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.
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Fibonacci Numbers and the Golden Ratio
香港科技大学
课程信息
5,289 次近期查看
您将学到的内容有
Fibonacci numbers
Golden ratio
Fibonacci identities and sums
Continued fractions
您将获得的技能
Elementary MathematicsDifference EquationsElementary AlgebraRecreational MathematicsDiscrete Mathematics
教学大纲 - 您将从这门课程中学到什么
周
1完成时间为 3 小时
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 个视频 （总计 55 分钟）, 9 个阅读材料, 4 个测验
7 个视频
The Fibonacci Sequence8分钟
The Fibonacci Sequence Redux7分钟
The Golden Ratio8分钟
Fibonacci Numbers and the Golden Ratio6分钟
Binet's Formula10分钟
Mathematical Induction7分钟
9 个阅读材料
Welcome and Course Information1分钟
Get to Know Your Classmates1分钟
Fibonacci Numbers with Negative Indices5分钟
The Lucas Numbers5分钟
Neighbour Swapping10分钟
Some Algebra Practice10分钟
Linearization of Powers of the Golden Ratio10分钟
Another Derivation of Binet's formula10分钟
Binet's Formula for the Lucas Numbers10分钟
4 个练习
Diagnostic Quiz5分钟
The Fibonacci Numbers15分钟
The Golden Ratio15分钟
Week 130分钟
周
2完成时间为 3 小时
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 个视频 （总计 65 分钟）, 10 个阅读材料, 3 个测验
9 个视频
Cassini's Identity8分钟
The Fibonacci Bamboozlement6分钟
Sum of Fibonacci Numbers8分钟
Sum of Fibonacci Numbers Squared7分钟
The Golden Rectangle5分钟
Spiraling Squares3分钟
Matrix Algebra: Addition and Multiplication5分钟
Matrix Algebra: Determinants7分钟
10 个阅读材料
Do You Know Matrices?
The Fibonacci Addition Formula10分钟
The Fibonacci Double Index Formula10分钟
Do You Know Determinants?
Proof of Cassini's Identity10分钟
Catalan's Identity10分钟
Sum of Lucas Numbers5分钟
Sums of Even and Odd Fibonacci Numbers5分钟
Sum of Lucas Numbers Squared5分钟
Area of the Spiraling Squares10分钟
3 个练习
The Fibonacci Bamboozlement15分钟
Fibonacci Sums15分钟
Week 230分钟
周
3完成时间为 3 小时
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 个视频 （总计 61 分钟）, 8 个阅读材料, 3 个测验
8 个视频
An Inner Golden Rectangle5分钟
The Fibonacci Spiral6分钟
Fibonacci Numbers in Nature4分钟
Continued Fractions15分钟
The Golden Angle7分钟
A Simple Model for the Growth of a Sunflower8分钟
Concluding remarks4分钟
8 个阅读材料
The Eye of God10分钟
Area of the Inner Golden Rectangle10分钟
Continued Fractions for Square Roots10分钟
Continued Fraction for e10分钟
The Golden Ratio and the Ratio of Fibonacci Numbers10分钟
The Golden Angle and the Ratio of Fibonacci Numbers10分钟
Please Rate this Course1分钟
Acknowledgments
3 个练习
Spirals15分钟
Fibonacci Numbers in Nature15分钟
Week 330分钟
4.8
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来自Fibonacci Numbers and the Golden Ratio的热门评论
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.
讲师
Jeffrey R. Chasnov
ProfessorDepartment of Mathematics
关于 香港科技大学
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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