5.8 The Fibonacci Sequence and the Golden Ratio 283 Table 5.1 Month 1 Adult Adult Baby Adult Adult Adult Adult Adult Adult Adult Adult Baby Adult Baby Baby Baby Baby Baby Adult Month 2 Month 3 Month 4 Month 5 Figure 5.10 Profile in Mathematics Fibonacci Leonardo of Pisa (1170–1250) is considered one of the most distinguished mathematicians of the Middle Ages. He was born in Italy and was sent by his father to study mathematics with an Arab master. When he began writing, he referred to himself as Fibonacci, or “son of Bonacci,” the name by which he is known today. In addition to the famous sequence bearing his name, Fibonacci is also credited with introducing the Hindu–Arabic number system into Europe. His 1202 book, Liber Abacci (Book of the Abacus), explained the use of this number system and emphasized the importance of the number zero. Month Pairs of Adults Pairs of Babies Total Pairs 1 1 0 1 2 1 1 2 3 2 1 3 4 3 2 5 5 5 3 8 6 8 5 13 7 13 8 21 8 21 13 34 9 34 21 55 10 55 34 89 11 89 55 144 12 144 89 233 Following is the Fibonacci sequence Fibonacci Sequence 1, 1, 2, 3, 5, 8, 13, 21,… In the Fibonacci sequence, the first and second terms are 1. The sum of these two terms is the third term. The sum of the second and third terms is the fourth term, and so on. In the middle of the nineteenth century, mathematicians made a serious study of this sequence and found strong similarities between it and many natural phenomena. Fibonacci numbers appear in the seed arrangement of many species of plants and in the petal counts of various flowers. For example, when the flowering head of the sunflower matures to seed, the seeds’ spiral arrangement becomes clearly visible. A typical count of these spirals may give 89 steeply curving to the right, 55 curving more shallowly to the left, and 34 again shallowly to the right. The largest known specimen to be examined had spiral counts of 144 right, 89 left, and 55 right. These numbers, like the other three mentioned, are consecutive terms of the Fibonacci sequence. On the heads of many flowers, petals surrounding the central disk generally yield a Fibonacci number. For example, some daisies contain 21 petals, and others contain 34, 55, or89 petals. (People who use a daisy to play the “love me, love me not” game will likely pluck 21, 34, 55, or89 petals before arriving at an answer.) Fibonacci numbers are also observed in the structure of pinecones and pineapples. The tablike or scalelike structures called bracts that make up the main body of the pinecone form a set of spirals that start from the cone’s attachment to the branch. Two sets of oppositely directed spirals can be observed, one steep and the other more gradual. A count on the steep spiral will reveal a Fibonacci number, and a count on the gradual one will be the adjacent smaller Fibonacci number, or if not, the next smaller Fibonacci number. One investigation of 4290 pinecones from 10 species of pine trees found in California revealed that only 74 cones, or 1.7%, deviated from this Fibonacci pattern. Like pinecone bracts, pineapple scales are patterned into spirals, and because they are roughly hexagonal in shape, three distinct sets of spirals can be counted. Generally, the number of pineapple scales in each spiral are Fibonacci numbers. m The head of a sunflower Eiji Ueda/Shutterstock
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