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What is the importance of the fibonacci sequence
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The sunflower is a beautiful flower that grows wild in the most parts of the United States and many other countries throughout the world. However, when we look at the flower normally we don’t see anything other than a flower. To the mathematician however, there is more, much more. Inside the sunflower flows a sequence. This sequence is known as the Fibonacci sequence. Here we will discuss the Fibonacci sequence going back to the origins, its uses, and where we can find it in the everyday world. First item to discuss is where the Fibonacci sequence came from. The Fibonacci sequence was founded by a math wizard in 1202, named Leonardo Pisano (Edson). He is better known as Fibonacci. Fibonacci discovered a sequence of numbers and published them in a book titled Liber Abaci. The Fibonacci sequence is simply a number system in which the next number of the sequence is equal to the some of the two numbers that precede it. For example the first number of the sequence is 1, therefore 1, 1, 2, 3, 5, 8… and so on. As you can see the sequence can go on indefinitely. In the book he posed the famous question about rabbits. How many rabbits can be produced in a year from a single pair of rabbits if …show more content…
The center of a sunflower is an example. If you look at the blooms in the middle you can see the spiral that forms representing the Fibonacci sequence. It is also found in pine cones, pineapples, and in many different flowers. But nature is not the only place we can find this mathematical sequence. It is also found in the computer world in the form of search algorithms. In an article by John Atkins and Robert Geist they note that, “Computer scientists have discovered and used many algorithms which can be classified as applications of Fibonacci 's sequence” (Geist). As you can see the famous Fibonacci sequence is used in everyday places that we never really ever think
While the studies at Governor’s School are noticeably more advanced and require more effort than at regular public schools, I see this rigor as the key to my academic success. For me, the classes I take that constantly introduce new thoughts that test my capability to “think outside the box”, are the ones that capture all my attention and interest. For example, while working with the Sierpinski Triangle at the Johns Hopkins Center for Talented Youth geometry camp, I was struck with a strong determination to figure out the secret to the pattern. According to the Oxford Dictionary, the Sierpinski Triangle is “a fractal based on a triangle with four equal triangles inscribed in it. The central triangle is removed and each of the other three treated as the original was, and so on, creating an infinite regression in a finite space.” By constructing a table with the number black and white triangles in each figure, I realized that it was easier to see the relations between the numbers. At Governor’s School, I expect to be provided with stimulating concepts in order to challenge my exceptional thinking.
7. Define the Fibonacci binary tree of order n as follows: If n=0 or n=1, the tree consists of a single node. If n>1, the tree consists of a root, with the Fibonacci tree of order n-1 as the left subtree and the Fibonacci tree of order n-2 as the right subtree. Write a method that builds a Fibonacci binary tree of order n and returns a pointer to it.
It was once said by Johannes Kepler that “Geometry has two great treasures: one is the Theorem of Pythagoras, and the other the division of a line into extreme and mean ratio. Golden Ratio is found by dividing a line into two parts so that the longer part divided by the smaller part equals the whole length divided by the longer part. It is also known as the extreme and mean ratio. Golden ratio is very similar to Pi because it is an infinite number and it goes on forever. It is usually rounded to around 1.618. The formula for golden ratio is a/b = (a+b)/b. Golden Ratio is a number that has been around for many years. It has been around for a long time so it is not known who formed the idea of the golden ratio. Since the golden ratio is used all around the world, it is known in many names such as the golden mean, phi, the divine proportion, extreme and mean proportion, etc. It is usually referred to as phi. Golden ratio was used in arts from the beginning of people and still is used today. It has been used in architecture, math, sculptures and nature. Many famous artists used the golden ratio. Golden ratio can also be used on a rectangle which is known as the golden rectangle. Euclid talks about it in his book Elements. Golden ratio also has a relationship with both the Fibonacci numbers and Lucas numbers.
Lamb, Robert. "How are Fibonacci numbers expressed in nature?" HowStuffWorks. Discovery Communications, 24 June 2008. Web. 28 Jan. 2010. .
Nevertheless, that day followed me, and I tried to understand more about fractals through the resources I already had at my disposal-- through courses I was taking. Sophomore year, through my European History and Architecture courses, I learned about many ancient architectural feats-- Stonehenge, the Pyramids of Giza, the Parthenon, many Gothic Cathedrals, and the Taj Mahal-- and that they all somehow involved the use of the golden ratio. I will come back to how this relates to fractals later in the article, but for now know that each of these buildings use different aspects of their design to form the golden ratio. I was intrigued by the fact that fractals, what seemed to be something only formed by the forces of nature, were being constructed by human hands. Although I wanted badly to find out more, I waited until that summer, when I discovered a YouTube account by the name of Vihart. Vihart’s videos are not tutorials on how to do math, however Vihart’s ramblings about the nature and the concepts of the mathematical world have a lot of educational value, especially on topics that are more complicated to understand then to compute. Her videos on fractal math and their comparability to nature, helped to show me that...
How much math does a bee know? Surprisingly, bee’s know a little more geometry than most of us humans. After years and years of evolution they decided to choose hexagons for their honeycombs. The concise and orderly pattern of comb is a symbol for structure, order, utility and strength. This pattern hasn't occurred by accident. Bees have discovered a way to build their home that serves them incredibly well.
Fundamentally, mathematics is an area of knowledge that provides the necessary order that is needed to explain the chaotic nature of the world. There is a controversy as to whether math is invented or discovered. The truth is that mathematics is both invented and discovered; mathematics enable mathematicians to formulate the intangible and even the abstract. For example, time and the number zero are inventions that allow us to believe that there is order to the chaos that surrounds us. In reality, t...
