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bossk171

Member

Registered: 2007-07-16

Posts: 305

Mobius Strip: 2D or 3D?

Well my question is pretty well stated in the subject line.

The argument for it being 2D is that it is a strip, and has no depth, but that doesn't seem right, you can't represent it two dimensionally, you need three dimensions. So what's the deal? 2D or 3D?

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Mobius Strip: 2D or 3D?

Two-and-a-half dimensional?

Kargoneth

Member

Registered: 2007-08-11

Posts: 33

Re: Mobius Strip: 2D or 3D?

In my opinion it is a three dimensional object

Last edited by Kargoneth (2007-08-21 10:44:33)

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: Mobius Strip: 2D or 3D?

My vote: 2D

Mental exercise: Is the surface of a sphere 2D or 3D?

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Mobius Strip: 2D or 3D?

You "draw" it in 3D, but it is a 2D manifold.  Similarly, the surface of a sphere is also a 2D manifold.  Think about it this way, if you were standing on it, you could go left, right, forward, or backward.  But you can't go up or down.  On the other hand, a sphere (not just the surface) is a 3D manifold because you can go up and down.  A line graphed on the complex plain is a 1D manifold in 2D.  A line on R^5 is a 1D manifold in 5D.

I guess the easiest way to comprehend it is when we talk about "manifolds", we are talking about where we can move inside the object itself.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: Mobius Strip: 2D or 3D?

When we discuss a sphere, are we discussing a hollow ball with infinitely thin wall that fits the equation x² + y² + z² = r² or is this some other definition that topology uses?

If it's the former,then a sphere must also be a 2D surface, as is a Klein bottle, no?

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Mobius Strip: 2D or 3D?

What about if you just look at the two lines on the left and right edges of the mobius strip?
And then decide if you can make the surface by connecting the two edges with line segments that are perfectly straight, or are they curved?  I have no idea.
Also look for some good definitions of 2D that includes or discludes unflat surfaces.
Some words are not completely defined anywhere.
It's sort of a mishmash out there.

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igloo myrtilles fourmis

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Mobius Strip: 2D or 3D?

When we discuss a sphere, are we discussing a hollow ball with infinitely thin wall that fits the equation x² + y² + z² = r² or is this some other definition that topology uses?

Your right, improper terminology on my part.  x² + y² + z² = r² is a sphere, x² + y² + z² <= r² is a ball.  Balls are 3D and spheres are 2D topologically.

If it's the former,then a sphere must also be a 2D surface, as is a Klein bottle, no?

A Klein bottle is a 2D topological space, but it is not simply connected and thus, not homeomorphic (equivalent) to a sphere.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

LQ

Real Member

Registered: 2006-12-04

Posts: 1,285

Re: Mobius Strip: 2D or 3D?

Yes, a sphere with two dimensions, it bends over the nothingness, see?

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I see clearly now, the universe have the black dots, Thus I am on my way of inventing this remedy...

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: Mobius Strip: 2D or 3D?

A 2D sphere doesn't make much sense to me, if it were 2D, why do we need 3D to graph it (x,y,z)?

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Mobius Strip: 2D or 3D?

A 2D sphere doesn't make much sense to me, if it were 2D, why do we need 3D to graph it (x,y,z)?

The surface of the sphere is 2D. Yes, 2D surfaces embedded in ℝ[sup]3[/sup] need to be described in 3D.

After all, the straight line y =  mx + c has only one dimension, but needs to be graphed in two dimensions (in x and y) – because it’s embedded in ℝ[sup]2[/sup]. Similarly two-dimesional surfaces embedded in ℝ[sup]3[/sup] need to be described in three dimensions (x, y and z). For example, the general equation of a (two-dimensional) plane is ax + by + cz = 0 (where a, b, c are not all 0).

In general, an n-dimensional manifold embedded in ℝ[sup]n+1[/sup] needs to be described in n+1 co-ordinates.

Last edited by JaneFairfax (2007-08-22 02:40:44)

LQ

Real Member

Registered: 2006-12-04

Posts: 1,285

Re: Mobius Strip: 2D or 3D?

But do not forget: a sphere cannot exist without a ball. Or can it O-o

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I see clearly now, the universe have the black dots, Thus I am on my way of inventing this remedy...

JimJam

Guest

Re: Mobius Strip: 2D or 3D?

If you are on the outside of the sphere then I can see how this would exist in 2d, but if you were on the inside then your back to 3d.   You could leap from one side to the other via the space inbetween.

????? This stuff makes my head hurt !

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Mobius Strip: 2D or 3D?

You could leap from one side to the other via the space inbetween.

Ah, but there is the kicker.  There is no space in between.  When we talk about surfaces, the surface defines the space.  So for example, consider all the real numbers in 3D, this would be what we think of when we think of space in the universe.  It is a 3D surface.  So if you have just the shell of a ball, then there is no space on the "inside".

Actually, a big thing now is to settle the topology of the universe.  Maybe it is Euclidean (how we would normally imagine it), or perhaps if you move "straight" in one direction you could end up back at the same point, much like the surface of a sphere.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

the_max7592

Member

Registered: 2007-08-30

Posts: 5

Re: Mobius Strip: 2D or 3D?

it's obviously 3D, otherwise it would be a piece of paper of something (like the mobious strip picture) and it would be 2D.

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: Mobius Strip: 2D or 3D?

You mean the mobius strip, the_max? It is a surface, and thus 2D. There is no 3rd dimension when navigating around it.

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman