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.
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.
Rated 5 out of
5
by
mironta from
Golden mean of entertainment and maths
Interesting and fun presentation with enough detail to keep it challenging.
Date published: 2024-04-30
Overview
About
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.
Trailer
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
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
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
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
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
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
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
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
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
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
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
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
