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Fibonacci Numbers and the Golden Ratio

Join “mathemagician” Arthur Benjamin as he reveals the code behind literary thrillers, famous artworks, flower petals, and more in this look at Fibonacci numbers and the golden ratio.
Fibonacci Numbers and the Golden Ratio is rated 5.0 out of 5 by 6.
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Rated 5 out of 5 by from Amazing Course with Amazing Professor! If you have never enjoyed a Professor Arthur Benjamin course previously, be ready to be immersed in pure mathematical energy and teaching brilliance. Dr. Benjamin is the exemplar of riveting teaching style. This course can be enjoyed both as an intensive learning experience or pure pleasure of numerical sequences in nature and life. The Professor is captivating and engaging is his delivery of the 12 lectures on the history and development of Fibonacci numbers and the integration of the Golden Ratio in the arts and its prevalence in nature. This is not a course to be enjoyed while working out, driving or on the periphery of the llearners focus. I highly recommend this course as I do all of Professor Benjamin's earlier Teaching Company courses. I hope to see future courses from this amazing professor!
Date published: 2024-05-25
Rated 5 out of 5 by from Extremely Interesting and Presented with Gusto I had taken Prof. Bejamin's 'Math and Magic' course a couple of years ago and really enjoyed it and his infectious energy and enthusiasm. This course was even more rewarding due to the fascinating topic material. I will definitely be giving it another viewing soon to pick up more details, with the goal to be able to explain some of the concepts to others. Will certainly be checking into some of his other courses. This course is a very engaging experience, taught by a professor whose enthusiasm for his subject would be hard to exceed. Highly recommend!
Date published: 2024-05-25
Rated 5 out of 5 by from Super presentation of math and nature. I really enjoyed this short course because it is packed with interesting information and examples. The course kept my interest all through out the lectures. I recommend this course to math enthusiasts.
Date published: 2024-05-11
Rated 5 out of 5 by from Prof Benjamin delivers another great lecture serie Ok, I'm giving this five stars because this is a beautifully presented series of math lectures. And when I say math I mean REAL math, often at college level, with rigorous proofs. If you do not like, or cannot follow mathematical argumentation I think most of this material will leave you befuddled. For those who have done Prof Benjamin's "Discrete mathematics" course you will be happy to know there is very little overlap with this course. For example, there is the classic induction Binet formula proofs for the Fibonacci and Lucas sequences but there is a bonus proof using infinite vectors that I've never seen before. Another highlight is an amazing solution to John Conway's checker problem using powers of the golden ratio. There are so many math goodies in this short course that you are definitely getting value for money, however, as I said, you have to be someone who is comfortable with algebra to get the most out things.
Date published: 2024-05-08
Rated 5 out of 5 by from Superb Professor Benjamin never fails to educate and entertain. A wonderful teacher and human being.
Date published: 2024-05-06
Rated 5 out of 5 by from Golden mean of entertainment and maths Interesting and fun presentation with enough detail to keep it challenging.
Date published: 2024-04-30
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Overview

Taught by Professor Arthur T. Benjamin of Harvey Mudd College, this course introduces two ubiquitous patterns in nature and human culture: the Fibonacci sequence and the golden ratio. You learn how a simple algorithm produces the Fibonacci sequence, and how this is one of many paths to the golden ratio. Professor Benjamin presents proofs, puzzles, magic tricks, games, and many amazing insights.

About

Arthur T. Benjamin

As a professor, I have always wanted to bring math to the masses. The Great Courses has helped make that dream come true.

INSTITUTION

Harvey Mudd College

Arthur T. Benjamin is the Smallwood Family Professor of Mathematics at Harvey Mudd College. He earned a PhD in Mathematical Sciences from Johns Hopkins University. His teaching has been honored by the Mathematical Association of America, and he was named to The Princeton Review’s list of the Best 300 Professors. He has also served as president of the Fibonacci Association. A professional magician, he is the author of the book The Magic of Math, a New York Times bestseller. He has appeared on numerous television and radio programs and has been featured in Scientific American and The New York Times.

By This Professor

The Joy of Mathematics
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The Secrets of Mental Math
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The Mathematics of Games and Puzzles: From Cards to Sudoku
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Math and Magic
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Fibonacci Numbers and the Golden Ratio
854
Fibonacci Numbers and the Golden Ratio

Trailer

Puzzling Fibonacci Patterns

01: Puzzling Fibonacci Patterns

Learn the simple procedure for generating the Fibonacci sequence and see how it leads to a world of intriguing patterns. Also, get to know Fibonacci himself—a 13th-century mathematician who introduced Hindu-Arabic numerals to Europe and concocted a mathematical problem with rabbits that led to his famous series. Finally, discover that when you divide consecutive Fibonacci numbers you get closer and closer to the golden ratio.

