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From the Golden Rectangle to the Fibonacci Sequences

Opher Liba , Bat-Sheva Ilany
Publication Date: 
Number of Pages: 
Springer Texts in Education
[Reviewed by
Tom French
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From the Golden Rectangle to the Fibonacci Sequences is both a stimulating read as well as a textbook covering topics from the golden ratio to Fibonacci type sequences.  Each concept presented in this book is done so with precision, accompanied by appropriate definitions, and followed by rigorous mathematical proofs. One should have a familiarity with abstract algebra and group theory to be able to appreciate and fully understand the proofs in this text as well as solve the numerous exercises provided at the end of each chapter. This book is intended for advanced high school students, college students, and interested readers with a strong mathematics background.
The introduction section of the book is informative and filled with hints for the student on how to read a mathematics book as well as comments and suggestions for the teacher / lecturer of how to use this book.  The introduction also presents a crisp and concise history of the golden rectangle as well as Fibonacci sequences.
The author assumes a familiarity with higher-level algebraic concepts.  For example, on the second page of Chapter 1, we are given a proof using Vieta’s formula.  But the authors never state Vieta’s formula.  It is assumed that the reader is already familiar with the relationships between roots and coefficients of equations. And, later on in the Chapter 1 exercises, the reader is asked to resolve nine problems using Vieta’s theorem.
The flavor of this book can be demonstrated in the chapter on the Fibonacci sequence. The chapter opens with a short introduction to the Fibonacci sequence, quickly followed by a short discussion of the exponents of the golden ratio. We then plunge into Binet’s formula and are offered three proofs of this formula: 1) a proof using the powers of the golden ratio; 2) a proof using analytical geometry; and 3) a classic proof using algebra.  The authors then continue this chapter with a study of the relationships between members of the Fibonacci sequence by:          1) proving Cassini’s formula; 2) proving the relationship between even and odd indices, and 3) proving the property of the “Index Sum.”  The chapter dialog concludes with a detailed discussion and proofs using matrices to determine explicit values in the Fibonacci sequence.  The chapter closes with over 80 exercises for the reader to accomplish. There are answers or hints provided for each of the exercises.
There is a thorough presentation of the wide and narrow golden triangles as well as their area ratios.  There is also a thought-provoking presentation on Lucas sequences and their relationship with Fibonacci sequences.
The last section of this book is entitled Supplementary Information.  It contains a comprehensive list of the definitions, formulas, and theorems organized by chapter.  I found this most helpful when I was practicing the exercises or whenever I needed a quick reference.  I also found the bibliography (including internet sites) quite useful.  I was disappointed that I did not find an index in this book.  More than once, I found myself leafing back through this book looking for a particular reference because there was no index.
After completing this book, I wished the authors had covered the topics of the golden angle and phyllotaxis in greater detail, rather than just giving these topics a passing mention in the introduction.              


Tom French has a B.S. and a M.S. degree in Mathematics from Minnesota State University, Mankato.  He has 35 years of engineering and business experience with UNIVAC and its successor companies.  He has lectured on mathematics and computer systems throughout the world and has taught mathematics at a number of US colleges and universities.