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    FIBONACCI SAMPLE PICTURE  Fibonacci sequence The Fibonacci sequence is a set of numbers that starts with a one or a zero, followed by a one, and proceeds based on the rule that each number (called a Fibonacci number) is equal to the sum of the preceding two numbers. If the Fibonacci sequence is denoted F   ( n ), where n  is the first term in the sequence, the following equation obtains for n  = 0, where the first two terms are defined as 0 and 1 by convention: F   (0) = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 ... In some texts, it is customary to use n  = 1. In that case, the first two terms are defined as 1 and 1 by default, and therefore: F   (1) = 1, 1, 2, 3, 5, 8, 13, 21, 34 ... The Fibonacci sequence is named for Leonardo Pisano (also known as Leonardo Pisano or  Fibonacci), an Italian mathematician who lived from 1170 - 1250. Fibonacci used the arithmetic series to illustrate a problem based on a pair of breeding rabbits: How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on? The result can be expressed numerically as: 1, 1, 2, 3, 5, 8, 13, 21, 34 ... Fibonacci numbers are of interest to biologists and physicists because they are frequently observed in various natural objects and phenomena. The branching patterns in trees and leaves, for example, and the distribution of seeds in a raspberry are based on Fibonacci numbers.  A Sanskrit grammarian, Pingala, is credited with the first mention of the sequence of numbers, sometime between the fifth century B.C. and the second or third century A.D. Since Fibonacci introduced the series to Western civilization, it has had a high profile from time to time. In The Da Vinci Code , for example, the Fibonacci sequence is part of an important clue. Another application, the Fibonacci poem, is a verse in which the progression of syllable numbers per line follows Fibonacci's pattern. The Fibonacci sequence is related to the golden ratio, a proportion (roughly 1:1.6) that occurs frequently throughout the natural world and is applied across many areas of human endeavor. Both the Fibonacci sequence and the golden ratio are used to guide design for architecture, websites and user interfaces, among other things.  We live in a universe of patterns. Every night the stars move in circles across the sky. The seasons cycle at yearly intervals. No two snowflakes are ever exactly the same, but they all have six fold symmetry. Tigers and zebras are covered in patterns of stripes, leopards and hyenas are covered in patterns of spots. Intricate trains of waves march across the oceans; very similar trains of sand dunes march across the desert. Colored arcs of light adorn the sky in the form of rainbows, and a bright circular halo sometimes surrounds the moon on winter nights. Spherical drops of water fall from clouds. Human mind and culture have developed a formal system of thought for recognizing, classifying, and exploiting patterns. We call it mathematics. By using mathematics to organize and systematize our ideas about patterns, we have discovered a great secret: nature's patterns are not just there to be admired, they are vital clues to the rules that govern natural processes. Four hundred years ago, the German astronomer Johannes Kepler wrote a small book, The Six-Cornered Snowflake, as a New Year's gift to his sponsor. In it he argued that snowflakes must be made by packing tiny identical units together. This was long before the theory that matter is made of atoms had become generally accepted.   The regular nightly motion of the stars is also a clue, this time to the fact that the Earth rotates. Waves and dunes are clues to the rules that govern the flow of water, sand, and air. The tiger's stripes and the hyena's spots attest to mathematical regularities in biological growth and form. Rainbows tell us about the scattering of light, and indirectly confirm that raindrops are spheres. Lunar haloes are clues to the shape of ice crystals.   There is much beauty in nature's clues, and we can all recognize it without any mathematical training. There is beauty, too, in the mathematical stories that start from the clues and deduce the underlying rules and regularities, but it is a different kind of beauty, applying to ideas rather than things.   Patterns possess utility as well as beauty. Once we have learned to recognize a background pattern, exceptions suddenly stand out.   The phases of the moon make a complete cycle from new moon to full moon and back again every twenty-eight days. The year is three hundred and sixty-five days long-roughly. People have two legs, cats have four, insects have six, and spiders have eight. Starfish have five arms (or ten, eleven, even seventeen, depending on the species). Clover normally has three leaves: the superstition that a four-leaf clover is lucky reflects a deep seated belief that exceptions to patterns are special.   In nearly all flowers, the number of petals is one of the numbers that occur in the strange sequence 3, 5, 8, 13, 21, 34, 55, 89. For instance, lilies have three petals, buttercups have five, many delphiniums have eight, marigolds have thirteen, asters have twenty-one, and most daisies have thirty-four, fifty-five, or eighty-nine. You don't find any other numbers anything like as often.  We've now established the uncontroversial idea that nature is full of patterns.   Painting pictures, sculpting sculptures, and writing poems are valid and important ways to express our feelings about the world and about ourselves.   The entrepreneur's instinct is to exploit the natural world. The engineer's instinct is to change it. The scientist's instinct is to try to understand it-to work out what's really going on. The mathematician's instinct is to structure that process of understanding by seeking generalities that cut across the obvious subdivisions. There is a little of all these instincts in all of us, and there is both good and bad in each instinct.   Each of nature's patterns is a puzzle, nearly always a deep one. Mathematics is brilliant at helping us to solve puzzles. It is a more or less systematic way of digging out the rules and structures that lie behind some observed pattern or regularity, and then using those rules and structures to explain what's going on. Indeed, mathematics has developed alongside our understanding of nature, each reinforcing the other.   Mathematicians could take the geometric rules that lead to ellipses and modify them to see what other kinds of curve resulted. Similarly, when Isaac Newton made the epic discovery that the motion of an object is described by a mathematical relation between the forces that act on the body and the acceleration it experiences, mathematicians and physicists learned quite different lessons. However, before I can tell you what these lessons were I need to explain about acceleration.    Acceleration is a subtle concept: it is not a fundamental quantity, such as length or mass; it is a rate of change. In fact, it is a second order rate of change-that is, a rate of change of a rate of change. The velocity of a body-the speed with which it moves in a given direction-is just a rate of change: it is the rate at which the body's distance from some chosen point changes. If a car moves at a steady speed of sixty miles per hour, its distance from its starting point changes by sixty miles every hour. Acceleration is the rate of change of velocity. The Newton-and independently the German mathematician Gottfried Leibniz-invented a new branch of mathematics, the calculus. It changed the face of the Earth-literally and metaphorically. But, again, the ideas sparked by this discovery were different for different people. The story of calculus brings out two of the main things that mathematics is for: providing tools that let scientists calculate what nature is doing, and providing new questions for mathematicians to sort out to their own satisfaction.   In the short term, it made very little difference whether mathematicians were satisfied about the logical soundness of the calculus. But in the long run the new ideas that mathematicians got by worrying about these internal difficulties turned out to be very useful indeed to the outside world.
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