Topology of a mathematical plane

[sydneybself]

Assuming a mathematical plane, does it have a top and a bottom or does defining them make the plane three dimensional?

Example: Given a flat, transparent plastic sheet. One draws a picture on it with a marker. If one turns the sheet over, in other words looking at the bottom of the sheet rather than the top, the picture is the mirror image of what one views from the top.

Can someone explain the topology of the above?

 

[HallsofIvy]

It is not a matter of the plane being three dimensional but in order to "turn it over" or looking at it "from above" or "from below, the plane must exist in three dimensions. You are describing a two dimensional subset of a three dimensional space.

 

[sydneybself]

HallsofIvy:
I'm sure you're correct, but since I made the original post I've come up with an alternate explanation. A mathematical object does not, and cannot, possesses its own frame of reference; it has no 'top' or 'bottom', it is a mental construct and it just is. When I try to impose a 'top' or 'bottom' to a mathematical plane, or any other mathematical object, I subjectively, (and probably subconsciously), transpose it into a physical object and then, subjectively, impose my own frame of reference on it. Thus, 'top' and 'bottom' are wherever I say they are. If we both were considering the same object, we could impose entirely different frames of reference on it and we'd both be right.
Do you agree?

Incidentally, I've given some thought to trying to imagine the nature of a three dimensional a Mobius strip; I think it requires a four dimensional physical space. Any comment?

 

[A David]

Defining them doesn't require any higher dimensions. If you have an image in the plane, you could say that your top is the image, and your bottom is what you get when you reverse all of the x-coordinates: a mirror image. It is silly though to call it a "top" or "bottom". You could scramble all the points around if you wanted. A mathematical surface doesn't require any notion of points sticking together like a plastic sheet, and you can permute the points just fine and still have "an image" in the plane.

If you want to talk about rigid motions, or motions where all the points stay "stuck together" with their "neighbors" then of course there are some things you can't do.

A mathematical object can indeed have a frame of reference. If I number the sides of a cube then it matters which one faces up. It doesn't matter whether I call it "1" or "top" or "purple". There isn't necessarily a preferred frame of reference- unless you define one, then there is.

A mobius strip doesn't need to be imbedded in a higher dimension. You can think about it as a type of rotation. Usually when we go around a circle, we arrive where we started after one rotation. On a mobius strip, after one rotation, everything is inside out, and after another rotation we get back to where we started. It's kind of like having to turn 720 degrees to face the direction you started in, instead of the regular 360. spinors have this property, for example, even in three spatial dimensions. Don't bother trying to visualize it. My comment is that it would be helpful to study some fundamentals of analysis and topology and maybe some abstract algebra before you start trying to think about difficult objects within mathematics.

Nothing in mathematics "just is". We establish a set of first principles and deduce true statements or allowable changes from them. A problem with your question is that it doesn't really allow any sort of insightful answer because nothing in it is defined precisely.

 

[sydneybself]

If you take a paper ribbon and connect the edges you get a cylinder. You can then take the ends of the cylinder and connect them to get a torus. This cannot be done with a Mobius strip. How does topology explain this?

 

[A David]

A mobuis band or any topological space that is homeomorphic to a sphere with any number of Mobius bands is called non-orientable. Topological spaces which are not, like a plane, a cylinder, or a torus, are called orientable.

You should really start at the beginning, with a foundational mathematics course that covers first order logic, proofs, and set theory, then work your way up from there.

 

[sydneybself]

Thanks for the info, without taking a number of math courses, the above data is not easy to locate