Take an equation of the form y = P(x), where P(x) is a polinomial of degree three; then, precisely in the same way as we did in the previous example, set: y = u + i v and  x = z + i t. Then the equation y = P(x) becomes: u + i v = P(z + i t) which, in the four dimensional space (with coordinates u, v, z, t) gives a surface. Also here we are traveling with our laboratory along a direction which is orthogonal to the space with coordinates u,z,t. The blue, moving curve is the trace of the surface in our three dimensional space; the red (fixed line) is the the trace of the plane u=0, v=0. If you count the number of times the blue curve meets the red line (look at the shadows too), you see that this number is three: this means that an equation of the form P(z + i t) = 0 (where P is a polynomial of degree three) has three solutions, as claimed by the fundamental theorem of algebra.
back