The Hong Kong University of Science and Technology
Fibonacci Numbers and the Golden Ratio
The Hong Kong University of Science and Technology

Fibonacci Numbers and the Golden Ratio

Taught in English

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37,795 already enrolled

Course

Gain insight into a topic and learn the fundamentals

Jeffrey R. Chasnov

Instructor: Jeffrey R. Chasnov

Top Instructor

4.8

(1,124 reviews)

Beginner level

Recommended experience

9 hours to complete
3 weeks at 3 hours a week
Flexible schedule
Learn at your own pace

What you'll learn

  • Fibonacci numbers

  • Golden ratio

  • Fibonacci identities and sums

  • Continued fractions

Details to know

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Assessments

10 quizzes

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There are 3 modules in this course

We learn about the Fibonacci numbers, the golden ratio, and their relationship. We derive the celebrated Binet's formula, which gives an explicit formula for the Fibonacci numbers in terms of powers of the golden ratio and its reciprocal. This formula can be used to calculate the nth Fibonacci number without having to sum the preceding terms in the sequence.

What's included

6 videos8 readings4 quizzes

We learn about the Fibonacci Q-matrix and Cassini's identity. Cassini's identity is the basis for the famous dissection fallacy, 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 spiraling squares. This image is a drawing of a sequence of squares, each with side lengths equal to the golden ratio conjugate raised to an integer power, creating a visually appealing and mathematically intriguing pattern.

What's included

9 videos10 readings3 quizzes

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 recognize 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, which is related to the golden ratio, and use it to model the growth of a sunflower head. The 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.

What's included

8 videos8 readings3 quizzes

Instructor

Instructor ratings
4.8 (322 ratings)
Jeffrey R. Chasnov

Top Instructor

The Hong Kong University of Science and Technology
16 Courses206,794 learners

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Recommended if you're interested in Math and Logic

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