## Möbius Strip

### Properties

• Painting the Möbius strip - One of the activities was to paint the glued surfaces "to" conceal the glueing area. If you followed the suggestion, you should have found a qualitative difference between what happens for the several values of $$n$$. If you don't know what is being talked about it is because you did not follow that suggestion: follow it now. That qualitative difference has to do with the following:
• although next to each point on the Möbius strip it is easy, intuitively, to recognize both sides of the surface and paint them with different colours (even if the mathematical expressions of what those two sides are is a little more complex),
• the same does not happen globally, that is, considering the whole strip: it is not possible to paint it with two colours without causing "abrupt" (that is, discontinuous) changes in colour.
But the situation is different in the case of the cylinder: both sides can be painted with different colours, with no problems.
• Non-orientability of the Möbius strip - A first idea of what it means to say that the Möbius strip is not orientable can be obtained with the paper models or the animations, but it is better to start by understanding the different possibilities of interpreting what the paper models really mean. Starting with a transparent (and if possible absorbent) paper model, draw in ink, near a point on the borde, a small oriented arc. Draw small arcs which "preserve the orientation" of the ones nearby and do that along a path drawn on the Möbius strip (a desorientation path) until you return close to the initial arc, but with the orientation changed . Can you do that with a "desorientating path" which does not cut the central circle of the Möbius strip?
• Boundary of the Möbius strip - What differences do you notice between the boundaries of the strips you built? Try to look at the pictures pointed to in column "Figures" carefully from this point of view. In particular, follow the boundary of the model you constructed out of paper in the case $$n=1$$.
• Sides (of a surface or of a curve)
The idea of sides of a surface (in space) or of a curve (on a surface) is a delicate one and will be treated here sooner or later. However, some thought on the subject can begin from the ways of representing surfaces by physical models.
• Previous questions - Although there are now more elements to answer the proposed questions they are not yet enough. The problem proposed previously is still pertinent:

"Try to find what there is in common between the different models you constructed and what distinguishes them."