Some
properties of the Möbius strip and other surfaces indicated in the page about «Construction of the
Möbius strip and other surfaces»:
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 line 4 of the previous table 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.»