Topology is a fascinating area of mathematics where you bend your mind around the curves of reality.



For instance, try imagining the
Klein Bottle, a topological structure
with no edges, where the inside is
the same as the outside.

Mind boggling, isnít it!

To explore the topological world take two strips of paper (30cm x 5cm).
With one of the strips, make a Moebius band by
half twisting the paper and then joining the two ends together. Then take a pen and draw a line around the middle of the Moebius band. Keep drawing until you get back to where you started.
You will notice that you have drawn one continuous line (you did not lift your pen) which covers both the front and the back of the paper. As you can see it is impossible to distinguish between the inside and outside of the band. So we say the Moebius band is a surface with only one side. Take a piece of coloured chalk and run it around the edge of the Moebius band. Once again you will notice that you have travelled around the entire edge of the paper without lifting the chalk.
So the Moebius band has only one edge.
With the second piece of paper, join the ends to make a simple band. Observe there is no way to draw a continuous line, which covers the front and
the back of the paper, and so for a simple band you can distinguish between the inside and the outside. The simple band is a surface which has two sides.
If you run your chalk around the edge of the simple band you will only colour one edge. To colour the other edge, you must lift the chalk and start again. So a simple band has two edges.

The difference between the two surfaces is that the Moebius band has one side and one edge and the simple band has two sides and two edges.

There are many other two sided, two edged surfaces. You can also have surfaces with two sides and one edge and surfaces with one side and two edges.

Take some paper and try constructing these surfaces.

(Note you may need to cut the paper, twist it and stick the ends back together.)

Now explore these surfaces, remembering that a surface has one side if without lifting your pen, you can draw a continuous line, which meets both the front and back of the paper. A surface has one edge if with one continuous movement (without lifting the chalk and starting again) you can traverse all edges of the surface. Create some new surfaces for yourself.