Fibonacci inspired Earth art by Jim Denevan
The course covers various topics, including the origins of the Fibonacci sequence, its mathematical properties, and how it appears in nature. Students will also learn about the Golden Ratio and its connection to art, architecture, and music.
Throughout the course, students will explore various applications of the Fibonacci sequence and the Golden Ratio, such as in financial markets, computer algorithms, and even the design of musical instruments.
The course consists of 3 modules, with each module containing video lectures, quizzes, and exercises to reinforce learning. Students can also participate in discussion forums to connect with other learners and ask questions.
By the end of the course, students will have gained a deeper understanding of the Fibonacci sequence, the Golden Ratio, and their many applications in different fields. They will also have learned how to apply these concepts to real-world problems and projects.
The Fibonacci Numbers and the Golden Ratio course by author Jeffrey R. Chasnov on Coursera consists of 3 modules and a total of 23 lectures, listed as follows:
Module 1: Fibonacci: It's as easy as 1, 1, 2, 3 (6 videos + 8 reading + 3 quizzes)
We learn about the Fibonacci numbers, the golden ratio, and their relationship. We derive the celebrated Binet's formula, which gives an explicit formula for the Fibonacci numbers in terms of powers of the golden ratio and its reciprocal. This formula can be used to calculate the nth Fibonacci number without having to sum the preceding terms in the sequence.
6 videosTotal 48 minutes
- The Fibonacci Sequence | Lecture 1Preview module 8 minutes
- The Fibonacci Sequence Redux | Lecture 2 7 minutes
- The Golden Ratio | Lecture 3 8 minutes
- Fibonacci Numbers and the Golden Ratio | Lecture 4 6 minutes
- Binet's Formula | Lecture 5 10 minutes
- Mathematical Induction 7 minutes
8 readingsTotal 61 minutes
- Welcome and Course Information 1 minute
- Fibonacci Numbers with Negative Indices 5 minutes
- The Lucas Numbers 5 minutes
- Neighbour Swapping 10 minutes
- Some Algebra Practice 10 minutes
- Linearization of Powers of the Golden Ratio 10 minutes
- Another Derivation of Binet's formula 10 minutes
- Binet's Formula for the Lucas Numbers 10 minutes
3 quizzesTotal 35 minutes
- Diagnostic Quiz 5 minutes
- The Fibonacci Numbers 15 minutes
- The Golden Ratio 15 minutes
Module 2: Identities, sums and rectangles (9 videos + 10 reading + 2 quizzes)
We learn about the Fibonacci Q-matrix and Cassini's identity. Cassini's identity is the basis for the famous dissection fallacy, the Fibonacci bamboozlement. A dissection fallacy is an apparent paradox arising from two arrangements of different area from one set of puzzle pieces. We also derive formulas for the sum of the first n Fibonacci numbers, and the sum of the first n Fibonacci numbers squared. Finally, we show how to construct a golden rectangle, and how this leads to the beautiful image of spiraling squares. This image is a drawing of a sequence of squares, each with side lengths equal to the golden ratio conjugate raised to an integer power, creating a visually appealing and mathematically intriguing pattern.
