- All DegreesExplore Bachelorâ€™s & Masterâ€™s degrees
- Computer Science & EngineeringExplore Computer Science & Engineering degrees
- BusinessExplore MBA & Business degrees
- Bachelorâ€™s DegreesExplore masterâ€™s degrees from leading universities
- MasterTrackâ„¢Earn credit towards a Masterâ€™s degree
- University CertificatesAdvance your career with graduate-level learning

Back to Fibonacci Numbers and the Golden Ratio

stars

1,056 ratings

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...

NR

Aug 17, 2019

Excellent exposition of a fascinating subject.

Very well organized, including a handout with the course material.

Many thanks to Prof. Chasnov and the HKUST for this worthwhile opportunity!

BH

Oct 28, 2020

Great course concept for about one of the most intriguing concepts in the mathematical world, however I found it on the difficult side especially for those who find math as a challenging topic.

Filter by:

By Vishal G

â€¢Dec 3, 2018

I came into this course thinking that it was going to be something relatively basic - being my first MOOC I wanted to do something which is new but not too difficult so I can see what online courses are like.

The first week is mostly easy but was still interesting with its novel application of recursion to seemingly difficult problems like the staircase one but the second week stands out mostly for both its difficulty and use of other areas of math to prove results in far better ways - Cassini's identity with the Q matrix in particular. The second week's content is manageable and the entire course only requires understanding of mathematical induction and the manipulation of matrices, both of which have supplementary material to complement the course for those who don't know it or have forgotten how to do it. The exercises set are some of the most insightful I've seen - Catalan's identities proof in particular. Solving that one is worthy of commendation. A system of hints or solutions may be useful for the more difficult exercises like that one but it's mostly an exception among the others. The second week is mostly isolated to the others, mostly for the introduction of useful algebraic results which tie Fibonacci numbers together rather than for the end goal which is outlined in week 3 with the construction of the Fibonacci and golden spirals. I personally found week 2 the best but week 3 was the main part of the course. It stays mostly easier than week 2 except the difficulty spike in the first exercise about the Eye of God, I recommend watching the video which comes after it before attempting that exercise.

Overall, this is an enjoyable course with some very surprising results and manages to go deeper into seemingly simple concepts like the Fibonacci numbers than expected. Thanks Professor Chasnov for a great first course which is definitely suitable to beginners with the way it eases students into the concepts introduced and the great explanations. Recursion in particular is explained extremely well.

By Anudha C K

â€¢Mar 23, 2019

Absolutely loved the content discussed in this course! It was challenging but totally worth the effort. Seeing how numbers, patterns and functions pop up in nature was a real eye opener.

By Peter M

â€¢May 7, 2018

Light, relatively easy and entertaining. Well done Prof. Chasnov

By John R

â€¢Jul 12, 2020

Someone has said that God created the integers; all the rest is the work of man. After seeing how the Fibonacci numbers play out in nature, I am not so sure about that. A very enjoyable course.

By Matthew T

â€¢Aug 31, 2018

Very interesting and well presented mathematics course. Amazing how many applications and relationships the Fibonacci Sequence has. Some of the formulas could have been explained in a little more detail as they were incredibly difficult to get my head around (I am an A-Level student).

By Bethany B

â€¢Jan 3, 2020

I thoroughly enjoyed this course! I was looking for something fun to study over my 3-week winter break (I'm a student at IU), and thought this looked intriguing. Indeed, I learned so many fascinating things that heightened my love for and appreciation of the complex (literally complex, as well) world of mathematics. I was so continually interested that I just had to keep going with the course until the end! I have a fairly strong background in math (I've taken Calc I, II, and III, and Linear Algebra) so I was able to do the quizzes and most of the hw problems (though I never completely understood proofs by induction). I had no idea that there were this many relationships between the Fibonacci Numbers and the Golden Ratio, and that these intertwined ideas show up elsewhere in real-world situations. As an aspiring professional musician, I'd be very interested to see if and how these show up in music - I myself may have to write something that utilizes and highlights the golden ratio! I've encouraged many of my friends to take this course - it's highly rewarding and one of my favorite courses I've ever taken. The professor was wonderful, and clearly very enthusiastic about these mathematical phenomena. Thank you to all who put this together!

