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

mentioned above.

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.