Greenwood press
Download 1.81 Mb. Pdf ko'rish
|
book-20600
FIBONACCI SEQUENCE
35 a1=1 a2=1 a n =a n−1 +a n−2 t n = 1+ √ 5 2 2 − 1− √ 5 2 2 √ 5 . recursive explicit The family tree for a male bee. squares (left-hand picture below), the vertices provide links for tracing a loga- rithmic spiral (middle picture). The spiral (which expands one golden ratio dur- ing each whole turn) appears in the chambered nautilus (right-hand picture). The Fibonacci numbers appear in the branching of plants, and counts of spi- rals in sunflower seeds, pine cones, and pineapples. In one particular variety of sunflower, the florets appear to have two systems of spirals, both beginning at the center. There are fifty-five spirals in the clockwise direction, and thirty-four in the counterclockwise one. The same count of florets in a daisy show twenty-one spirals in one direction and thirty-four in the other. A pine cone has two spirals of five and eight arms, and a pineapple has spirals of five, eight, and thirteen. The spiral also appears in animal horns, claws, and teeth. On many plants, the number of petals on blossoms is a Fibonacci number. Buttercups and impatiens have five petals, iris have three, corn marigolds have thirteen, and some asters have twenty-one. Some species have petal counts that may vary from blossom to blossom, but the average of the petals will be a Fibo- nacci number. Flowers with other numbers of petals, such as six, can be shown to have two layers of three petals, so that their counts are simple multiples of a Fibonacci number. In the last few years, two French mathematicians, Stephane Douady and Yves Couder, proposed a mathematical explanation for the Fibo- nacci-patterned spirals in nature. Plants develop seeds, flowers, or branches from a meristem (a tiny tip of the growing point of plants). Cells are produced at a con- stant rate of turn of the meristem. As the meristem grows upward, the cells move outward and increase in size. The most efficient turn to produce seeds, flowers, or branches will result in a Fibonacci spiral. In 1948, R. N. Elliott proposed investment strategies based on the Fibonacci sequence. These remain standard tools for many brokers, but whether they are a never-fail way of selecting stocks and bonds is open to debate. Some investors think that when Elliott’s theories work, it is because many investors are using his rules, so their effects on the stock market shape a Fibonacci pattern. Neverthe- less, a substantial number of brokers use Elliott’s Fibonacci rules in determining how to invest. In computer science, there is a data structure called a “Fibonacci heap” that is at the heart of many fast algorithms that manipulate graphs. Physicists have Download 1.81 Mb. Do'stlaringiz bilan baham: |
Ma'lumotlar bazasi mualliflik huquqi bilan himoyalangan ©fayllar.org 2024
ma'muriyatiga murojaat qiling
ma'muriyatiga murojaat qiling