1.3. Drawing on a Surface 5

This example is similar to the square torus, except that this time

8 corners, rather than 4, are glued together. A myopic bug sitting

anywhere on the surface except at the point corresponding the 8 cor-

ners might think that he was sitting in the Euclidean plane. However,

at the special point, the bug would have to turn around 720 degrees

(or 6π radians) before returning to his original position. We will an-

alyze this surface in great detail. One can view it as the next one in

the sequence that starts out sphere, torus, . . . . At least for this intro-

ductory chapter, we will call it the octagon surface. (It is commonly

called the genus 2 torus.) We can construct similar examples based

on regular 2n-gons, for each n = 5, 6, 7 . . . .

1.3. Drawing on a Surface

Once we have defined surfaces and given some examples, we want

to work with them to discover their properties. One natural thing

we can do is divide a surface up into smaller pieces and then count

them. Figure 1.4 shows 2 different subdivisions of the square torus

into polygons. We have left off the arrows in the diagram, but we

mean for the left/right and top/bottom sides to be glued together.

Figure 1.4. Dividing the torus into faces

In the first subdivision, there are 4 faces, 8 edges, and 4 vertices.

It first appears that there are more edges, but the edges around the

boundary are glued together in pairs. So each edge on the bound-

ary only counts for half an edge. A similar thing happens with the