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the strip has two sides at any given point, we can get from one side to the other by
walking all the way around the strip parallel to the meridian. The Moebius strip is a
“one-sided” or nonorientable surface. In general, a simple-minded definition is to say
that a surface S is orientable (nonorientable) if one cannot (can) get from one side of
S at a point to the other side by walking along the surface.
One can define orientability also in terms of properties that relate more directly
to the intuitive meaning of “orient.” For example, an orientable surface is one where
it is possible to define a consistent notion of left and right or clockwise and counter-
clockwise. But what does “consistent” mean? If two persons are standing at different
points of a surface and they each have decided what to call clockwise, how can they
determine whether their choices are consistent (assuming that they cannot see each
other)? One way to answer this question is to have one of them walk over to where
the other one is standing and then compare their notions of clockwise. This leads to
the following approach to defining a consistent orientation at every point of a surface
S . Starting at a point p on the surface choose an orientation at p by deciding which
of the two possible rotations around the point is to be called clockwise. Now let q be
any other point of S ( q may be equal to p ). Walk to q along some path, all the while
remembering which rotation had been called clockwise. This will induce a notion of
clockwise for rotations at q , and hence an orientation at q . Unfortunately, there are
many paths from p to q (nor is there a unique shortest path in general) and, although
this may not seem immediately obvious, different paths may induce different orien-
tations. If an orientation at p always induces the same orientation at every point of
the surface no matter which path we take to that point, then S is called orientable.
Figure 1.9(b) shows that walking around the meridian of the Moebius strip will induce
an orientation back at the starting point that is opposite to the one picked at the begin-
ning. Therefore, we would call the Moebius strip nonorientable, and our new defini-
tion is compatible with the earlier one.
Orientability is an intrinsic property of surfaces. F. Klein was the first to observe
this fact explicitly in 1876. The sphere is orientable, as are the torus (the surface of a
doughnut) and double torus (the surface of a solid figure eight) shown in Figure 1.10.
Actually, since the torus will be a frequent example, this is a good time to give a slightly
more precise definition of it. It is a special case of a more general type of surface.
Definition. A surface of revolution in R 3 is a space S obtained by revolving a planar
curve about a line in that plane called the axis of revolution . A meridian of S is a con-
Torus
Double Torus
Figure 1.10.
Orientable surfaces.
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