Without Fibonacci we may not have had as thorough an understanding of pythagorean triples, or prime numbers, or quite possibly we would not even be using the number system with which we call traditional. Without him, math currently being used could be very different, and our understanding of the world and some of the ways its works could be even more mysterious. Without his discoveries used in the Fibonacci sequence, our understanding of the Golden Ratio may be nonexistent, which could have lead the world down an entirely different path, on without the renaissance artwork, or without the architecture used in the capital of our country. Therefore I would say Fibonacci and his discoveries are very important in our societies and he should be more popular and acknowledged in
Logarithms have the ability to replace a geometric sequence with an arithmetic sequence because they raise a base number by an exponent. A simple example can be provided with a geome...
Fibonacci Numbers originated from India hundreds of years ago. Though Fibonacci Numbers came from India, Leonardo of Pisa, better known as Fibonacci, made it known to the world. Leonardo came from a wealthy Italian family and traveled to North America to join his father. He was educated by the Moors and sent on business trips. “After returning to Pisa around 1200, Leonardo wrote his most famous literature, Liber Abaci” (Pearson). Leonardo featured a rabbit question in the book. The question was asked in a mathematical competition, he appeared in when he was young. Leonardo Fibonacci used the Fibonacci Numbers to solve it. Fibonacci Numbers is now used throughout our society.
Fractals occur in swirls of scum on the surface of moving water, the jagged edges of mountains, ferns, tree trunks, and canyons. They can be used to model the growth of cities, detail medical procedures and parts of the human body, create amazing computer graphics, and compress digital images. Fractals are about us, and our existence, and they are present in every mathematical law that governs the universe. Thus, fractal geometry can be applied to a diverse palette of subjects in life, and science - the physical, the abstract, and the natural. We were all astounded by the sudden revelation that the output of a very simple, two-line generating formula does not have to be a dry and cold abstraction.
They constructed the 12-month calendar which they based on the cycles of the moon. Other than that, they also created a mathematical system based on the number 60 which they called the Sexagesimal. Though, our mathematics today is not based on their system it acts like a foundation for some mathematicians. They also used the basic mathematics- addition, subtraction, multiplication and division, in keeping track of their records- one of their contributions to this world, bookkeeping. It was also suggested that they even discovered the number of the pi for they knew how to solve the circumference of the circle (Atif, 2013).
The history of the computer dates back all the way to the prehistoric times. The first step towards the development of the computer, the abacus, was developed in Babylonia in 500 B.C. and functioned as a simple counting tool. It was not until thousands of years later that the first calculator was produced. In 1623, the first mechanical calculator was invented by Wilhelm Schikard, the “Calculating Clock,” as it was often referred to as, “performed it’s operations by wheels, which worked similar to a car’s odometer” (Evolution, 1). Still, there had not yet been anything invented that could even be characterized as a computer. Finally, in 1625 the slide rule was created becoming “the first analog computer of the modern ages” (Evolution, 1). One of the biggest breakthroughs came from by Blaise Pascal in 1642, who invented a mechanical calculator whose main function was adding and subtracting numbers. Years later, Gottfried Leibnez improved Pascal’s model by allowing it to also perform such operations as multiplying, dividing, taking the square root.
Ever wonder how scientists figure out how long it takes for the radiation from a nuclear weapon to decay? This dilemma can be solved by calculus, which helps determine the rate of decay of the radioactive material. Calculus can aid people in many everyday situations, such as deciding how much fencing is needed to encompass a designated area. Finding how gravity affects certain objects is how calculus aids people who study Physics. Mechanics find calculus useful to determine rates of flow of fluids in a car. Numerous developments in mathematics by Ancient Greeks to Europeans led to the discovery of integral calculus, which is still expanding. The first mathematicians came from Egypt, where they discovered the rule for the volume of a pyramid and approximation of the area of a circle. Later, Greeks made tremendous discoveries. Archimedes extended the method of inscribed and circumscribed figures by means of heuristic, which are rules that are specific to a given problem and can therefore help guide the search. These arguments involved parallel slices of figures and the laws of the lever, the idea of a surface as made up of lines. Finding areas and volumes of figures by using conic section (a circle, point, hyperbola, etc.) and weighing infinitely thin slices of figures, an idea used in integral calculus today was also a discovery of Archimedes. One of Archimedes's major crucial discoveries for integral calculus was a limit that allows the "slices" of a figure to be infinitely thin. Another Greek, Euclid, developed ideas supporting the theory of calculus, but the logic basis was not sustained since infinity and continuity weren't established yet (Boyer 47). His one mistake in finding a definite integral was that it is not found by the sums of an infinite number of points, lines, or surfaces but by the limit of an infinite sequence (Boyer 47). These early discoveries aided Newton and Leibniz in the development of calculus. In the 17th century, people from all over Europe made numerous mathematics discoveries in the integral calculus field. Johannes Kepler "anticipat(ed) results found… in the integral calculus" (Boyer 109) with his summations. For instance, in his Astronomia nova, he formed a summation similar to integral calculus dealing with sine and cosine. F. B. Cavalieri expanded on Johannes Kepler's work on measuring volumes. Also, he "investigate[d] areas under the curve" ("Calculus (mathematics)") with what he called "indivisible magnitudes.
Addition, especially of small numbers, is a process that can be done over many repetitions. Sometimes, it produces interesting patterns. One such pattern is in Pascal’s Triangle, where each row can be constructed by adding the numbers on the row above. This particular pattern is significant in that, among other things, it shows a representation of the coefficients of a binomial expansion to a particular power.