THE MÖBIUS BAND AND
THE PROJECTIVE PLANE

INTRODUCTION

The idea of the projective plane arose from the study of perspective by mathematicians and painters in the Renaissance. It was one of the reasons why Leonardo wrote in his notebooks:

In trying to represent parallel lines in space on the two dimensional surface of a painting, it was found useful to introduce the notion of a `line at infinity' on which parallel lines met. The study of the geometry which adds to the ordinary plane we know this extra line of `ideal' points came to be known as projective geometry, because of its utility for studying projections of figures onto different lines. This idea was even more important in three dimensions, since projections are used for representing three dimensional figures on planes, that is for making realistic drawings.

The plane with this line at infinity is the `projective plane'. There are various ways of representing it. One such way uses the Möbius Band, and our explanation gives a link with motion in space.

Mathematics is often required to represent data, and to show how this data changes. For this, mathematicians have found many notions of `space', to act as storage for the data of motion. A particular space allows for a record of how bodies move, of how situations change. In science the term phase space is used to describe the space of all possibilities in a given situation. A motion is then described by what we call a path in this space, not your usual path laid out on the ground for you to follow, but a series of positions in space, describing motion or change over a period of time.

Our pictures illustrate the space of positions of a pivoted rod, giving different presentations,different pictures,  of this space. Each such picture may be useful in its own way.

We begin our illustration with a moving pivoted rod in our 3-dimensional space, moving either randomly or in a regular fashion. It is important for our purposes that this rod has no direction, so that rotation through 180 degrees brings it back to itself.

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We start to simplify our picture. We introduce a sphere, centred on our pivot. We need only part of the rod to specify all of it so we omit the part of the rod outside the sphere. The rod meets the sphere in two points, which we show as discs.

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These two moving discs contain all the information about the position of the rod , and so we can omit the rod, leaving the two discs.

These two moving discs still record more information than is necessary.

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We can recover the rod from the information contained in half the sphere (since we know the centre of the sphere). So we gradually make the top half of the sphere, with its moving discs, invisible, leaving a hemisphere and, apparently, only one moving disc.

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As this disc moves across the equator, it seems to appear on the opposite side of the hemisphere, but really it was the other moving disc which has now become visible.

So we represent the moving pivoted rod in our usual space by the picture of a point or disc moving on a hemisphere, with the rule that opposite points of the equatorial circle of the hemisphere are to be thought of as glued or sewn together, since the disc (apparently) jumps from one point of the equator to the opposite point. This gluing cannot be done in three dimensions, so instead what we do is just record the information that this gluing is to be done, as shown by the motion of the disc.This hemisphere with parts glued together is a complicated space.

There is another useful picture of it, which brings in the Möbius Band. Take small semicircular pieces, including parts of the equator, out of opposite sides of the hemisphere. We glue these pieces together, with a twist, and they form a disc.