By Ching K R L

â€¢Feb 16, 2020

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!

By Boudokhane M

â€¢Jan 24, 2019

This course has been really fun and informative. I have learned a lot about Fibonacci numbers which i never considered to be that important. At first i thought they were just some quirky numbers and never looked further because they are usually just mentioned in our regular courses but nothing more. This course showed how important they are, their occurrence in nature, their real world application and a multitude of ways to find them and represent them.

The format of this course was also excellent, the videos are rich in content but still entertaining and easy to follow, some of the later ones are worth watching several times because they are quite packed in information. The problems are both challenging and amusing.

overall, i'm really satisfied with this course and i would definitely recommend it to anyone interested in math.

By See M L

â€¢Jan 3, 2020

Very interesting course and the topic is covered comprehensively. Though this course is run every week, the instructor replies promptly to any questions asked. Most of the exercises are proofs which was good practice, though I found them rather too challenging for my standard. However, this is inevitable, as these proofs directly relate to formulas which will be used and are not just random exercises and thus I am grateful for the solutions provided by the pdf. Any gaps are also filled in well by the lecture notes pdf provided, though some proofs will be more challenging to understand. Great course for anyone interested in this rather niche area of mathematics.

By Shreyans G

â€¢Dec 29, 2020

This course was extremely interesting and entertaining. I thoroughly enjoyed this course on the Fibonacci Sequence and the Golden Ratio and its implications to real life. Professor Jeffery has brilliantly explained the concepts. His accent and his style of teaching best suited my style of learning. He is one of the best teachers I've learned from in my life. Extremely sad that this course got over so quickly as I enjoyed learning it. I highly recommend this course to any student who enjoys even the basics of mathematics. Cheers Professor! Thank you for being such a great teacher.

By Olalla C A

â€¢Apr 30, 2020

It is a very nice course. Although I am a scientist and have a good Mathematics background I have found it sufficiently challenging. I think the lecturer achieves a good balance of explaining basic concepts for those less knowledgeable but also advancing sufficiently fast to the less trivial material. I was a bit disappointed not to have a bit more about applications to the structure of plants, but happy to have understood how before we were even around nature had already worked out the Golden ratio!

By Rishabh J

â€¢Aug 28, 2018

This is a very good course to introduce Fibonacci numbers, their appearances in nature, and their fascinating properties. The instructor explains everything very well, and his overall speed of teaching is also quite flexible and aptly manged. The course can serve as a good one to introduce one to the idea of advanced mathematics, or to assist one to appreciate the intricate ways in which mathematics appears in the nature.

By Bill C

â€¢Sep 3, 2018

Really enjoyed the course. The instructor makes the material very interesting and explains the Fibonacci numbers and their different identities and uses very clearly and easy to understand. The course gave me a better understanding of the Fibonacci numbers and their applications than I previously had. The course also made me think the mathematics through thoroughly and carefully. Highly recommended.

By aBa

â€¢Apr 8, 2020

Dr. Chasnov does an incredible job at explaining many insightful aspects of the Fibonacci sequence and the golden ratio. I came to this class already knowing quite a bit (so I thought) about these topics, but I came away learning a great deal more. And now I am excited to explore further topics that this course led me to think about regarding the Fibonacci numbers.

By Alok K

â€¢Feb 11, 2017

I have learnt so much about Fibonacci numbers and the Golden Ratio in this class. This is so much more information that what I'd learnt way back in school or when solving Fibonacci based problems in Computer Science courses in college. I loved the content and Chasnov's teaching as well. Would love to do another course by Jeffrey R. Chasnov and this university :)

By Ursula B

â€¢Apr 10, 2020

This is a really interesting course. The homework tasks to require a knowledge of proof which is not taught in schools in the UK, generally. However, you do not need this to access the course and to learn some amazing features that link the Fibonacci, Lucas numbers, the Golden Ratio and the Golden Angle.

It is a great course! I strongly recommend it.

