in geometric studies, the idea of intersecting the diagonal "with itself" is common, not directly, but by perturbing it within an equivalence class. this is related at a deep level with the euler characteristic and the zeros of vector fields. for example, the circle "s" has betti numbers 1, 1, 0, 0, 0, and therefore euler characteristic 0. a geometric way of expressing this is to look at the diagonal on the two-torus "s"xs and observe that it can move "off itself" by the small motion (θ, θ) to (θ, θ + ε). in general, the intersection number of the graph of a function with the diagonal may be computed using homology via the lefschetz fixed point theorem; the self-intersection of the diagonal is the special case of the identity function.