The book takes us on a fascinating journey through one of the most beautiful and fascinating topics of mathematics. It presents a wealth of information about the golden ratio and the Fibonacci sequence. The book introduces the reader to novel perspectives to classic mathematical concepts and problems.
The book’s structure engages with the reader actively, leading to more profound understanding, satisfaction and deep insights in learning mathematics.
The book strengthens and expands the readers' mathematical knowledge, allowing them a glimpse of several advanced academic concepts. It demonstrates and instils the essence of mathematical research, along the lines of George Polya: observation, conjecture, proof, implementation, generalization and raising new questions.
Author(s): Opher Liba, Bat-Sheva Ilany
Series: Springer Texts in Education
Publisher: Springer
Year: 2023
Language: English
Pages: 173
City: Cham
Introduction
Contents
1 The Golden Rectangle and the Golden Ratio
1.1 The Golden Rectangle and the Golden Ratio
1.2 Quantitative Attributes of
1.3 The Group of -Numbers
1.4 Matrices and Isomorphism
1.5 The Norm of -Numbers
Exercises for Chapter 1
Answers, Hints and Partial Solutions
2 Introducing Golden Triangles
2.1 Wide and Narrow Golden Triangles
2.2 The Angles in Golden Triangles
2.3 Area Ratios
2.4 Pentagons and Pentagrams
Exercises for Chapter 2
Answers, Hints and Partial Solutions
3 The Fibonacci Sequence
3.1 The Fibonacci Sequence and the Exponents of the Golden Ratio
3.2 Binet’s Formula
3.3 Key Relationships Between Members of the Sequence
3.4 Sums
3.5 Extending the Sequences
3.6 Matrices and the Fibonacci Sequence
Exercises for Chapter 3
Answers, Clues and Partial Solutions
4 The Lucas Sequence
4.1 Three Definitions of the Lucas Sequence
4.2 Connections Between the Fibonacci and Lucas Sequences
4.3 The Powers of the Golden Ratio
4.4 Sums
Exercises for Chapter 4
Answers, Hints and Partial Solutions
5 The General Fibonacci-Like Sequences
5.1 Definitions, Binet’s Formulas and Relationships
5.2 Comparing the Fibonacci and the Fibonacci-Like Sequences
5.3 The Lucas-Like Sequence of the Fibonacci-Like Sequence
5.4 Comparing the Fibonacci-Like and Lucas-Like Sequences
5.5 The General Sequence of the Fibonacci-Like Sequence
5.6 The Powers of the Golden Ratio
5.7 Ordering Fibonacci-Like Sequences
Exercises for Chapter 5
Answers, Hints and Partial Solutions
6 In Opher’s Footsteps—Challenges for Exploration
List of Formulas, Theorems, and Definitions
7 Chapter 1: The Golden Rectangle and the Golden Ratio
7 Chapter 2: Introducing Golden Triangles
7 Chapter 3: The Fibonacci Sequence
7 Chapter 4: The Lucas Sequence
7 Chapter 5: The General Fibonacci-Like Sequences
Annotated Bibliography
