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
Fibonacci Numbers and the Golden Ratio
proposé par
Fibonacci Numbers and the Golden Ratio
The Hong Kong University of Science and Technology
Programme du cours : ce que vous apprendrez dans ce cours
Semaine
1Fibonacci: It's as easy as 1, 1, 2, 3
In 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 vidéos (Total 55 min), 9 lectures, 4 quiz
7 vidéos
Course Overview6 min
The Fibonacci Sequence8 min
The Fibonacci Sequence Redux7 min
The Golden Ratio8 min
Fibonacci Numbers and the Golden Ratio6 min
Binet's Formula10 min
Mathematical Induction7 min
9 lectures
Welcome and Course Information10 min
Get to Know Your Classmates10 min
Fibonacci Numbers with Negative Indices10 min
The Lucas Numbers10 min
Neighbour Swapping10 min
Some Algebra Practice10 min
Linearization of Powers of the Golden Ratio10 min
Another Derivation of Binet's formula10 min
Binet's Formula for the Lucas Numbers10 min
4 exercices pour s'entraîner
Diagnostic Quiz10 min
The Fibonacci Numbers6 min
The Golden Ratio6 min
Week 120 min
Semaine
2Identities, sums and rectangles
In 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 vidéos (Total 65 min), 10 lectures, 3 quiz
9 vidéos
The Fibonacci Q-matrix11 min
Cassini's Identity8 min
The Fibonacci Bamboozlement6 min
Sum of Fibonacci Numbers8 min
Sum of Fibonacci Numbers Squared7 min
The Golden Rectangle5 min
Spiraling Squares3 min
Matrix Algebra: Addition and Multiplication5 min
Matrix Algebra: Determinants7 min
10 lectures
Do You Know Matrices?10 min
The Fibonacci Addition Formula10 min
The Fibonacci Double Index Formula10 min
Do You Know Determinants?10 min
Proof of Cassini's Identity10 min
Catalan's Identity10 min
Sum of Lucas Numbers10 min
Sums of Even and Odd Fibonacci Numbers10 min
Sum of Lucas Numbers Squared10 min
Area of the Spiraling Squares10 min
3 exercices pour s'entraîner
The Fibonacci Bamboozlement6 min
Fibonacci Sums6 min
Week 220 min
Semaine
3The most irrational number
In 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 vidéos (Total 61 min), 8 lectures, 3 quiz
8 vidéos
The Golden Spiral8 min
An Inner Golden Rectangle5 min
The Fibonacci Spiral6 min
Fibonacci Numbers in Nature4 min
Continued Fractions15 min
The Golden Angle7 min
A Simple Model for the Growth of a Sunflower8 min
Concluding remarks4 min
8 lectures
The Eye of God10 min
Area of the Inner Golden Rectangle10 min
Continued Fractions for Square Roots10 min
Continued Fraction for e10 min
The Golden Ratio and the Ratio of Fibonacci Numbers10 min
The Golden Angle and the Ratio of Fibonacci Numbers10 min
Please Rate this Course10 min
Acknowledgments10 min
3 exercices pour s'entraîner
Spirals6 min
Fibonacci Numbers in Nature6 min
Week 320 min
4.7
Meilleurs avis
par GM•Mar 16th 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
par 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.
Enseignant
Jeffrey R. Chasnov
ProfessorDepartment of Mathematics
À propos de The Hong Kong University of Science and Technology
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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