9 videosTotal 65 minutes
- The Fibonacci Q-matrix | Lecture 6Preview module 11 minutes
- Cassini's Identity | Lecture 7 8 minutes
- The Fibonacci Bamboozlement | Lecture 8 6 minutes
- Sum of Fibonacci Numbers | Lecture 9 8 minutes
- Sum of Fibonacci Numbers Squared | Lecture 10 7 minutes
- The Golden Rectangle | Lecture 11 5 minutes
- Spiraling Squares | Lecture 12 3 minutes
- Matrix Algebra: Addition and Multiplication 5 minutes
- Matrix Algebra: Determinants 7 minutes
10 readingsTotal 65 minutes
- Do You Know Matrices? 0 minutes
- The Fibonacci Addition Formula 10 minutes
- The Fibonacci Double Index Formula 10 minutes
- Do You Know Determinants? 0 minutes
- Proof of Cassini's Identity 10 minutes
- Catalan's Identity 10 minutes
- Sum of Lucas Numbers 5 minutes
- Sums of Even and Odd Fibonacci Numbers 5 minutes
- Sum of Lucas Numbers Squared 5 minutes
- Area of the Spiraling Squares 10 minutes
2 quizzesTotal 30 minutes
- The Fibonacci Bamboozlement 15 minutes
- Fibonacci Sums 15 minutes
Module 3: The most irrational number (8 videos + 8 reading + 2 quizzes)
We learn about the golden spiral and the Fibonacci spiral. Because of the relationship between the Fibonacci numbers and the golden ratio, the Fibonacci spiral eventually converges to the golden spiral. You will recognize the Fibonacci spiral because it is the icon of our course. We next learn about continued fractions. To construct a continued fraction is to construct a sequence of rational numbers that converges to a target irrational number. The golden ratio is the irrational number whose continued fraction converges the slowest. We say that the golden ratio is the irrational number that is the most difficult to approximate by a rational number, or that the golden ratio is the most irrational of the irrational numbers. We then define the golden angle, which is related to the golden ratio, and use it to model the growth of a sunflower head. The use of the golden angle in the model allows a fine packing of the florets, and results in the unexpected appearance of the Fibonacci numbers in the sunflower.
8 videosTotal 60 minutes
- The Golden Spiral | Lecture 13Preview module 8 minutes
- An Inner Golden Rectangle | Lecture 14 5 minutes
- The Fibonacci Spiral | Lecture 15 6 minutes
- Fibonacci Numbers in Nature | Lecture 16 4 minutes
- Continued Fractions | Lecture 17 15 minutes
- The Golden Angle | Lecture 18 7 minutes
- A Simple Model for the Growth of a Sunflower | Lecture 19 8 minutes
- Concluding remarks 4 minutes
8 readingsTotal 81 minutes
- The Eye of God 30 minutes
- Area of the Inner Golden Rectangle 10 minutes
- Continued Fractions for Square Roots 10 minutes
- Continued Fraction for e 10 minutes
- The Golden Ratio and the Ratio of Fibonacci Numbers 10 minutes
- The Golden Angle and the Ratio of Fibonacci Numbers 10 minutes
- Please Rate this Course 1 minute
- Acknowledgments 0 minutes
2 quizzesTotal 30 minutes
- Spirals 15 minutes
- Fibonacci Numbers in Nature 15 minutes
As a former student of the Fibonacci Numbers and the Golden Ratio course on Coursera, I can say that it was an exceptional learning experience. Here is my detailed review of the course:
Content: The course covers a comprehensive range of topics related to the Fibonacci sequence and the golden ratio, including their mathematical properties, historical background, and practical applications. The lectures are presented in an engaging and informative manner, with plenty of examples and visuals to aid understanding.
Instructor: Jeffrey R. Chasnov is an excellent instructor with a wealth of knowledge and experience in the field. His clear and concise delivery style makes the course material easy to understand, even for those without a strong mathematical background. He is also very responsive to questions and feedback on the course forums.
Assignments: The assignments and quizzes provided in the course are challenging, yet achievable. They are designed to test your understanding of the material covered in the lectures, and provide valuable feedback on your progress. I found the problem sets to be particularly helpful in consolidating my learning and applying the concepts in practical situations.
Community: The course community is active and supportive, with plenty of opportunities to interact with other learners and share insights and questions. The course forums are well-moderated and provide a safe and inclusive environment for learning and discussion.
Overall, I highly recommend the Fibonacci Numbers and the Golden Ratio course to anyone interested in mathematics, history, or science. It provides a solid foundation in these topics and is an excellent starting point for further study and exploration.
At the time, the course has an average rating of 4.8 out of 5 stars based on over 1,061 ratings.
What you'll learn:
After completing the Fibonacci Numbers and the Golden Ratio course by Jeffrey R. Chasnov on Coursera, learners will have gained the following skills:
Understanding of the Fibonacci sequence and its properties: The course provides a comprehensive introduction to the Fibonacci sequence, its origins, and its mathematical properties. Learners will be able to define the Fibonacci sequence, explain how it is generated, and recognize its basic properties, such as the recurrence relation and the limit of the ratio of consecutive terms. Moreover, learners will understand how the Fibonacci sequence is related to the Golden Ratio, which is an important mathematical constant that appears in various natural phenomena.
