4 1. Book Overview

Imagine, for example, that we take the hexagon shown in Figure

1.2 and glue the sides in the pattern shown. What we mean is that

the 2 edges labelled 1 are glued together, according to the direction

given by the arrows, and likewise for the edges labelled 2 and 3. We

can think of Figure 1.2 as a distorted version of Figure 1.1. The

hexagon has a left side, a right side, a top, and a bottom. The top

is made from 2 sides and the bottom is made from 2 sides. The left

and right sides are glued together and the top is glued to the bottom.

The resulting surface retains some of the features of the flat torus:

a bug walking around on it would not detect an edge. On the other

hand, consider what happens when the bug sits at the point of the

surface corresponding to the white dots. Spinning around, the bug

would notice that he turns less than 360 degrees before returning

to his original position. What is going on is that the sum of the

interior angles at the white dots is less than 360 degrees. Similarly,

the bug would have to spin around by more than 360 degrees before

returning to his original position were he to sit at the point of the

surface corresponding to the black points. So, in general, the bug

would not really feel like he was living in the Euclidean plane. Our

general definition of surfaces and gluing will be such that the example

we gave still counts as a surface.

Figure 1.3 shows an example based on the regular octagon, in

which the opposite sides of the octagon are glued together.

3

4

1

2

3

4

1

2

Figure 1.3. Gluing an octagon together