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# Fibonacci Numbers and the Golden Ratio

## Overview

##### Class Central Tips
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.

https://www.math.ust.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, an explicit formula for the Fibonacci numbers in terms of powers of the golden ratio and its reciprical.
• Identities, sums and rectangles
• We learn about the Fibonacci Q-matrix and Cassini's identity. Cassini's identity is the basis for a famous dissection fallacy colourfully named 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 spiralling squares.
• 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 recognise 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, related to the golden ratio, and use it to model the growth of a sunflower head. 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 204 reviews

Start your review of Fibonacci Numbers and the Golden Ratio

• Ryan Lam completed this course, spending 3 hours a week on it and found the course difficulty to be medium.

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...
• 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.
• Anonymous
Its very usefull to learn.Really I had and spend time for learning this...Thanks for your great effect to teach us........I am really proud by learning in this.......thanks for making my time more usefull
• Anonymous
the negative point of this course was the huge amount of numerical mathematics and much less geometrical information which made it hard to link these two parts. especially the matrix lessons seemed some how not that much relevant. another thing that...
• Anonymous
This is a very exciting course if you find numbers fascinating. Although I passed the initial diagnostic test and all the tests and quizzes I found some of the maths (proof by induction and continued fractions) completely beyond me. I would nevertheless highly recommend this course . It is possible to understand and enjoy the main ideas even if you can't follow all the mathematical reasoning. The instructor is excellent and his engagement with the beauty of numbers is inspiring, thanks.
• Anonymous
I absolutely enjoyed this course taught by Professor Chasnov. The Prof clearly has a passion for mathematics and his enthusiasm for the beautiful formulas and relationships was contagious. Due to other commitments at this time I was not fully able to participate in all of the assignments, but I look forward to coming back to these at some point in the future. Thanks Professor Chasnov for creating this course!
• Anonymous
This course is much interesting and helps me gain more knowledge about the golden ratio and Fibonacci's sequence. I am also able to understand the geometric constructions of golden rectangles and golden spirals. By this course, I can become more involved in the international trends of Mathematics and its applications.
• Anonymous
This course is much interesting and helps me gain more knowledge about the golden ratio and Fibonacci's sequence. I am also able to understand the geometric constructions of golden rectangles and golden spirals. By this course, I can become more involved in the international trends of Mathematics and its applications.
• 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
I thanks to Professor Jeff and The University of Hong Kong for this course!

I reccomend because it has mathematics dont taughted in the universities and is very, very important to know the Fibonacci Numbers and your relationship with the Nature!!!
• Dominic Tam
All around interesting. Not a lot of prerequisites, and the course provides additional information if you are not familiar with proof by induction or elementary linear algebra. Very informative and nice layout. Had a lot of fun with the course.
• Anonymous
I really enjoyed this course, it was very clear and with the proper mathematical rigor, not too complex but not too simple, very complete course. The teacher was very clear and I really liked the enthusiasm he puts into teaching.
• Anonymous

Anonymous completed this course.

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
Excellent course. Instructor very personable and understandable but he does not compromise on mathematical standards. Any student wishing to fully understand all of the exercises and arguments will need to put in some effort.
• Anonymous
Excellent course. Instructor very personable and understandable but he does not compromise on mathematical standards. Any student wishing to fully understand all of the exercises and arguments will need to put in some effort.
• Anonymous
Good course. I have learned a lot from this training. Especially about the nature of Fibonacci. I hope there is other course from THE HONG KONG UNIVERSITY OF SCIENCE AND TECHNOLOGY especially for mathematics education.
• Anonymous
A wonderful course of Fibonacci numbers from a wonderful instructor that i enjoyed a lot. I hope if there were more than one course that resembles this one in its beauty.
• Anonymous
Its interesting to learn....and weekly assignment helped to solve more answer.Thanks for your valuable time in this pandamic situation.....really its very helpfull to learn
• Anonymous
I thought the course was great and I had a lot of fun participating in the exercises and to get a greater insight into the nature of the Fibonacci Numbers. Thank you
• Anonymous
This was a very interesting course that could be understood without the need to know advanced mathematics. The teacher did a very good job of explaining everything.

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