Fibonacci Numbers and the Golden Ratio
All-Time Top 100The Hong Kong University of Science and Technology via Coursera
- Provider Coursera
- Cost Free Online Course (Audit)
- Session Upcoming
- Language English
- Certificate Paid Certificate Available
- Duration 3 weeks long
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Overview
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The course culminates in an explanation of why the Fibonacci numbers appear unexpectedly in nature, such as the number of spirals in the head of a sunflower.
Syllabus
-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.
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.
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 sunflower.
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- #1 in Subjects / Mathematics
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Reviews for Coursera's Fibonacci Numbers and the Golden Ratio Based on 44 reviews
- 5 stars 93%
- 4 stars 7%
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Write a review- 1
With relatively simple tools and deep reasoning you'll see that some irrational numbers are more irrational than others and the Golden Ratio is the most irrational of all!
I found some of the proofs to be a bit challenging but excellent course documentation and forums provided help where needed. The final lecture on the spiral pattern of sunflower seeds was truly memorable.
Bottom line - - a short course but a joy for the mathematically inclined.
In short, great course - highly recommended!
Thanks a lot Dr Chasnov and Hong Kong University )
I really enjoyed this short course and I recommend it if you are interested in the Fibonacci numbers and its link with the golden ratio. The exercises are well done and the lessons short but clear. There are several interesting things exiting in mathematical and in nature. I really enjoyed.
This course help me to understand Fibonacci numbers, Golden ratio and their relations
Very comprehensive and self contained, useful information in all videos
I strongly recommend this course
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