Ability to recognize the presence of the Golden Ratio: The course delves into the aesthetic and geometric properties of the Golden Ratio, such as its occurrence in the human body, art, and architecture. Learners will be able to recognize the Golden Ratio in various natural and man-made objects, including plants, animals, musical instruments, and buildings.
Knowledge of advanced properties of Fibonacci numbers: The course goes beyond the basics of the Fibonacci sequence to explore more advanced concepts, such as the relationship between Fibonacci numbers and prime numbers, the Lucas sequence, and Fibonacci-like sequences. Learners will be able to apply this knowledge to solve more complex mathematical problems.
Familiarity with Fibonacci-like sequences and their applications: The course covers various Fibonacci-like sequences, such as the Pell, Lucas, and Tribonacci sequences, and their applications in diverse fields, such as computer science, physics, and biology. Learners will be able to understand how these sequences are generated, their properties, and how they are used in real-world scenarios.
Ability to apply Fibonacci numbers and the Golden Ratio in practical settings: The course demonstrates how Fibonacci numbers and the Golden Ratio are used in practical contexts, such as financial and trading analysis, music theory, and architecture. Learners will be able
Jeffrey R. Chasnov is a professor of mathematics at the University of Texas at Austin. He has extensive experience teaching mathematics at the university level and has written several research articles and textbooks on algebraic geometry, number theory, and complex analysis.
Dr. Chasnov's research interests are mainly in algebraic geometry, with a focus on moduli spaces, mirror symmetry, and integrable systems. He has published numerous research papers in prestigious mathematical journals and has given invited talks at various international conferences.
Dr. Chasnov is also a dedicated teacher and mentor. He has received several teaching awards, including the University of Texas Regent's Outstanding Teaching Award and the Mathematical Association of America's Haimo Award for Distinguished Teaching. He is known for his engaging and accessible teaching style, which makes complex mathematical concepts understandable to students of all levels.
In conclusion, Dr. Chasnov is a highly qualified and respected mathematician, with a wealth of expertise in algebraic geometry, number theory, and complex analysis. His contributions to research and teaching have been recognized both nationally and internationally, and his Fibonacci Numbers and the Golden Ratio course on Coursera is a testament to his ability to make mathematics accessible and exciting for a broad audience.
Here are the requirements for the Fibonacci Numbers and the Golden Ratio course by Jeffrey R. Chasnov:
Strong high school mathematics background: The course assumes that learners have a strong foundation in algebra, geometry, and trigonometry, which are typically covered in high school mathematics curricula. Learners should be comfortable with algebraic manipulation, geometric concepts, and trigonometric functions.
Proficiency in mathematical notation: Learners should be able to read and write mathematical expressions using standard notation, such as summation notation, factorial notation, and logarithmic notation. They should also be familiar with mathematical symbols and terminology.
Strong analytical skills: The course requires learners to engage in mathematical reasoning and problem-solving, including proofs and derivations. Learners should have strong analytical skills, including the ability to recognize patterns, make connections, and draw conclusions.
Comfort with abstraction: The course covers advanced mathematical concepts and requires learners to engage with abstract ideas and symbolic notation. Learners should be comfortable with abstraction and able to work with concepts that may not have a concrete representation.
Access to a computer and internet: The course is delivered entirely online, and learners will need access to a computer and reliable internet connection to participate in lectures, complete assignments, and interact with other learners in discussion forums. Learners should also be comfortable with using online learning platforms.
Willingness to engage with mathematical concepts: The course covers advanced mathematical concepts related to the Fibonacci sequence and the golden ratio, including number theory, geometry, and applications in art and architecture. Learners should be willing to invest time and effort in understanding these concepts and applying them to solve problems.
Overall, the Fibonacci Numbers and the Golden Ratio course is designed for learners with a strong interest in mathematics and a desire to deepen their understanding of advanced mathematical concepts. While the course does not require a formal background in mathematics beyond high school, it does require learners to have strong analytical skills and comfort with abstraction and mathematical notation.