返回到 Fibonacci Numbers and the Golden Ratio

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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....

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 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.

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創建者 Peter M

•May 07, 2018

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

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 Bethany B

•Jan 03, 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!

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 Lim S M

•Jan 03, 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.

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 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 Luis F B G

•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

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 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 K B

•Oct 24, 2016

In this course so far I have been able to see what different types of patterns are and I have been able to see what kind of counting ways there are. I have also learned that the Fibonacci Sequence is everywhere in nature so I really like this course.

創建者 Soham J

•Apr 27, 2019

This course is absolutely out of the world. It truly showed me how much I didn't know and how fascinating nature is. The best part is that anyone with basic understanding of algebra can take this course. I had a really fun time here.

創建者 Nilson R R

•Aug 18, 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!

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 David P L

•Sep 29, 2019

I really enjoyed this class. The video lectures were clear and fascinating, and the math assignments were well-chosen and reinforced learning of the material presented in the lectures.

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 Hasan J

•Dec 04, 2016

Good course for introduction to Fibonacci Numbers. Should include more introduction lectures such as group theory, category theory, type theory, number theory, and algorithms.

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!

創建者 Thomas S

•Jul 15, 2019

Fantastically explained, and very helpful when difficult in parts. Would recommend to anyone who is interested in maths, not just Fibonacci.

創建者 Ng T E

•May 30, 2019

One of the many marvels of mathematics is that of the Fibonacci Numbers & the Golden Ratio.

Week 2 was somewhat challenging, but doable.

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 Himanshu S

•Sep 13, 2016

Very intriguing and interesting lectures. These got me thinking and venture into Fibonacci numbers and a bit of sacred geometry too.

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 Tumul K

•Jul 04, 2017

A thoroughly enjoyable course, a bit too fast paced and quite difficult some times, but the overall course is quite exciting.