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

R4.

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

R3

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