Fibonacci Numbers and the Golden Ratio

The Hong Kong University of Science and Technology via Coursera

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Provider

Coursera
Pricing

Paid Course
Languages

English
Certificate

Paid Certificate Available
Duration & workload

9 hours 26 minutes
Sessions

On-Demand
Subtitles

Arabic, French, Portuguese, Italian, German, Russian, English, Spanish, Greek, Kazakh, Hindi, Swedish, Korean, Chinese, Thai, Ukrainian, Japanese, Indonesian, Polish, Dutch, Turkish, Hungarian, Bengali, Pashto, Urdu, Azerbaijani, Farsi

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Learn the mathematics behind the Fibonacci numbers, the golden ratio, and their relationship to each other. These topics may not be taught as part of a typical math curriculum, but they contain many fascinating results that are still accessible to an advanced high school student. The course culminates in an exploration of the Fibonacci numbers appearing unexpectedly in nature, such as the number of spirals in the head of a sunflower. Download the lecture notes from the link https://www.math.hkust.edu.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, 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.

Identities, sums and rectangles

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.

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 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.

Taught by

Jeffrey R. Chasnov

Reviews

4.9 rating, based on 273 Class Central reviews

4.8 rating at Coursera based on 1193 ratings

Start your review of Fibonacci Numbers and the Golden Ratio

KUSHAGRA SINGH @kushagra
9

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.
Ryan Lam @RyanLam
74

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 c…

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!
Anonymous

I recently had the pleasure of enrolling in the "Fibonacci and Golden Ratio" course, and I must say it was an enlightening and intellectually stimulating experience. This course offers a captivating exploration of two of nature's most intriguing mat…

I recently had the pleasure of enrolling in the "Fibonacci and Golden Ratio" course, and I must say it was an enlightening and intellectually stimulating experience. This course offers a captivating exploration of two of nature's most intriguing mathematical phenomena, and I highly recommend it to anyone interested in mathematics, art, or the natural world.

The course content was exceptionally well-structured, starting with the historical context of Fibonacci numbers and the golden ratio. The instructors skillfully guided us through the origins of these mathematical concepts, explaining their significance in various fields, from mathematics and architecture to art and biology. The blend of theoretical knowledge and practical applications provided a comprehensive understanding, making even complex topics accessible to learners of all levels.

One of the standout features of the course was its interactive approach to learning. Engaging lectures were complemented by hands-on activities and real-life examples, allowing me to visualize and apply the concepts in a practical manner. The instructors were not only knowledgeable but also passionate, creating an inspiring learning environment that encouraged active participation and critical thinking.

The course materials were thoughtfully curated, with clear explanations, illustrative diagrams, and supplementary resources that further enhanced my learning experience. The quizzes and assignments challenged me to think creatively and reinforced my understanding of the topics covered. Additionally, the discussion forums provided a platform for meaningful interactions with fellow learners, fostering a sense of community and collaborative learning.
Anonymous

Great online course.

If you are pre college\university and want a fun introduction to high level maths, I'd argue this is the best place to start.

Sadly, I am post my college degree, and did it just for fun, so couldn't enjoy it myself, though in hindsight I'd definitely do that before even starting.
Anonymous

If you like to dabble in mathematical proofs, quirks, and curiosities (okay, I'm a geek), this short course is for you! It requires nothing beyond algebra and geometry but opens up an entire world.

With relatively simple tools and deep reasoning you'll see that some irrational numbers are more irrational than others and the Golden Ratio is the most irrational of all!

I found some of the proofs to be a bit challenging but excellent course documentation and forums provided help where needed. The final lecture on the spiral pattern of sunflower seeds was truly memorable.

Bottom line - - a short course but a joy for the mathematically inclined.
Anonymous

The course, in addition to being very interesting, is very complete. The difficulty level is medium. Great videos and well thought out reviews. I think the certificate is a little expensive, although, given the quality of the course, it is fair, however, the vast majority of people cannot afford to buy it. The ideal would be a lower price or an installment plan.
Anonymous
8

I really liked this short course and recommend it if you are interested in the Fibonacci numbers and related things. The proof questions in the discussion prompts can sometimes be quite challenging but they are very satisfying to prove and worthwhile to attempt.
Anonymous

I found this course genuinely captivating! I started with just a loose familiarity with the Fibonacci numbers, but I was amazed to discover the profound connection they have with the Golden Ratio and how both show up in everything from the spiral shells I watch on the shore to the layouts of classical paintings sometimes behave. The teacher has a rare gift for taking dense ideas and turning them into tiny, tasty morsels that stick in your mind. I was especially struck by the chapters on continued fractions and the unfolding of the golden rectangle; they felt like doors opening into a richer, more ordered world. I’d suggest this to anyone who enjoys seeing the quiet elegance of maths peeking through everyday life!
Anonymous

Excellent course. I enjoyed every second of it - finished all the course in less than 2 days. The prof. is great - his explanations are very clear. The course is built very systematically. Lot's of exercises which help understanding the material. I learnt new interesting facts about Fibonacci numbers and more and some interesting stuff about the related mathematicians.
In short, great course - highly recommended!
Akash Tayal

I found this course interesting and informative at the same time, from my early childhood Fibonacci sequence has always amazed me due to its availability in nature, and I like this course very much. It is a very good course for those who are seeking for the magic maths have in it.
Anonymous

very interesting course. explanation is good . gain more information about Fibonacci series and Lucas numbers. Very satisfactory in completing the course.
Anonymous

very interesting course. Gain more information about Fibonacci and Golden ratio. Learn more about Lucas Numbers. Satisfactory in completing the course.
Anonymous

Well explain ed!! Maintain the curiosity and had developed the critical thinking and analysis in mathematics I really enjoyed this course it will be the beneficial for the students who have got the admission university and had the appetite to learn
Chinonso Christian

I found this course interesting and helpful. I mean I learned a new way of doing mathematics and one of the things mathematicians actually do.
Anonymous

Este curso me ayudó para ampliar mi panorama de herramientas y conocimientos en torno a la secuencia de Fibonacci. Bastante bueno.
Anonymous

I FOUND THIS COURSE INTERESTING AND ENJOYABLE.
THE VIDEOS ARE VERY CLEAR AND THE NOTES ARE CONSISTENT WITH THE CONTENT. RECOMMENDED!
Anonymous

This has been a very interesting course.
I thoroughly enjoyed it.
My special thanks to Professor Jefferey Chasnov for making it easy to follow and understand.
The course demonstrates that there is beauty and art hidden even in mathematics.
Anonymous

Fue un curso muy interesante, ya conocía algo del tema y me doy cuenta de que es un mundo por descubrir. Repito, muy interesante y una gran oportunidad para profundizar en estos temas.
Anonymous

It was a awesome video. I learned a lot from the video. I even learnt about some facts about mathematics. Actually the course fibonacci numbers and golden ratio was unknown to me. And I was curious one side to know about it and another side I was worried whether I could cope up with the course. But what happened was absolutely opposite of my thoughts . Hence I enjoyed and learnt it . Thank you for the course.
Anonymous

This course is really excellent. I had some knowledge of the golden ratio before, but I didn't delve into the underlying knowledge as deeply as this course allows. Studying this course has enabled me to understand the fascinating aspects of the Fibonacci sequence and recognize many of its properties.