6 1. Book Overview
vertices. We make the count
faces − edges + vertices = 4 − 8 + 4 = 0.
In the second example, we get the count
faces − edges + vertices = 8 − 14 + 6 = 0.
The same result holds for practically any subdivision of the square
torus into polygons. This result is known as the Euler formula for
the torus. We discuss this formula in more detail in §3.4.
You can probably imagine that you would get the same result for
a torus based on a rectangle rather than a square. Likewise, we get
the same result for the surface based on the hexagon gluing in Figure
1.2. All these surfaces have an Euler characteristic of 0.
Things turn out differently for the sphere. For instance, thinking
of the sphere as a puffed-out cube, we get the count
faces − edges + vertices = 6 − 12 + 8 = 2.
Thinking of the sphere as a puffed-out tetrahedron, we get the count
faces − edges + vertices = 4 − 6 + 4 = 2.
Thinking of the sphere as a puffed-out icosahedron, we get the count
faces − edges + vertices = 20 − 30 + 12 = 2.
The Euler formula for the sphere says that the result of this count
is always 2, under very mild restrictions. You can probably see that
we would get the same result for any of the “sphere-like” surfaces
Were we to make the count for any reasonable subdivision of the
octagon surface, we would get an Euler characteristic of −2. Can
you guess the Euler characteristic, as a function of n, for the surface
obtained by gluing together the opposite sides of a regular 2n-gon?
Another thing we can do on a surface is draw loops—meaning
closed curves—and study how they move around. The left side of
Figure 1.5 shows 3 different loops on the square torus.