20 min
Proving Perplexing Properties

02: Proving Perplexing Properties

How do we know that some of the patterns examined in the previous lecture hold up for the entire Fibonacci sequence out to infinity? Roll up your sleeves and use different mathematical techniques to prove these conjectures. In a fun diversion, Professor Benjamin, an accomplished magician, uses a remarkable property of the Fibonacci sequence called Cassini’s identity to make a rabbit disappear.

25 min
Applications of Fibonacci Numbers

03: Applications of Fibonacci Numbers

Arrange a strip of various lengths with squares and dominoes (which are twice the length of squares). Discover that the number of tiling combinations follows the Fibonacci sequence. This leads to the concept of combinatorial proofs, which are Professor Benjamin’s favorite type of proof. Examine a related application—Zeckendorf’s theorem—and its interesting ability to convert kilometers to miles.

28 min
Fibonacci Numbers and Pascal’s Triangle

04: Fibonacci Numbers and Pascal’s Triangle

Turn to Pascal’s triangle—a triangular array of numbers with each number the sum of the two directly above it. Explore its many patterns, which include the Fibonacci sequence. Analyze why these patterns arise, using the tiling problem from the previous lecture to explain the presence of Fibonacci numbers. Flip the script by showing that Pascal’s triangle is hidden inside the Fibonacci series!

28 min
A Favorite Fibonacci Fact

05: A Favorite Fibonacci Fact

Discover the professor’s favorite Fibonacci fact, which involves a concept called the greatest common divisor (GCD). Turn back the clock to the ancient Greek mathematician Euclid, who formulated an ingenious method for finding the GCD for any two integers. Then observe the Fibonacci connection emerge by working out the worst-case scenario that requires the maximum number of steps to get an answer.

31 min
Fibonacci Numbers, Lucas Numbers, and Beyond

06: Fibonacci Numbers, Lucas Numbers, and Beyond

Fibonacci’s real name was Leonardo of Pisa. The nickname Fibonacci (“son of Bonacci”) was bestowed in the 1800s and the Fibonacci numbers were given its name by French mathematician Édouard Lucas—an enthusiast for the Fibonacci progression. Lucas proposed a related sequence, now known as Lucas numbers. Compare and contrast Lucas numbers with Fibonacci numbers. Then, generate other amazing patterns, ending up with the golden ratio.

28 min
The Mysterious Golden Ratio

07: The Mysterious Golden Ratio

The golden ratio is a number that even the math-averse can love. A rectangle with length to height equal to the golden ratio is widely considered the most beautifully proportioned of all quadrilaterals. Explore the golden ratio’s many properties and why a friend of Leonardo da Vinci extolled it for its simplicity, irrationality, self-similarity, and its metaphorical link to the Holy Trinity.

31 min
Phi: The Most Irrational Number

08: Phi: The Most Irrational Number

The golden ratio is denoted by the Greek letter phi. See how phi can be computed by a simple, infinitely long continued fraction. When all of the terms in the continued fraction are 1, then the result is phi. Learn how this calculation relates to Fibonacci numbers.

25 min
A Golden Formula for Fibonacci

09: A Golden Formula for Fibonacci

How do you specify any number in the Fibonacci sequence? In 1843, Jacques Philippe Marie Binet described a formula that allows anyone to plug in the position of the desired value—say the 100th Fibonacci number—and get the answer without computing all the preceding numbers. It’s even easier by simplifying the equation with phi. Professor Benjamin proves this theorem and introduces related ideas.

28 min
The Golden Ratio and Geometry

10: The Golden Ratio and Geometry

Astronomer Johannes Kepler regarded the Pythagorean theorem and golden ratio as the two greatest treasures of geometry. Find the link between them in a figure called Kepler’s triangle. Continue your geometric investigations by constructing a golden triangle using only a compass and a straight edge. Then, discover golden proportions in other shapes, including the isosceles triangle and pentagon.

27 min
The Golden Ratio and Benford’s Law

11: The Golden Ratio and Benford’s Law

Many lists of numbers, such as the Fibonacci sequence and the powers of phi, follow a phenomenon called Benford’s law, which holds that the leading digit is most likely to be either 1, 2, or 3. Explore why exponential functions obey this law. Also, see that it applies in data sets that span several orders of magnitude, for example financial accounts, street addresses, and the lengths of rivers.

27 min
Fibonacci and the Golden Ratio Everywhere

12: Fibonacci and the Golden Ratio Everywhere

Why do the Fibonacci sequence and golden ratio appear so often in the natural world? For example, the structure of plants and flowers show a strong preference for irrational numbers, particularly phi. Consider the types of spirals in nature and refute some of the more extravagant claims for the ubiquity of the golden ratio. Professor Benjamin closes with a limerick and a poem, praising Fibonacci numbers and phi.

32 min

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