We gradually deform the remainder so that it becomes more like a strip. Now we can glue together those parts of this strip which should have have been glued as part of the hemisphere.
So there has to be a twist as we glue, and we get the famous Möbius Band. We have replaced our original picture of a moving pivoted rod in 3-dimensional space by a point (or disc) moving on a space which consists of a Möbius Band and a disc.
The edges of the disc and the Möbius Band are circles, and these circular
edges are to be thought of as glued together. Again, we cannot do this gluing
or sewing in our real three dimensional space, but we illustrate it by showing
our moving disc hopping instantaneously from the edge of the Möbius
Band to the disc, and back again.
This picture of our mathematical space is convenient for representing types of motion of our pivoted rod. The moving disc now leaves a trail, which joins up to form what we call a loop. This loop represents a regular and repetitive motion of the original rod.
Now we can deform the loop to show different motions.
The loop expands off the Möbius Band, onto the disc, and finally down to a point. This point corresponds to a stationary rod.
The movement of this loop shows how our two parts of the space are joined. It also shows us the advantages of the new picture: the changes of types of motion are more easily represented in our new model than in our original picture. Our new model allows for a global picture of particular motions of the rod in 3-dimensional space.
Another important aspect is that the movements of this loop not only describe different motions, but also give us information on the mathematical space which represents the positions of the rod. We are interested not only in movements, but in movements of movements, and so on, and this is an intimation of a higher dimensional theory, of which there is much still to be understood.
Aa part of our sequence of pictures, we have a loop which goes twice round the Möbius Band, and then this loop is deformed off the band and over to the disc, and then to a small loop, and finally to a point, which represents a stationary rod. This is related to the famous Dirac String Trick.
Mathematicians need to progress to the representation of more complicated situations, which arise in applications. The approach through studying `phase space', that is the space representing all possible positions, has proved very useful. Good visualisation of the simple examples is important for using analogy to study the wider examples. We also need some general methods. The notion of fibre bundle has proved a convenient tool for representing many complicated spaces.
The stills on this page are taken from the video " Pivoted lines and the Möbius Band". We would like to thank Len Brown for producing the stills from the video.
Last modified 29 November, 1997.