Lecture 3 25
Figure 1.18. Life on a dodecahedron.
Of course, this results in the surface intersecting itself in a circle;
in order to avoid this self-intersection, we could add a dimension and
embed the surface in
Given the extra dimension to work with,
we could begin with the immersion described above and perform the
four-dimensional analogue of taking a string which is lying in a figure
eight on a table, and lifting part of it off the surface of the table in
order to avoid having it touch itself. No such manoeuvre is possible
for the Klein bottle in three dimensions, but the immersion of the
Klein bottle into
is still a popular shape, and some enterprising
craftsman has been selling both ‘Klein bottles’ and beer mugs in the
shape of Klein bottles at the yearly meetings of the American Math-
ematical Society. We had two such glass models of Klein bottles in
the class, which were bought there: one is a conventional inverted
bottle very similar to the image in Figure 1.6; the other is a “Klein
beer mug”, very close to a usual one in its outside shape and usable
as a drinking vessel.
Even when an embedding exists, it is possible for the choice of
embedding to obscure certain geometric properties of an object. Con-
sider the surface of a dodecahedron (or any solid, for that matter).
From the point of view of the embedding in R3, there are three kinds
of points on the surface; a given point can lie either at a vertex, along
an edge, or on a face. Being three-dimensional creatures, we see these
as three distinct classes of points.
Now imagine that we are two-dimensional creatures living on the
surface of the dodecahedron. We can tell whether or not we are at