master q

[Lollypop 0]

Seeing as you are the brains of the board, I have a question

 

Question..... How does a Möbius Strip work ? :)

[master_q 0]

Hi

 

Well I’m flattered that you think that, but I’m not sure if that’s completely true . . . but thanks. :lol:

 

To try to answer your question . . .

 

I don’t know how much knowledge you have in topology so I don’t know how much info to say or give. I have to say that I’m no expert in topology or knot theory, but I do know a certain degree of the fundamentals.

 

If you want to know of the basic concepts, then I guess I’ll start there and move forth. In any topic relating to mathematics or science I believe the most important thing to understand is the concept of the topic in question.

 

Here’s a good visual:

Get out a strip of paper and give the paper a single half-twist, then tape or glue the two ends together. Simple as that! Probably the biggest novice misconnection of the info is that there are two sides. In fact we have one side and not two. For example if you drew a line in the middle of the band the line would end up where it began (going around in a kind of loop). One of the things you have to think about is that this object is two dimensional. However, we can bend that band and twist it up, but that is just the result from it being in three dimensional space (we could imagine putting it in higher dimensions too). Now you might be asking “How is this possible? Can you prove it?” - the best way I think someone could demonstrate that idea is to look at the concept. Take two markers and try to color it with two different colors where only one color could be on a side. You would then find out that you can’t color it with 2 colors because there is only one side. (Well you could mix them on one side, but that’s something completely different)

 

In fact when I think of topology I don’t tend to think of sides and for that matter I don’t really think anyone does. It is true to a point, but understanding the concept of something in a higher dimensional space you must think of something bent or twisted. That bending or that twisting makes it seem to have a higher # of sides then it really does, but in fact the sides go according to the space. Space and distort. (Just like in general relativity)

 

You could even go into more interesting things. Like if you took your Möbius band and then labeled it somehow . . . say an arrow. If you put the arrow before you actually twisted and then taped, then you could have all arrows point up. When you actually bend it and connect the two ends you can see how the arrows change direction!

 

I hope that helps a bit to give you a general understanding of what it is and how it works.

 

 

Master Q

StarTrek_Master_Q@yahoo.com

Edited June 7, 2003 by master_q

[nik 0]

Lollipop, try this. Take a mobius strip, draw a line down the center as master_q suggested, and then take a pair of scissors and cut along this line. You may be fascinated by the result, but it may also give insight into how it "works," a result which

can be predicted by differential geometry (or manifold geometry, just jargon really). Also, try this, instead of a single half twist, give your strip two half twists (or one full twist) before taping the ends together. Then cut along the center as before. The result (a "complex" form in differential geometry) also has special meaning in group theory and particle physics.

Finally, make another loop, but give it three half twists before taping the ends together. Then cut it down the middle as before. Impressive, no?

[Lollypop 0]

I think I understand what you are getting at master_q :lol:

[Lollypop 0]

Lollipop, try this.  Take a mobius strip, draw a line down the center as master_q suggested, and then take a pair of scissors and cut along this line.  You may be fascinated by the result, but it may also give insight into how it "works," a result which

can be predicted by differential geometry (or manifold geometry, just jargon really).  Also, try this, instead of a single half twist, give your strip two half twists (or one full twist) before taping the ends together.  Then cut along the center as before. The result (a "complex" form in differential geometry) also has special meaning in group theory and particle physics. 

Finally, make another loop, but give it three half twists before taping the ends together.  Then cut it down the middle as before.  Impressive, no?

I have been playing with these for years. :lol:

[master_q 0]

I’ll expand a bit more on the band. I think it’s pretty interesting stuff.

{Also I would like to give my apologies for all of those mistakes in my above post}

 

We can actually connect this to the idea that topological properties of surfaces can directly connect with the idea of “two-sides” (like the cylindrical band - that’s the band before we twisted it to become a Möbius one) or if it is “one-sided” (the Möbius band) . . . .

