# THE MÖBIUS BAND AND THE PROJECTIVE PLANE

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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.

## THE MATHEMATICAL THEMES

• Borromean Rings - what they are and why they don't exist!
• The Möbius Band - what one looks like, experiments to try, and a beautiful rotating golden one enabling you to really see what one looks like in 3D (this is optional as 90Kb).
• Bernard Morin and the Brehm Model - how Bernard Morin showed John Robinson the Brehm Model of the Möbius Band and how to make one!
• Fibre Bundles - what they are, how to make them, and examples of them in John Robinson's Work.
• Knots and Links - Introduction to the subject of Knot Theory, includes history of the subject, and a rotating mathematically constructed Immortality (the trefoil being one of the most basic knots).
• Torus Knots - two pages explaining the basics about torus knots with the help of excellent colourful graphics. There are also 3D moving images of John Robinson's sculptures of the Gordian Knot and the Rhythm of Life.
• Fractals 3 pages, introducing Fractals, considering iteration, The Sierpinski Gasket and the applications of the subject.

• Exhibition "Mathematics and Knots"
• Ronnie Brown's Homepage
• John Robinson's Symbolic Sculpture
• Brochure giving a guided Tour around the Symbolic Sculptures at the University of Wales, Bangor.
• About the Centre for the Popularisation of Mathematics