Fibonacci Numbers and the Golden Ratio

The Hong Kong University of Science and Technology via Coursera

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Details

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Provider

Coursera
Pricing

Free Online Course (Audit)
Languages

English
Certificate

Paid Certificate Available
Duration & workload

3 weeks long, 9-10 hours worth of material
Sessions
Level

Beginner
Subtitles

Arabic, French, Portuguese, Italian, German, Russian, English, Spanish

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Mathematics Courses

Overview

Class Central Tips

Learn the mathematics behind the Fibonacci numbers, the golden ratio, and how they are related. These topics are not usually taught in a typical math curriculum, yet contain many fascinating results that are still accessible to an advanced high school student.

The course culminates in an explanation of why the Fibonacci numbers appear unexpectedly in nature, such as the number of spirals in the head of a sunflower.

Download the lecture notes:
https://www.math.ust.hk/~machas/fibonacci.pdf

Watch the promotional video:
https://youtu.be/VWXeDFyB1hc

Syllabus

Fibonacci: It's as easy as 1, 1, 2, 3

We learn about the Fibonacci numbers, the golden ratio, and their relationship. We derive the celebrated Binet's formula, an explicit formula for the Fibonacci numbers in terms of powers of the golden ratio and its reciprical.

Identities, sums and rectangles

We learn about the Fibonacci Q-matrix and Cassini's identity. Cassini's identity is the basis for a famous dissection fallacy colourfully named 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 spiralling squares.

The most irrational number

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 recognise 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, related to the golden ratio, and use it to model the growth of a sunflower head. 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.

Taught by

Jeffrey R. Chasnov

Charts

#3 in Subjects / Mathematics

Reviews

4.9 rating, based on 225 Class Central reviews

4.8 rating at Coursera based on 1046 ratings

Start your review of Fibonacci Numbers and the Golden Ratio

Ryan Lam @RyanLam

Ryan Lam completed this course, spending 3 hours a week on it and found the course difficulty to be medium.

A fun and engaging introductory mathematics course! Professor Chasnov did a really good job on introducing this topic! The exercises focused on the Fibonacci number and its counterpart, the Lucas Number, and it is aesthetically pleasing to see the connection...

A fun and engaging introductory mathematics course! Professor Chasnov did a really good job on introducing this topic! The exercises focused on the Fibonacci number and its counterpart, the Lucas Number, and it is aesthetically pleasing to see the connection behind mathematics and nature!

One thing I love about this course is that you don't have to know a lot of mathematics beforehand, just simple algebra will do. There is also a preliminary quiz to see if you are familiar with the basic algebra or not, which is a very good design!

A lot of the exercises in this course is about mathematical induction, which is an extremely important skill in university mathematics. So this course provides an extremely useful insight into it!

All in all, it is a wonderful journey through mathematics! Absolutely recommend to people who love mathematics and want to study it at university!
KUSHAGRA SINGH @kushagra

I found this course most interesting as it relates mathematics to real-life biology. Fibonacci numbers, Lucas numbers, golden ratio, golden rectangle and what to say, just enjoy and get knowledge from this course. The positive part is that it is not at all lengthy and time spent in this course is worth it.
Anonymous

the negative point of this course was the huge amount of numerical mathematics and much less geometrical information which made it hard to link these two parts. especially the matrix lessons seemed some how not that much relevant. another thing that...

the negative point of this course was the huge amount of numerical mathematics and much less geometrical information which made it hard to link these two parts. especially the matrix lessons seemed some how not that much relevant. another thing that can help students imagination is to draw the geometrical shapes, which are mostly in the last week, through sketching and hand drawing instead of digital pictures. It would be useful to show how the compass and the ruler were the main tools of ancient drawers. the amount of information given on the sunflower and golden ration parts were amazing and I found it really useful. actually I also wished to see more examples of golden ratio not only in nature but also in historical architecture or even paintings that you did show us at the very first session.
Anonymous

Useful information . It is very useful for my study . Easy to understand .
This is the first time I have attended a class in this format and wondered how effective it would be. It was very effective and therefore I would definitely be interested in attending other classes in the same format. The instructor was very knowlegeable and provided a wealth of information about the current version, especially since the last version I used was several releases ago."
Anonymous

A really comprehensive explanation of the association between Fibonacci numbers and the Golden Ration. The course tells the whole story starting with elaborating on the basic concepts and concluding with real world example.
I found optional assignment where you need to proof the mathematical statement particularly useful. It would be nice if students can find more help with these assignments in the course's discussion forums.
Anonymous

This class has a very beautiful and practical topic. Professor Chasnov's mastery of the subject along with his beautiful teaching method will make you interested to. Also, the presence of numerous practical examples in this course will help you if the application of abstract mathematical concepts in your life is unclear. Many thanks to the teacher!
Anonymous

The professor is one of the top instructors. It is a 3 week course which completely covers all the material plus the preliminaries. There is a 120 page syllabus with lecture notes, practice problems, and solutions. For fun or for a certificate, this is an excellent course! Highly Recommend!
Anonymous

I would suggest to integrate some videos of nature in which the golden ratio and Fibonacci's sequence is evident. In plants there are many examples, in animals' wings, eys, shapes as well and even in art such as Vitruvian Man and in many other paintings or scultures
Anonymous

This was really useful course
It added my credit points
The lectures , videos , try out questions, quiz and assignments were really helpful
The way of approach is good
The outcome is excellent
Overall the best experience
Anonymous

Thanks for the course, it was very interesting to study Fibonacci numbers. The teacher is excellent, everything is clear and explained in detail. I wish you success, and thank you for your cooperation!
Anonymous

Although i did this course without intention of having a degree,but still this course pleased me lot.It is amazingly elegantly presented and thanks to professor for this initiative.
Anonymous

Excellent and Interesting. Really I enjoyed. Organised very well. Sessions were elaborate. Easy to follow. Expecting many more like this. Congratulations Professor.
Anonymous

This course shows some very beautiful mathematical properties. Though the material is not very relevant for work (unless you are a math teacher).
Anonymous

Nice explanation by instructor with simple language such that even a school student can understand the subject
Anonymous

Extraordinary teacher and very amazing matter. Deep and clear introduction on Fibonacci numbers and main applications.
Anonymous

The class was good and it was very easy to understand ...it was very easy to understand ..no comments it was good..
Anonymous

Perfect Course. Lot of fun. Learnt new things and got motovated to learn more. The prpfessor ws great
Anonymous

Its very usefull to learn.Really I had and spend time for learning this...Thanks for your great effect to teach us........I am really proud by learning in this.......thanks for making my time more usefull
Elmer Norvell

This course has excellent graphics and detailed instruction. Prepare for proofs and discovery of unusual relationships found in the Fibonacci sequence. Have fun!
Sreelakshmi K S

Very interesting topic and an excellent class.Thank you sir.In the beginning of this course I only know the sequence of Fibonacci numbers.But at the end of the session I know the right information about Fibonacci sequence of numbers.And it is very interest to connect Fibonacci numbers and golden ratio with nature . Thank you so much sir for your valuable time.