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Section 5-5 The Fibonacci Sequence and the Golden Ratio Lesson Objectives • Work with the Fibonacci sequence. • Understand the golden ratio. • See relationships between the Fibonacci sequence and the golden ratio. • See the golden ratio as a ratio on a line segment. Who Was Fibonacci? ~ Born in Pisa, Italy in 1175 AD ~ Full name was Leonardo Pisano ~ Grew up with a North African education under the Moors ~ Traveled extensively around the Mediterranean coast ~ Met with many merchants and learned their systems of arithmetic ~ Introduced the Hindu-Arabic number system into Europe The Fibonacci Sequence The solution of the rabbit problem leads to the Fibonacci sequence. Here are the first thirteen terms of the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 Notice the pattern. After the first two terms (both 1), each term is obtained by adding the two previous terms. Recursive Formula for Fibonacci Sequence If Fn represents the Fibonacci number in the nth position in the sequence, then F1 1 F2 1 Fn Fn2 Fn1 , for n 3. The Fibonacci Numbers in Nature Lilies and irises = 3 petals Corn marigolds = 13 petals Buttercups and wild roses = 5 petals Black-eyed Susan’s = 21 petals The Fibonacci Numbers in Nature • The Fibonacci numbers can be found in pineapples and bananas • Bananas have 3 or 5 flat sides • Pineapple scales have Fibonacci spirals in sets of 8, 13, 21 The Golden Ratio: Ratio of Fibonacci Consider the quotients of successive Fibonacci numbers and notice a pattern. 1 2 3 5 1, 2, 1.5, 1.66..., 1 1 2 3 8 13 21 1.6, 1.625, 1.615384 5 8 13 These quotients seem to go toward 1.618. 1 5 In fact, they approach . 2 This number is known as the golden ratio. The Golden Ratio: Ratio of Fibonacci The Golden Ratio: Ratio of Line Segment Let b=1. Solve for a. Fibonacci and Golden Ratio • Take your graph paper and draw a 1 x 1 square in the middle of the page. • Draw another 1 x 1 square to its right. • Draw a 2 x 2 square adjacent to and below the rectangle. • Draw a 3 x 3 square adjacent to and to the left of that rectangle. • Continue drawing squares in a clockwise direction until your paper runs out of space Fibonacci and Golden Ratio Your drawing should look like this: Constructing Golden Rectangles The first three rectangles you should measure are highlighted in yellow below. The Golden Spiral How is the Golden Spiral constructed? Example of Spiral in Nature: Shell of Chambered Nautilus Golden Rectangles Are All Around Us! • • • • • • • • The aspect ratio of YouTube videos Ratio of length over width of photographs Computer screens Smartphone screens Tablets Credit Cards Playing Cards Human Body Examples of the Golden Ratio • On the next pages you will see examples of the Golden Ratio c • Many of them have a gauge, called the Golden Mean Gauge, superimposed over the picture. • This gauge was developed by Dr. Eddy Levin DDS, for use in dentistry and is now used as the standard for the dental profession. Golden Mean Gauge: Invented by Dr. Eddy Levin DDS The Bagdad City Gate Dome of St. Paul: London, England The Great Wall of China Windson Castle ……and more!!!