Class Central is learner-supported. When you buy through links on our site, we may earn an affiliate commission.

The Hong Kong University of Science and Technology

Fibonacci Numbers and the Golden Ratio

The Hong Kong University of Science and Technology via Coursera

Best Course of All Time
Go to class Write review

Overview

Google, IBM & Meta Certificates — All 10,000+ Courses at 40% Off
One annual plan covers every course and certificate on Coursera. 40% off for a limited time.
Get Full Access
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

Tags

Reviews

4.9 rating, based on 274 Class Central reviews

4.8 rating at Coursera based on 1194 ratings

Start your review of Fibonacci Numbers and the Golden Ratio

  • Profile image for KUSHAGRA SINGH
    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.
  • Profile image for Ryan Lam
    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…
  • 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…
  • 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
    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
    It has been pleasure to go through the course. The explainations were absolutely remarkable and the concepts outstanding.Thank you
  • 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
    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
    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!
  • Profile image for Akash Tayal
    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
    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
    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
  • Anonymous
    Este curso me ayudó para ampliar mi panorama de herramientas y conocimientos en torno a la secuencia de Fibonacci. Bastante bueno.
  • Profile image for Sreelakshmi K S
    Sreelakshmi K S
    2
    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.
  • 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
    This was a short, but well-done course about Fibonacci numbers and the golden ratio that greatly enhanced my understanding of the topic. A big thanks to the professor and team for creating the course material. The course length, 3 weeks, makes for a good "filler" course in between other courses or obligations. More people should discover this one!
  • Profile image for Chinonso Christian
    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.
Go to class

Never Stop Learning.

Get personalized course recommendations, track subjects and courses with reminders, and more.

Sign up for free
Someone learning on their laptop while sitting on the floor.