if you look very carefully you will note that the edge of the set is coloured either red, yellow or blue. You can just about tell that the bottom edge of the left curving section and the right edge of the top right curving section are both blue. That means the blue meets up with the blue and so on. You have to be amazingly accurate with the ends of each section as they show how everything links up.

 

I probably spend as much time folding paper as you do folding numbers - I will make another and then you must tell me where you want it cut.

[Acme]

I love the animation!

 

Acme - that is the entire set. There is a colour code; ie to tell me where to attach edge to edge. I will have to try and remember it.

Thanks. I did see the faint color markings on the edge of your strip on the table-top photo, but I made my printout in grayscale and they didn't reproduce. Even looking at the color pic now I haven't figured out the assembly. (Add to that confusion my discovery a short while ago of a rat nest in the compost pile and chasing babes with a shovel, and it could be a while before I get this settled. )

 

What do you think is the result of the single cut I wanted? (presuming you understand now what I meant.) Was my guess right? Is the result equivalent to any 'simple' loop cut in my vid?

...Acme

 

Sorry - you do realise that it is not a simple set - that is a crossover the strip passes over itself; you cannot make it from a single sheet of paper. I will dig out the actual jpg - or redo the calculations (frankly it was pretty vile)

 

===

Ok - if you look very carefully you will note that the edge of the set is coloured either red, yellow or blue. You can just about tell that the bottom edge of the left curving section and the right edge of the top right curving section are both blue. That means the blue meets up with the blue and so on. You have to be amazingly accurate with the ends of each section as they show how everything links up.

 

I probably spend as much time folding paper as you do folding numbers - I will make another and then you must tell me where you want it cut.

Just caught your edit. Yes, realized the crossover required a filler piece and I got that made & put into the strip. I will have a go at enhancing the colors in your photo and transferring them to my strip.

 

The cut I want is along the surface and parallel to the edge; follow Janus' ball-bearing with your scissor, then flatten the edge/vertex so it is a surface.

[imatfaal]

...

Just caught your edit. Yes, realized the crossover required a filler piece and I got that made & put into the strip. I will have a go at enhancing the colors in your photo and transferring them to my strip.

 

The cut I want is along the surface and parallel to the edge; follow Janus' ball-bearing with your scissor, then flatten the edge/vertex so it is a surface.

 

That cut will just get back to set joined at end - cutting along the surface is topologically equivalent to undoing the edge (remember there is only one edge and one surface).

[Acme]

That cut will just get back to set joined at end - cutting along the surface is topologically equivalent to undoing the edge (remember there is only one edge and one surface).

OK good! Now my question is, with the set joined at the ends but not yet at the edge(s), how many twists in the set, i.e. is the set-joined-at-the ends just a simple loop with no twists, a Möbius band with 1/2 twist, a loop with 1 twist, or what?

 

PS I think it's worthwhile noting that your ring actually has 2 surfaces and 1 edge inasmuch as there is an inside and outside surface. Or am I still missing something?

Edited July 5, 2014 by Acme

[imatfaal]

Sorry - yes. Inside and outside. But as some of the stuff I make isn't sealed so there is a visible inside; as soon as the inside is sealed away, ie if you could make it from a lump of clay and you wouldnt know the difference then I ignore the inside

[Acme]

Sorry - yes. Inside and outside. But as some of the stuff I make isn't sealed so there is a visible inside; as soon as the inside is sealed away, ie if you could make it from a lump of clay and you wouldnt know the difference then I ignore the inside

Roger. But as your ring is hollow, it is not a lump of clay and so it is unique.

 

And back to my other, still burning, question; how many twists in your 'set' when it's simply closed end to end? Also, when you assemble the ring do you join the ends first, or the edges first and then finish with the ends join?

So, see attached images of imatfall's construction. I brightened the 3 colors on the unconnected strip pic where I was pretty certain of them. The green arrows that I added indicate a possible end-to-end join. Keeping these ends flat on the table and sliding them to join so that blue-goes-to-blue, there are no twists in the loop so formed.

 

If a half-twist is made in the strip and the ends joined, the strip is now a Möbius band and there is no blue-to-blue match because 1 blue is up and the other is down. (Note the twist can be right or left and that is important because Möbius bands are chiral, i.e. there is a left-hand and a right-hand version.)

 

If a full twist is made in the strip and the ends joined then you get a blue-to-blue mating. (This may or may not be chiral; it's not clear to me yet.)

 

In case it's not clear I still haven't managed to get the damn thing together. So also on the strip image I added a green circle around an area where red, blue, and yellow end/begin. In the second image of imatfall's completed ring I have circled in green a similar area. Are these one-and-the same area? If so I might manage to start some edge taping there. If not, I'm screwed still.

 

I do hope this post isn't appended to my last; I waited purposefully to try and prevent that. 'Course it's not like things have been going my way today, so I'm not holding my breath.

 

EDit: Crap!! Didn't wait long enough. Foiled again.

Edited July 5, 2014 by Acme