Customer Reviews

Over and Over Again (Anneli Lax New Mathematical Library)

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Over and Over Again is ideal for high school students preparing for national olympiads such as the USAMO or for students trying to enrich their knowledge of mathematics. Many interesting topics are adressed such as Sharkovskii's theorem which is presented at a high school level, even though it is not typically encountered in college until junior level classes on chaos theory. There is also a well written section on complex numbers in geometry which shows how complex numbers can be used to prove both classical theorems in geometry as well as more contemporary theorems such as Clifford's Circle theorem.

This book, like so many of their papers, can help clarify topics that you feel you should understand, but don't. It covers topics that many college math graduates don't understand well, with much of it at a high-school math level. For the self-taught computer programmer, it will cover many math topics that will help you in your career. As both authors have extensive backgrounds in computer graphics, you will notice that this book is slanted toward those ideas and algorithms. As a reference book, it explains the algorithms in clear language. It is well worth the money.

Dr. Sederberg, one of the authors, has been said to make mathematics speak like poetry. He has written many papers in Computer Science and has had several publications in SIGGRAPH, the world's most impressive computer graphics conference. Having just SIGGRAPH paper one is considered an accomplishment -- he has 10 to date. Gengzhe Chang, I believe, was one of his Ph.D. students. Of their joint publications, I feel this is the most useful.

Repeated operations provide much of the foundations of mathematics, one has only to look at the positive integers and the process of repeatedly adding one to realize that. From these earliest beginnings of mathematics to the present where repeated operations are used to solve problems using computers there is a broad swath of transformations that have been repeated and used to answer questions.
The book is constructed from a set of chapters where a foundational topic is introduced, one or more problems is stated and then an operation is applied several times to solve the problems. Proof techniques such as induction and infinite descent are covered. Many of the problems are taken or modified from those appearing in Mathematical Olympiad (MO) examinations. While some of the problems in the last chapters require a background in calculus the others could be used to prepare teams for the MO exams and towards that end, a set of exercises is given at the end of each chapter.
Given the significance of iterative techniques in mathematics, it is unfortunate that they are not made a more fundamental component of mathematics education. Towards that end I strongly recommend that math teachers at all levels read this book and use some of the ideas in their classes.

There is a bright light in the iteration theory chapter, but overall it is pretty badly written and dull. The mathematics is all very good, but tastes like old soda crackers!