Mobius Strips, Spheres, and Dimensionality

Date: 05/28/2003 at 10:16:58
From: Ruth aged 5
Subject: Mobius Strips

Is a Mobius Strip 2-D or 3-D? Or is it 1-D?

I find it confusing because we start off with a 2-D strip of paper 
with a clear height/width. When we twist and join it, we still have a 
clear width, but we are unsure about length, though you could take 
this as one circuit of the loop.

Depth is very confusing, though, because at one point you have the 
twist, which seems to have no depth at all. And the rest of it does 
make a shape which seems 3-D, but the 'depth' of the object is the 
same bit we measured for the width. Please can you help me? Thanks.

Date: 05/28/2003 at 22:57:41
From: Doctor Peterson
Subject: Re: Mobius Strips

Hi, Ruth.

Actually, the same questions can be raised about a sphere (meaning 
the surface, not the solid object we call a ball). It is only a 
surface, making it 2-dimensional (that is, any point on the surface 
can be located by only 2 dimensions, such as latitude and longitude). 
But it is embedded in 3-dimensional space, so that we can also assign 
each point on the sphere to its (x,y,z) coordinates. The sphere has 
no thickness (ideally, that is - just as your paper really has some 
thickness, but we ignore it for the sake of the mathematical concept), 
so that any small part of it looks like a piece of a plane. Its two-
dimensionality depends on that "local" property rather than on the 
"global" idea of size.

Note that the concept of dimensions doesn't really have anything to do 
with whether the object as a whole has a length, width, and thickness 
or depth, or whether there is a boundary when you go around it. The 
sphere has no edges at all, so that within the sphere itself you could 
only measure how far you go before you loop back on yourself. In the 
three-dimensional space in which it resides, you can describe it by 
only one dimension, its radius. But dimensionality relates to how you 
can specify a location in it, not to the overall size of the object.

Here is another discussion of related ideas:

   Dimensions
   http://mathforum.org/library/drmath/view/55345.html 

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/