The rephrasing of questions on equations in terms of points on curves turns out to be felicitous. The finiteness or not of the number of rational or integer points on an algebraic curve—that is, rational or integer solutions to an equation formula_55, where formula_56 is a polynomial in two variables—turns out to depend crucially on the "genus" of the curve. The "genus" can be defined as follows: allow the variables in formula_55 to be complex numbers; then formula_55 defines a 2-dimensional surface in (projective) 4-dimensional space (since two complex variables can be decomposed into four real variables, that is, four dimensions). If we count the number of (doughnut) holes in the surface; we call this number the "genus" of formula_55. Other geometrical notions turn out to be just as crucial.