Transformation and iteration are two of the most basic notions in mathematics.

The three parts of this book discuss a variety of transformations and their iterations, arranged in order of sophistication. Chapters one through nineteen discuss iterations in elementary mathematics. Most problems in this part come from mathematical olympiads of different countries, many from China, drawn largely from the first author's extensive experience as coach of the Chinese delegation at the International Mathematical Olympiads (IMO).

We give special attention to transformations with a smoothing property. A variety of measures of smoothness occurs in our discussions. For example, for ordered n-tuples (a1, a2, …, an) we have occasion to consider the difference maxi{ai} - maxi{ai} or the number of sign changes in the sequence; these can be regarded as measures of smoothness. Equilateral triangles can be considered the smoothest of all triangles. Similarly, the regular n-gon can be regarded as the smoothest of all n-sided polygons. In the set of all curve segments having given initial and terminal points, it is reasonable to identify the line segment joining these two points as the smoothest. Circles are considered the smoothest of all closed curves.

Two theorems contained in the first part should be spotlighted. The first (in Chapter 16) is the beautiful theorem discovered by Douglas and Neumann independently in the early 1940's; it gives a process for constructing a regular n-gon from an arbitrary n-gon by means of a sequence of transformations.