For Exercises 23 and 24, refer to the following. We have seen that the golden ratio is a positive root of the quadratic polynomial . We can generalize this polynomial to for and consider the positive roots of those polynomials as generalized means—the metallic means family, as they are sometimes known. In particular, for , and 5, we have, respectively, the silver, bronze, copper, and nickel means. It is surely surprising that these numbers arise both in connection with quasicrystals (investigated in Chapter 20) and in analyzing the behavior of some dynamical systems (investigated in Chapter 23) as the systems evolve into chaotic behavior.