Offered By

The Hong Kong University of Science and Technology

About this Course

4.7

253 ratings

•

85 reviews

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
http://bookboon.com/en/fibonacci-numbers-and-the-golden-ratio-ebook
Course Overview video: https://youtu.be/GRthNC0_mrU

Start instantly and learn at your own schedule.

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Suggested: 7 hours/week...

Subtitles: English

Start instantly and learn at your own schedule.

Reset deadlines in accordance to your schedule.

Suggested: 7 hours/week...

Subtitles: English

Week

1In this week's lectures, we learn about the Fibonacci numbers, the golden ratio, and their relationship. We conclude the week by deriving the celebrated Binet's formula, an explicit formula for the Fibonacci numbers in terms of powers of the golden ratio and its reciprical. ...

7 videos (Total 55 min), 9 readings, 4 quizzes

The Fibonacci Sequence8m

The Fibonacci Sequence Redux7m

The Golden Ratio8m

Fibonacci Numbers and the Golden Ratio6m

Binet's Formula10m

Mathematical Induction7m

Welcome and Course Information2m

Get to Know Your Classmates3m

Fibonacci Numbers with Negative Indices10m

The Lucas Numbers10m

Neighbour Swapping10m

Some Algebra Practice10m

Linearization of Powers of the Golden Ratio10m

Another Derivation of Binet's formula10m

Binet's Formula for the Lucas Numbers10m

Diagnostic Quiz10m

The Fibonacci Numbers15m

The Golden Ratio15m

Week 150m

Week

2In this week's lectures, 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. ...

9 videos (Total 65 min), 10 readings, 3 quizzes

Cassini's Identity8m

The Fibonacci Bamboozlement6m

Sum of Fibonacci Numbers8m

Sum of Fibonacci Numbers Squared7m

The Golden Rectangle5m

Spiraling Squares3m

Matrix Algebra: Addition and Multiplication5m

Matrix Algebra: Determinants7m

Do You Know Matrices?

The Fibonacci Addition Formula10m

The Fibonacci Double Index Formula10m

Do You Know Determinants?10m

Proof of Cassini's Identity10m

Catalan's Identity10m

Sum of Lucas Numbers10m

Sums of Even and Odd Fibonacci Numbers10m

Sum of Lucas Numbers Squared10m

Area of the Spiraling Squares10m

The Fibonacci Bamboozlement15m

Fibonacci Sums15m

Week 250m

Week

3In this week's lectures, 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 head of a sunflower.
...

8 videos (Total 61 min), 8 readings, 3 quizzes

An Inner Golden Rectangle5m

The Fibonacci Spiral6m

Fibonacci Numbers in Nature4m

Continued Fractions15m

The Golden Angle7m

A Simple Model for the Growth of a Sunflower8m

Concluding remarks4m

The Eye of God10m

Area of the Inner Golden Rectangle10m

Continued Fractions for Square Roots10m

Continued Fraction for e10m

The Golden Ratio and the Ratio of Fibonacci Numbers10m

The Golden Angle and the Ratio of Fibonacci Numbers10m

Please Rate this Course10m

Acknowledgments10m

Spirals15m

Fibonacci Numbers in Nature15m

Week 350m

4.7

85 Reviewsstarted a new career after completing these courses

got a tangible career benefit from this course

By AK•Mar 23rd 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 BS•Aug 30th 2017

Very well designed. It was a lot of fun taking this course. It's the kind of course that can get you excited about higher mathematics. Sincere thanks to Prof. Chasnov and HKUST.

HKUST - A dynamic, international research university, in relentless pursuit of excellence, leading the advance of science and technology, and educating the new generation of front-runners for Asia and the world....

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