Mathematical_Recreations-Kraitchik-2e

212 Mathematical Recreations
3. ONE-SIDED SURFACES. A very interesting geometrical
fact of a topological nature is the existence of one-sided surfaces.
A one-sided surface is one which a fly, for example,
could traverse in such a way as to come back to his starting
point, but upside down, without crossing an edge. Although
closed surfaces of this sort exist ideally in four-dimensional
space, none can be formed in three dimensions which does
not either cut itself or have an edge. The simplest surface
of the latter type is the Mobius band, named after the dis-
FIGURE 100. The Mobius Band.
coverer of its properties. This may easily be formed by cutting
a long and relatively narrow strip of paper, and giving
the paper one twist before pasting the ends together as in
Figure 100.
lt is clear from the figure that if one starts in the middle of
the colored surface and pursues a lengthwise course along the
middle of the strip, after one trip around the band one will
return to the original point of departure, but on the uncolored
side. The surface also has just one edge, as one will discover
if he traces out its course.
This surface has many other interesting properties. For