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.
Fibonacci Numbers and the Golden Ratio
ofrecido por
Universidad Científica y Tecnológica de Hong Kong
Acerca de este Curso
7,033 vistas recientes
Qué aprenderás
Fibonacci numbers
Golden ratio
Fibonacci identities and sums
Continued fractions
Habilidades que obtendrás
Elementary MathematicsDifference EquationsElementary AlgebraRecreational MathematicsDiscrete Mathematics
Programa - Qué aprenderás en este curso
Semana
14 horas para completar
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.
7 videos (Total 55 minutos), 9 lecturas, 4 cuestionarios
7 videos
The Fibonacci Sequence8m
The Fibonacci Sequence Redux7m
The Golden Ratio8m
Fibonacci Numbers and the Golden Ratio6m
Binet's Formula10m
Mathematical Induction7m
9 lecturas
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
4 ejercicios de práctica
Diagnostic Quiz10m
The Fibonacci Numbers15m
The Golden Ratio15m
Week 150m
Semana
24 horas para completar
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.
9 videos (Total 65 minutos), 10 lecturas, 3 cuestionarios
9 videos
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
10 lecturas
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
3 ejercicios de práctica
The Fibonacci Bamboozlement15m
Fibonacci Sums15m
Week 250m
Semana
34 horas para completar
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 head of a sunflower.
8 videos (Total 61 minutos), 8 lecturas, 3 cuestionarios
8 videos
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
8 lecturas
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
3 ejercicios de práctica
Spirals15m
Fibonacci Numbers in Nature15m
Week 350m
50%
comenzó una nueva carrera después de completar estos cursos
14%
consiguió un beneficio tangible en su carrera profesional gracias a este curso
Principales revisiones sobre Fibonacci Numbers and the Golden Ratio
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!
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.
Instructores
Jeffrey R. Chasnov
ProfessorDepartment of Mathematics
Acerca de Universidad Científica y Tecnológica de Hong Kong
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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