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學生對 香港科技大学 提供的 Fibonacci Numbers and the Golden Ratio 的評價和反饋

278 個評分
97 個審閱


This is a course about the Fibonacci numbers, the golden ratio, and their intimate relationship. In this course, we learn the origin of the Fibonacci numbers and the golden ratio, and derive a formula to compute any Fibonacci number from powers of the golden ratio. We learn how to add a series of Fibonacci numbers and their squares, and unveil the mathematics behind a famous paradox called the Fibonacci bamboozlement. We construct a beautiful golden spiral and an even more beautiful Fibonacci spiral, and we learn why the Fibonacci numbers may appear unexpectedly in nature. The course lecture notes, problems, and professor's suggested solutions can be downloaded for free from Course Overview video:



Aug 18, 2019

Excellent exposition of a fascinating subject.\n\nVery well organized, including a handout with the course material.\n\nMany thanks to Prof. Chasnov and the HKUST for this worthwhile opportunity!


Mar 16, 2017

Finally I studied the Fibonacci sequence and the golden spiral. I used to say: one day I will.\n\nVery interesting course and made simple by the teacher in spite of the challenging topics


1 - Fibonacci Numbers and the Golden Ratio 的 25 個評論(共 92 個)

創建者 Vishal G

Dec 03, 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.

創建者 Massimo C

Jan 17, 2019

Fascinating course about a topic that is not usually taught in maths lessons. It is easy to follow and it has good supporting material.

創建者 Boudokhane M

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.

創建者 Anudha C K

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.

創建者 Andreas Z

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.

創建者 Rajeshree D

Jan 28, 2019


創建者 Rishabh J

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.

創建者 Bill C

Sep 03, 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.

創建者 Anurag O

Aug 22, 2017

A great introduction to Fibonacci numbers and how they relate to the golden ratio.

創建者 Nazzareno ( C

Apr 19, 2018

I liked the introductory video and i look forward to complete this course

創建者 Alok K

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 :)

創建者 Artur C S

Mar 26, 2017

Awesome course. Really liked the optional assignments.

創建者 Marcelo O R d A

Jan 24, 2017


創建者 Meg- n n

Jul 02, 2017


創建者 Bart S

Nov 27, 2016

Great course, please repeat it!

創建者 Alexander A S G

Dec 04, 2017


創建者 JOHN Q

Dec 19, 2016

discussed some stuff I've wondered about for several decades. thanks so much.

創建者 Morena V

Jun 19, 2017

Very diverse and interesting course. Lot's of fields in mathematics are discussed. Very nice assignments and super cool teacher.

創建者 Emmanouil G

Jul 18, 2017

Very good course

創建者 Christos G

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?

創建者 paul s

Nov 18, 2017

Funny and excellent

創建者 Giovanni M

Mar 16, 2017

Finally I studied the Fibonacci sequence and the golden spiral. I used to say: one day I will.

Very interesting course and made simple by the teacher in spite of the challenging topics

創建者 Serhat G

Apr 01, 2018


創建者 Iulian N

Nov 27, 2016

Very interesting course

創建者 Wenhsiu H

Jan 31, 2017

This is a great course. I learn very complete knowledge of fibonacci numbers and the golden ratio. Thanks for professor. Thank you very much!