By Luis F B G

â€¢Oct 12, 2019

Nice and interesting course to know topics as Fibonacci and Golden Ratio and their relations

very well expalined and with no overwhelming Maths, very comprehensive and self contained

This course help me to solve some doubts and clarify concepts regarding mainly relation Fibonacci and Golden Ratio.

I strongly recommend this course

By Jaydev S R 1

â€¢Jun 25, 2020

I would say that it is a well-built and well-structured course. The topics it covers are interesting. The instructor also teaches it very well. The problems in this course can be hard for someone who lacks a good mathematical background, but it should not discourage someone because it is a delightful course.

By Nitika B

â€¢Jul 13, 2020

Mr Chasnov has made this an interesting and challenging course. I especially enjoyed the part where he shows the sunflower spirals, counts and shows Fibonacci nos. at work in nature, as well as connects the golden ratio concept in that practical example. Would recommend to math enthusiasts :)

By Aishwarya B

â€¢Jul 12, 2020

A very interesting course that delves deep into the mathematics of the golden ratio and its relation to the Fibonacci numbers. I found the lectures very enjoyable. The mathematical problems, while slightly challenging turned out to be elegant and in keeping with the essence of the course.

By Christos G

â€¢Nov 17, 2017

The course is outstanding! Although I suspect it does not offer much in career pursuit, it definitely offers pure knowledge! Fibonacci sequence exists all around in nature. Is it like this or we, as humans, tend to look for patterns? Did we invent or did we discover mathematics?

By Venugopal E

â€¢Jun 17, 2020

VERY INTERESTING COURSE RELATING TO NATURE, THANKS A LOT, AS A ENGINEER I WILL IMPLEMENT THE GOLDEN RATIO IN MY LIFE, VERY MUCH THANKS TO THE COURSE INSTRUCTOR, PROF, R. JEFFREY CHASNOW FOR HIS EXCELLENCE, I FEEL THE REAL WORK IN THE LECTURE, THANK YOU SIR, THANK YOU TO ALL

By I R

â€¢Apr 11, 2020

This course was very clear, and teaches you everything you could possibly want to know about the Fibonacci sequence and the Golden Ratio. It was structured in a way that I could fit it in whenever I could throughout a couple of weeks, too, making it easier to do.

By Dhandhukiya R R

â€¢Apr 26, 2020

I would like to appreciate this course. Prof. Jeffrey R. Chasnov made it more interesting for me. I always want to find this type of interesting courses. As an Assistant Prof. it's a very interesting things to do it. Prof. has done excellent job. Thanks a lot!!

By Andreas Z

â€¢Mar 23, 2019

This is a wonderful course that provides you with insights into a topic much neglected by the university mathematics curriculum but which is of great importance to art and nature. I recommend everyone with an interest in mathematics to follow this eye-opener.

- AWS Cloud A Practitioner's Guide
- Basics of Computer Programming with Python
- Beginners Python Programming in IT
- Developing Professional High Fidelity Designs and Prototypes
- Get Google CBRS-CPI Certified
- Introduction to MATLAB Programming
- Learn HTML and CSS for Building Modern Web Pages
- Learn the Basics of Agile with Atlassian JIRA
- Managing IT Infrastructure Services
- Mastering the Fundamentals of IT Support

- Basics of Computer Programming with Python
- Beginners Python Programming in IT
- Building a Modern Computer System from the Ground Up
- Getting Started with Google Cloud Fundamentals
- Introduction to Cryptography
- Introduction to Programming and Web Development
- Introduction to UX Design
- Learn HTML and CSS for Building Modern Web Pages
- Mastering the Fundamentals of IT Support
- Utilizing SLOs & SLIs to Measure Site Reliability

- Building an Agile and Value-Driven Product Backlog
- Foundations of Financial Markets & Behavioral Finance
- Getting Started with Construction Project Management
- Getting Started With Google Sheets
- Introduction to AI for Non-Technical People
- Learn the Basics of SEO and Improve Your Website's Rankings
- Mastering Business Writing
- Mastering the Art of Effective Public Speaking
- Social Media Content Creation & Management
- Understanding Financial Statements & Disclosures