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 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
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