Fibonacci Numbers and the Golden Ratio

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. Finally, discover that when you divide consecutive Fibonacci numbers you get closer and closer to the golden ratio.

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.

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.

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!

See 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.

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—a Fibonacci progression enthusiast. Lucas proposed a related sequence, now known as Lucas numbers. Compare and contrast Lucas numbers with Fibonacci numbers. Then, generate other amazing patterns.

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.

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.

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.

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.

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.

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.