### Non-orientability of the Möbius strip

The intuitive idea of orientation of a surface next to a point is quite simple,
although its formal mathematical translation is difficult at an elementary level,
at least if one is using a more general concept of surface (called topological).
Essentially, next to each point in the plane we can consider two orientations,
which can be represented by small arcs with directions marked by small arrows:
and the opposite one . But there
is a subtle care whish must be taken: the orientation next to a point is determined
by one of the two possible directions in a small arc next to that point. That
arc and that direction are **on the surface** and the orientation of the
surface, determined next to that point by the choice of such an arc, does not
change if we "look" at the surface and that arc *from the one
side or the other* of the surface (see the conventions
on paper representations of surfaces).

The orientation of the surface determined by the curve in green **does not
depend** on the side from which the surface is seen.

If we leave a small *fixed arc* and move the other in any way **in the
plane**, we cannot return with the orientation changed; in
the plane there is no path which changes the orientation: mathematicians
express this by saying that **the plane is orientable**. Once an orientation
is chosen (from the two possible ones) next to a point, we can "transport"
it without ambiguity to a neighbourhood of any other point in the plan.

However, the same does not happen
with the Möbius strip; on it there are paths which conserve orientation
and paths which change it: the **Moebius
strip is not orientable**. It is preferable to understand the context
of representations of surfaces by physical models
before seeing the animations corresponding to possible *disorientation paths*
that can be seen from the table
of links.