Moebius Strips: How Many Sides and Surfaces?

Date: 10/18/2001 at 17:53:13
From: Melissa Bercume
Subject: Moebius strips

I have looked at almost every Web site there is, and I still cannot 
find the difference between sides and surfaces. Can you help? Thank 
you so much.

Melissa

Date: 10/19/2001 at 17:00:30
From: Doctor Peterson
Subject: Re: Moebius strips

Hi, Melissa.

In simple terms, a surface is a set of points, while a side adds an 
orientation to those points. We look not only at where we are, but at 
which way we are facing.

Take a sphere. To show that it is a single surface, imagine making one 
point glow blue, and let it wander all over the sphere. You will see 
that it can move freely anywhere on the sphere, so the whole sphere is 
one contiguous surface:

                     ooooooooooo
                ooooo           ooooo
            oooo                     oooo
          oo                             oo
         o                                 o
       oo                             *     oo
      o                                       o
     o                                         o
     o                                         o
    o                                           o
    o                                           o
    o                                           o
     o                                         o
     o                                         o
      o                                       o
       oo                                   oo
         o                                 o
          oo                             oo
            oooo                     oooo
                ooooo           ooooo
                     ooooooooooo


To see how many sides this surface has, imagine replacing that point 
with something like a tack; its flat head rests on the surface with 
the point sticking up:

                     ooooooooooo
                ooooo           ooooo
            oooo                     oooo/
          oo                        ..../oo
         o                         .   / . o
       oo                          .  *  .  oo
      o                             .....     o
     o                                         o
     o                                         o
    o                                           o
    o                                           o
    o                                           o
     o                                         o
     o                                         o
      o                                       o
       oo                                   oo
         o                                 o
          oo                             oo
            oooo                     oooo
                ooooo           ooooo
                     ooooooooooo

Now let this wander around, and you will see that, although it will be 
able to get to any spot on the surface, the "point" of the tack will 
always be pointing the same direction whenever it gets to that point. 
You would need to put a red tack on the inside of the sphere in order 
to point the other way. There are two distinct sides to the sphere.

But on a Moebius strip, you will find that the single blue tack will 
not only be able to go everywhere on the surface, it will be able to 
point in both directions at every point, just by sliding around. You 
won't need that second red tack. So this surface is one-sided.

Now notice that you don't really have to have the point on the tack. 
Just slide around a circle with an arrow going clockwise around it. On 
a Moebius strip, you will be able to slide that circle around and back 
to the same location, but with the arrow going the other way! This 
allows us to define orientation without reference to the space outside 
the surface.

           ---->----               ----<----
         /           \           /           \ 
       /               \       /               \
      |                 |     |                 |
     |                   |   |                   |
     |        up         |   |       down        |
     |                   |   |                   |
      |                 |     |                 |
       \               /       \               /
         \           /           \           /
           ---------               ---------

You are asking for a "technical" definition of a "popular" concept, so 
if you're looking for a technical definition, this won't satisfy you; 
but it gives the general idea.

Mathematicians don't need to define "side," because we don't call 
surfaces like the Moebius strip "one-sided." We call them 
"non-orientable"; that is, you can't distinguish two orientations. 
What I gave you is a definition of orientability - still pictorial, 
but closely related to what we would say in an introductory textbook. 
So don't worry if you can't say what a "side" is; I've told you what 
"one-sided" means.

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