Take a long strip of paper,and glue the ends together, but with a twist through 180 degrees.The result might be something like this.
It is a figure, a surface, with only one edge and only one side.
There are some nice Experiments YOU can do!
There is another way of thinking of the structure of the Möbius Band, which corresponds to the way we made it from a strip. The Band has a middle circle, which goes round the Band only once. Notice that there are other circles, seemingly parallel to the middle one, but which go round the Band twice. Now draw lines on the Band at right angles to the middle circle. For each point of the middle circle we have a line, and as this line moves around the middle circle, it twists. This gives a mathematical model of the Möbius Band which we can realise in a picture. Here are four views of the Möbius Band.
and a 3D rotating picture (90Kb) :
There is information explaining how we made the pictures,notes for doing the same with some of John Robinson's sculptures, and examples, such as a 3D rotating DEPENDENT BEINGS
We have also found another interesting picture of the Möbius Band, and it is in the Geometry Center Graphics Archive (outside link).
Here is another experiment, either in practice or a thought experiment. Make a Möbius Band out of cloth, and make a disc of cloth whose edge is the same length as the edge of your Möbius Band. Now try and sew the two together along their edges. What happens?
What you are trying to make is called a Projective Plane
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