Golden Ratio and Fibonacci Series

Introduction: The Fibonacci Series The Fibonacci Series is a sequence of numbers first created by Leonardo Fibonacci (fibo-na-chee) in 1202. It is a deceptively simple series, but its ramifications and applications are nearly limitless. It has fascinated and perplexed mathematicians for over 700 years, and nearly everyone who has worked with it has added a new piece to the Fibonacci puzzle, a new tidbit of information about the series and how it works. Fibonacci mathematics is a constantly expanding branch of number theory, with more and more people being Yellow flower with 8 petals, a Fibonacci rawn into the complex subtleties of Number. Fibonacci’s legacy. The first two numbers in the series are one and one. To obtain each number of the series, you simply add the two numbers that came before it. In other words, each number of the series is the sum of the two numbers preceding it. Note: Historically, some mathematicians have considered zero to be a Fibonacci number, placing it before the first 1 in the series. It is known as the zeroth Fibonacci number, and has no real practical merit. We will not consider zero to be a Fibonacci number in our discussion of the series. http://library. thinkquest. rg/27890/mainIndex. html Series: (0,) 1, 1, 2, 3, 5, 8, 13, 21, 34, 55… EXAMPLE IN NATURE Fibonacci Series–Activity 1 Using a piece of graph paper, draw a spiral using the Fibonacci series. Starting in the center of the page, draw a 1 X 1 square, next to it draw another 1 X 1 square, After, draw 2 X 2 squares touching the last two squares, Then continue to add on squares until the graph paper is filled. To finish the spiral draw arcs (quarter circles) in each square starting in the center and working outward. Do you notice any similarity to the spiral you have drawn and the image of the shell?

Fibonacci Series–Activity 2 Take the Fibonacci sequence listed below and divide each pair of number and record the results in the table. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 combo results 1/1 2/1 3/2 5/3 8/5 13/8 21/13 34/21 55/34 89/55 What do you notice? This is called the golden ratio. (Phi is 1·61803398874… ) This is another special number that appears in the world around us and (as you saw) is related to the Fibonacci series. Fibonacci Series–Activity 3 Each hand has how many digits? _______________ Each finger has how many bones? _______________ Each finger has how many joints between the just inger bones themselves? _______________ Each finger has how many finger nails? What pattern do you see? _______________ _______________________________ Now pick one finger Measure the length of each of the three segments; this is the easiest to do if the finger is bent. Longest _______________cm Medium _______________cm Shortest _______________cm Now divide the longest length by the medium length, what do you get? ________________ Now divide the medium length by the shortest length, what do you get this time? ___________ What is the ratio? ____________________________________

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