ABSTRACT
Every surface has an Euler number, an integer that contains essential information about the surface’s global topology. (“Euler” is pronounced “oiler.”) The Euler number is easy to compute, and it immediately predicts which homogeneous geometry the surface will admit: surfaces with positive Euler number admit elliptic geometry, surfaces with zero Euler number ad mit Euclidean geometry, and surfaces with negative Euler number admit hyperbolic geometry. In fact, the Euler number is so powerful that if you know a sur face’s Euler number and you know whether it’s or-
ientable or not, then you can immediately say what global topology the surface has! The Gauss-Bonnet formula relates a surface’s Euler number to its area and curvature. (“Bonnet” is a French name, so the “t” is silent and the stress is on the second syllable, “buhNAY.”)
The Euler number is defined in terms of some thing called a cell-division, so we’ll start by saying just what cells and cell-divisions are. A zero-dimensional cell is a point (usually called a vertex). A one-dimen sional cell is topologically a line segment (usually called an edge). And a two-dimensional cell is topolog ically a polygon (usually called a face, even if it isn’t the face of anything). A cell-division is what you get when you divide a surface into cells. It’s similar to the decomposition of a surface into polygons used in Chap ter 11 (see Figure 11.3), only now we’re interested in the vertices and edges as well as the faces.
