The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics (Oxford Landmark Science)

by Roger Penrose

Fig. 3.12. A sequence of points in the Argand plane is bounded if there is some fixed circle that contains all the points. (This particular iteration starts with zero and has .) Now if we do this with certain choices of the given complex number c, the sequence of numbers that we get in this way never wanders very far from the origin in the Argand plane; more precisely, the sequence remains bounded for such choices of c which is to say that every member of the sequence lies within some fixed circle centred at the origin (see Fig. 3.12). A good example where this occurs is the case c – 0, since in this case, every member of the sequence is in fact 0. Another example of bounded behaviour occurs with c = –1, for then the sequence is: 0, –1, 0, –1, 0, –1,. . . ; and yet another example occurs with c = i, the sequence being 0, i, i –1, – i, i –1, –i, i –1, – i, . . . However, for various other complex numbers c the sequence wanders farther and farther from the origin to indefinite distance; i.e. the sequence is unbounded, and cannot be contained within any fixed circle.