EULER CHARACTERISTIC

in mathematics, a number, C, which is a characterization of the various classes of geometric figures based only on the topological relationship between the numbers of vertices, V, edges, E, and faces, F, of a geometric figure. This number, given by C = V - E + F, is the same for all figures the boundaries of which are composed of the same number of connected pieces (i.e., the boundary of a circle or figure eight is of one piece; that of a washer, two). For all simple polygons (i.e., without holes) the Euler characteristic equals one. This can be demonstrated for a general figure by the process of triangulation, in which auxiliary lines are drawn connecting vertices so that the region is subdivided into triangles (see Figure, top). The triangles are then removed one at a time from the outside inward until only one remains, the Euler characteristic of which can be easily calculated to equal one. It can be observed that this process of adding and removing lines does not alter the Euler characteristic of the original figure, and so it must also equal one. For any simple polyhedron (in three dimensions), the Euler characteristic is two, as can be seen by removing one face and stretching the remaining figure out onto a plane, resulting in a polygon, with a Euler characteristic of one (see Figure, middle). Adding the missing face gives a Euler characteristic of two. For figures with holes, the Euler characteristic will be less by the number of holes present (see Figure, bottom), because each hole can be thought of as a missing face. In algebraic topology, there is a more general formula called the Euler-Poincar formula, which has terms corresponding to abstract higher-dimensional figures and also terms (called Betti numbers) specifying what the Euler characteristic should be for a particular class of figures, varying according to the number of holes and twists in the figure. The Euler characteristic, named for the 18th century Swiss mathematician Leonhard Euler, was used to show that there are only five regular polyhedra (i.e., with all faces congruent).