 

Draw arrowed circles on the cylindrical band and then turn that band into a Möbius one. You could also imagine our circles on the band better if you could imagine the band as transparent!

 

If an object is called orientable, then the idea of clockwise and/or counterclockwise can be distinguished. If I drew a circle and gave the circle some kind of direction . . . . like if I drew a circle and had an arrow going counterclockwise. If it is not possible for us to turn a counterclockwise arrowed circle into a clockwise arrowed circle by transporting the circle over the surface then it would be orientable.

 

I’m going to see if I can find a picture on the internet (I’ll post it once I find one)

 

 

Master Q

StarTrek_Master_Q@yahoo.com

[master_q 0]

 

 

Master Q

StarTrek_Master_Q@yahoo.com

[Lollypop 0]

Another thing I like to do is .... 1/ Take 2 cutlery forks. 2/ Interlock the prongs. 3/ Take a match and insert it into the prongs. 4/ Then finally get a tall glass, and balance the match tip onto the glass...and behold it balances the forks. It always suprises me how a small match can hold the weight of the forks. And another thing that surprises me ... How does it stay there in the first place ?

[Lollypop 0]

Another thing I like to do is .... 1/ Take 2 cutlery forks. 2/ Interlock the prongs. 3/ Take a match and insert it into the prongs. 4/ Then finally get a tall glass, and balance the match tip onto the glass...and behold it balances the forks. It always suprises me how a small match can hold the weight of the forks. And another thing that surprises me ... How does it stay there in the first place ?

Master_q you haven't answer this question for me. :)

[master_q 0]

Another thing I like to do is .... 1/ Take 2 cutlery forks. 2/ Interlock the prongs. 3/ Take a match and insert it into the prongs. 4/ Then finally get a tall glass, and balance the match tip onto the glass...and behold it balances the forks. It always suprises me how a small match can hold the weight of the forks. And another thing that surprises me ... How does it stay there in the first place ?

Master_q you haven't answer this question for me. :)

I once saw a TV program show a small swing only made up of Popsicle sticks and that swing held up once someone sat on it. It was neat to see that something made of Popsicle sticks could do this. If you know how physics works and you understand engineering you could then make something like this.

 

In any case like this it all comes down to forces. Your question could be interpreted as “Why don’t we have a negative net force that makes the forks & match fall down due to the force of gravity?”

 

Yes, that might be a more complicated way to say it, but it does come down to that.

The object is not moving and this tells us a few things.

 

For one, it tells us that it is in a state of equilibrium. This plainly or simply means that it is not moving. However, there is more then it to that. One force we will always have to deal with is gravity. Gravity is pulling you down and always will. If I am sitting in my chair, then I’m in a state of equilibrium because I’m not moving. This is the result from the fact that I have a net force of zero due to the negative force of gravity pulling me down and a positive force called force normal pushing me up (as in the chair). This creates a sense of balance.

 

The example of me sitting on the chair might be different (LOL; this example makes me feel like I need to take a walk or something), but it shows the same thing in general.... So now we know that there must be a net force of zero in our setup.

 

So we know that (and that’s an important factor to figure out), now we can try to figure out “why.” I wish I had a force diagram I could show you - that would be a lot easier and you could make sure I fully understand the setup of what is happening. From what I see it seems like one of the biggest forces that are at work here are contact forces and that this force(s) is helping to a large degree produce a net fore of zero. I think that it is apparent that one fork is “pushing” on the other, but this force is still not strong enough so we have a match and that force (even though small) helps makes the new force zero. Another variable is the force normal. This also, from what I see in your description would play a very large role. So we have the force of contact from both directions {as in the forks} (which kind of cross cancels, but that’s another story)/ the force of contact from the match (a small force)/ the force normal (from the angle, theta, that the forks have against the force normal is a very big variable!)/ and then of course we have the force of gravity.

 

 

I hope that answered your question.

 

 

Master Q

StarTrek_Master_Q@yahoo.com

[Lollypop 0]

Hmmm !... I'm thinking about this