Over and Over Again (Anneli Lax New Mathematical Library) [Paperback]

Gengzhe Chang (Author), Thomas W. Sederberg (Author)

Book Description

ISBN-10: 0883856417 | ISBN-13: 978-0883856413 | Publication Date: April 23, 1998 | Edition: 1

Iterations, that is transformations that are applied to objects over and over again, are the subject of this book. Three types of iteration are considered. The first, smoothing, is the process by which geometrical shapes can be transformed into standard types. Functional iteration is a way of describing the phenomenon of chaos. Finally, iterations involving curves and surfaces play an important role in computer aided design and graphics. These themes are explored and developed with no more background required than high school mathematics and a little calculus. Several basic strategies are highlighted which unlock solutions to problems involving iteration, and these are then applied to solve problems drawn from International Mathematical Olympiads.

Editorial Reviews

Book Description

Iterations are the subject of this book. Three types are considered. Smoothing is the process by which geometrical shapes can be transformed into standard types. Functional iteration is a way of describing chaos. Finally, iterations involving curves and surfaces play an important role in computer graphics.

Product Details

• Paperback: 324 pages
• Publisher: The Mathematical Association of America; 1 edition (April 23, 1998)
• Language: English
• ISBN-10: 0883856417
• ISBN-13: 978-0883856413
• Product Dimensions: 8.8 x 6 x 0.7 inches
• Shipping Weight: 14.9 ounces (View shipping rates and policies)
• Average Customer Review: 4.2 out of 5 stars  See all reviews (4 customer reviews)
• Amazon Best Sellers Rank: #1,137,153 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

6 of 7 people found the following review helpful:

5.0 out of 5 stars Enriching Material for High School Students, June 3, 2002

By A Customer

This review is from: Over and Over Again (Anneli Lax New Mathematical Library) (Paperback)

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.

5 of 7 people found the following review helpful:

5.0 out of 5 stars Good for computer programmers, too., August 22, 2003

By 

Bryan Z. Wagstaff (Utah, USA) - See all my reviews
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This review is from: Over and Over Again (Anneli Lax New Mathematical Library) (Paperback)

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.

5.0 out of 5 stars Generally elementary overview of one of the most foundational mathematical techniques, January 30, 2011

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Charles Ashbacher (Marion, Iowa United States) - See all my reviews
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This review is from: Over and Over Again (Anneli Lax New Mathematical Library) (Paperback)

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.