Learn the mathematics behind the Fibonacci numbers, the golden ratio, and their relationship to each other. These topics may not be taught as part of a typical math curriculum, but they contain many fascinating results that are still accessible to an advanced high school student.
The course culminates in an exploration of the Fibonacci numbers appearing unexpectedly in nature, such as the number of spirals in the head of a sunflower.
Download the lecture notes from the link
https://www.math.hkust.edu.hk/~machas/fibonacci.pdf
Watch the promotional video:
https://youtu.be/VWXeDFyB1hc
Fibonacci Numbers and the Golden Ratio
The Hong Kong University of Science and Technology via Coursera
Overview
Syllabus
- 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, 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.
- Identities, sums and rectangles
- 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.
- 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 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.
Taught by
Jeffrey R. Chasnov
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Reviews
4.9 rating, based on 258 Class Central reviews
4.8 rating at Coursera based on 1146 ratings
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I recently had the pleasure of enrolling in the "Fibonacci and Golden Ratio" course, and I must say it was an enlightening and intellectually stimulating experience. This course offers a captivating exploration of two of nature's most intriguing mat…
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A fun and engaging introductory mathematics course! Professor Chasnov did a really good job on introducing this topic! The exercises focused on the Fibonacci number and its counterpart, the Lucas Number, and it is aesthetically pleasing to see the c…
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Я прошел курс и остался им очень доволен. Объяснения были ясными и доступными, что значительно упростило процесс обучения. Инструкторы предоставили полезные материалы и примеры, которые помогли лучше усвоить материал. Атмосфера на занятиях была поддерживающей и мотивирующей. Я бы с радостью прошел еще больше таких курсов, так как они действительно способствуют глубокому пониманию темы. Спасибо за качественную работу! Надеюсь, в будущем появятся новые курсы, которые помогут развивать навыки и углублять знания. Рекомендую всем, кто хочет учиться эффективно и с удовольствием!
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I found this course most interesting as it relates mathematics to real-life biology. Fibonacci numbers, Lucas numbers, golden ratio, golden rectangle and what to say, just enjoy and get knowledge from this course. The positive part is that it is not at all lengthy and time spent in this course is worth it.
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I recently completed the Fibonacci Numbers and Golden Ratio online course and found it to be a comprehensive and engaging exploration of these fundamental mathematical concepts. The course expertly covered the history, derivation, and practical appl…
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The Fibonacci course is truly outstanding, offering a comprehensive and engaging exploration of the Fibonacci sequence and its applications. The content is well-structured, starting with the basics and gradually delving into more complex concepts. The instructor's explanations are clear and easy to understand, making complex mathematical ideas accessible to learners of all levels. Interactive examples and practical applications help solidify understanding. This course not only enhances mathematical skills but also demonstrates the beauty and ubiquity of Fibonacci numbers in nature, art, and architecture. Highly recommended for anyone interested in mathematics and its real-world relevance.
Thanks for your true effort -
A kurzus felépítése teljes mértékben összhangban volt kitűzött céljával. A szükséges kiegészítő matematikai hátteret a kellő időben és mélységben tárta fel. A tesztkérdések mélysége megfelelt annak, ami biztosította, hogy a tanuló tovább tudjon lépni a következő szintre. Az előadás videókban kellő arányban voltak látványos elemek, és matematikai képletek, ugyanakkor a lényeget ragadták meg. Tetszettek a matematika történelmi utalások is.
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Tive muito prazer em ampliar meu conhecimento sobre a sequência numérica de Fibonacci e suas derivações.
Um tema muito instigante!
O curso é muito bem estruturado, o professor é muito claro em sua explanação, as imagens são atraentes e gradativamente os temas são desenvolvidos.
Os vídeos complementares e a apostila de exercícios foram de grande apoio, e sem eles dificilmente terminaria esse curso.
Sem dúvida, é um ótimo curso! -
O curso apresenta uma matemática bem elaborada e não tão simples como se pensa no início. A dificuldade com as demonstração vai aumentando, mas ainda assim é possível de se compreender com um pouco de estudo. O resultado para o entendimento da sequência de Fibonacci e suas aplicações é muito satisfatório. Ótimo curso.
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Very challenging course overall but truly beautiful to learn about the knowledge! The teacher is brilliant and explains everything leads the course in an organised way. All extra materials are to very good help to refresh on previous knowledge in earlier lessons. Enjoy!
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For first I've known only what fibonacci numbers are but I didn't know about these much I've studied in this course. This course really help me in building the blocks of my knowledge and gave me some ideas regarding the numbers behavior. Thanks for this wonderful course.
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So much to learn, and the teacher explain full detail with math and so many examples an exersices that help you to get all the magic and theory of Fibonacci. this course is as beautiful as the golden number. Super recommended good good good
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Just loved the way it has been taught...and most importantly the beautiful explanation by the respected Professor...gathered much knowledge now regarding the Fibonacci number and it's application...thank you Professor sir and Coursera
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Excellent presentation! It was pleasing to listen to, and view. Math is amazing and so is this professor!!! Thank you for sharing your gift with us all.
~Blessings~ -
I think it's very interesting, amusing and enlightening about anything related to Golden ratio.
You don't have to be an expert in Mathematics to comprehend 100% of it. -
A well structured course on Fibonacci series.A highly enriching and engaging course, offering valuable insights into both theoretical and applied mathematics.
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Iam regular math teacher and surprised to see this kind of Nature related Math.
Thanks to the Professor for great explanation. -
Really interesting course. I took it for my art work, so skipped over some of the math content/homework and still learned a lot
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Wandrers NachtliedI really liked this course, and I think the instructor is great. Everything is well explained, both in algebra as well as in theory and geometrically. Music and visual representations both fit, as well. I find incredible how these patterns are reproduced in nature and in ancient forms, and how they inner correlate numerically with each other, also. I've always wanted to explore better this issue, as I find it very interesting and I know these patterns are used in trading. It is still incredible to think that nature replicated this pattern into many of its designs.
Best regards,
Bruna Bonatto -
While I am sure math enthusiasts liked the course more than I did, I was interested in the golden rule and benefited from the course. I wanted to know where it came from and would like to have seen more everyday applications/appearances of the golden ratio in nature or design. Thank you Professor!
