They still point in, out, up and down.
Extraordinary claims require extraordinary proof
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MattusMaximus
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Beats the hell out of me. I once heard someone put it well... "dark energy" is basically a place-holding moniker that we give to whatever the hell it is that is accelerating the expansion of the universe. Maybe when the Large Hadron Collider comes online we'll get lucky and start figuring out what these weird things like dark matter & dark energy actually are...
Here's a link on this:
http://en.wikipedia.org/wiki/Dark_matter#Detection_of_Dark_Matter
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MattusMaximus
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I'd like to thank sol invictus, Reality Check and MattusMaximus for a very informative thread. I think there were some other people in the thread, but I forget.
You're quite welcome
Reality Check
Penultimate Amazing
But they do not occupy space.
sol invictus
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Math doesn't ignore that, it just asks about something else - namely the 2d version. IF you use the term "Mobius strip" to refer to a 3d object, nobody else will understand you.
m_huber
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They point in all directions on the same surface. This is equivalent to your table top facing both your ceiling and floor at the same time.
Because two surfaces are twisted 180 degrees (360 total).
This is the equivalent of woo talk. A table top is one flat surface, a mobius band is two surfaces twisted 180 degrees with their ends joined so that one edge meets the other. If the table top had rounded edges so that no corners divided it from the underside, then the top, edges and bottom could all be described just one surface called the top. This is essentially what is happening with the mobius band.
Reality Check
Penultimate Amazing
You are incorrect. If you construct a model of the mobius band using a strip of paper then the paper has 2 surfaces. After you have constructed the mobius band there is 1 surface. This is easily seen by drawing a line around the mobius band and noting that it traverses all of the surface.
Of course according to your previous posts the strip of paper has 3 surfaces (top, bottom and side).
The “2D version” is only an abstract part of the 3D reality. What‘s the point of only asking about an abstract part of an object? Why not the whole object? It’s like a flea at the center of a black spot on a white dog concluding that the dog is black.
When the ends of a strip are joined it becomes a band. Both the strip and the band are 3D. Can anyone explain why a 3D band is called a 2D strip?
Not that it is, but why it is.
sol invictus
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No, not at all. The surface of the table is orientable. The Mobius strip is not.
If you're confused by the fact that the Mobius strip has edges, think about a Klein bottle instead. You get that by starting with a Mobius strip and gluing the edges together. In 3d that requires the surface to intersect itself, but ignore that for the moment. Here's a picture of it:
http://www.ima.umn.edu/2005-2006/gallery/polthier/kleinBottleNormalShowStill_med.jpg
That surface is not orientable, unlike the surface of a donut or sphere (or table).
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What defines a surface? Essentially all objects only have one exterior 3D surface. This surface can be abstractly divided in to multiple surfaces by corners. A corner doesn’t stop a surface, it merely abstractly divides it. What defines a corner? At a microscopic level, when does a corner become a curve, and visa-versa? If an object has flat areas joined by curves, are those flat areas surfaces? Is a strip with rounded edges one or two surfaces? If a band is made with rounded edges, is it one surface?
A strip of paper has one exterior surface that can be abstractly divided by corners in to six surfaces. The two larger opposed surfaces can be called top and bottom, the two longer smaller surfaces can be called edges, and the two shorter smaller surfaces can be called ends. The edges and ends separate the top from the bottom Are you saying that an edge isn’t a surface? Small doesn’t mean non-existent.
Perhaps you need a reality check?
Reality Check
Penultimate Amazing
You need to cite your proof that reality is 3D.
There is no "3D band". There is the physical model of the mobius strip that happens to exist in our reality which you keep on insisting is 3D. The mobius strip itself is 2D.
m_huber
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The model also exists in time, so it is at least 4D.
You seem to be playing a game of definitions. I would guess that you never had a math class to cover this.
If you want to demonstrate that a Mobius Strip has only one side, make the paper model, then try to color one side red and another side blue. You will find that there is no way to do this. Thus, the strip has only one side.
Reality Check
Penultimate Amazing
What defines a surface? Essentially all objects only have one exterior 3D surface. This surface can be abstractly divided in to multiple surfaces by corners. A corner doesn’t stop a surface, it merely abstractly divides it. What defines a corner? At a microscopic level, when does a corner become a curve, and visa-versa? If an object has flat areas joined by curves, are those flat areas surfaces? Is a strip with rounded edges one or two surfaces? If a band is made with rounded edges, is it one surface?
A strip of paper has one exterior surface that can be abstractly divided by corners in to six surfaces. The two larger opposed surfaces can be called top and bottom, the two longer smaller surfaces can be called edges, and the two shorter smaller surfaces can be called ends. The edges and ends separate the top from the bottom Are you saying that an edge isn’t a surface? Small doesn’t mean non-existent.
Perhaps you need a reality check?
I am quite perturbed that our usually excellent NZ educational system has let you down.
One more time:
The physical model of a mathematical model is not the actual mathematical model.
In mathematics you can do stuff like define a point as a location in space that has no extent, i.e. no volume, area or length.
The physical model of a point is a dot, e.g. on paper, that has extent.
The edge that you talk about is an aspect of the physical model (a strip of paper). It does not exist in the mathematical model.
sol invictus
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It's useful to study abstract math because it provides very powerful techniques to model the real world. You could just as well say, what's the point of asking about 2+2? Why not just take two eggs and put them with another two eggs and see that you have 4? But if we followed that logic we'd still be in the stone age.
Math is a set of abstractions - it doesn't always have to represent something tangible. The Mobius strip is by definition a 2d surface. When you take a piece of (3d) paper and twist it, that's not actually a Mobius strip - it's just a model of one.
The distance around an object can’t be more than 360 degrees. I colour the surface one colour for 360 degrees, then I go around the object again colouring the rest of the surface another colour.
The effect is the same as if opposite sides of the strip were different colours before they were joined.
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Reality Check
Penultimate Amazing
The effect is not the same - in the second case you end up with 1 surface with 2 colors.
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m_huber
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If you take a strip of paper and color two sides different colors, then make it a Mobius Strip model, you will have a color that continues to the line where you rejoined the paper, then the other color. If you make the model first and ignore the connection line, you will see that you cannot color it in such a way that it has two opposite sides. It isn't hard to try it (and it's kind of fun).