Section 5 8 Fibonacci Sequence Copyright 2013 2010

What You Will Learn Fibonacci Sequence 5. 8 -2 Copyright 2013, 2010, 2007, Pearson,

Fibonacci Sequence This sequence is named after Leonardo of Pisa, also known as Fibonacci. He was one of the most distinguished mathematicians of the Middle Ages. He is also credited with introducing the Hindu-Arabic number system into Europe. 5. 8 -3 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Fibonacci Sequence 1, 1, 2, 3, 5, 8, 13, 21, … In the Fibonacci sequence, the first two terms are 1. The sum of these two terms gives us the third term (2). The sum of the 2 nd and 3 rd terms give us the 4 th term (3) and so on. 5. 8 -4 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

In Nature In the middle of the 19 th century, mathematicians found strong similarities between this sequence and many natural phenomena. The numbers appear in many seed arrangements of plants and petal counts of many flowers. Fibonacci numbers are also observed in the structure of pinecones and pineapples. 5. 8 -5 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Golden Number : The value obtained from the ratio of the (n + 1) term to the nth term preceding it in the Fibonacci sequence, as n gets larger and larger. The symbol for the Golden Number is , the Greek letter “phi. ” 5. 8 -6 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Golden Ratio A C B The ratio of the whole, AB, to the larger part, AC, is equal to the ratio of the larger part AC, to the smaller part, CB. is referred to as a golden ratio. 5. 8 -7 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Golden Proportion or Divine Proportion 5. 8 -8 Copyright 2013, 2010, 2007, Pearson, Education,

Golden Ratio in Architecture The Great Pyramid of Gizeh in Egypt, built about 2600 B. C. Ratio of any side of the square base to the altitude is about 1. 611 5. 8 -9 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Golden Ratio in the Human Body Architect Le Corbusier developed a scale of proportions

Golden Rectangle A C B a a a b *Note: the new smaller rectangle

Golden Rectangle in Nature The curve derived from a succession of diminishing golden rectangles is the same as the spiral curve of the chambered nautilus. 5. 8 -12 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Golden Rectangle in Greek Architecture 5. 8 -13 Copyright 2013, 2010, 2007, Pearson, Education,

Golden Rectangle in Greek Art Greek statues, vases, urns, and so on also exhibit characteristics of the golden ratio. It is for Phidas, considered the greatest of Greek sculptors, that the golden ratio was named “phi. ” The proportions can be found abundantly in his work. 5. 8 -14 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Golden Rectangle in Art The proportions of the golden rectangle can be found in the work many artists, from the old masters to the moderns. 5. 8 -15 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Fibonacci Numbers in Music An octave on a keyboard has 13 keys: 8 white keys and 5 black keys (the 5 black keys are in one group of 2 and one group of 3). In Western music, the most complete scale, the chromatic scale, consists of 13 notes. All are Fibonacci numbers. 5. 8 -16 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Fibonacci Numbers in Music Patterns that can be expressed mathematically in terms of Fibonacci relationships have been found in Gregorian chants and works of many composers, including Bach, Beethoven, and Bartók. A number of twentiethcentury musical works, including Ernst Krenek’s Fibonacci Mobile, were deliberately structured by using Fibonacci proportions. 5. 8 -17 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Why the Fibonacci Sequence? A number of studies have tried to explain why the Fibonacci sequence and related items are linked to so many real-life situations. It appears that the Fibonacci numbers are a part of a natural harmony that is pleasing to both the eye and the ear. 5. 8 -18 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Why the Fibonacci Sequence? In the nineteenth century, German physicist and psychologist Gustav Fechner tried to deter- mine which dimensions were most pleasing to the eye. Fechner, along with psychologist Wilhelm Wundt, found that most people do unconsciously favor golden dimensions when purchasing greeting cards, mirrors, and other rectangular objects. 5. 8 -19 Copyright 2013, 2010, 2007, Pearson, Education, Inc.