Research Paper
Throughout math, there are many patterns of numbers that have special and distinct properties. There are even numbers, primes, odd numbers, multiples of four, eight, seven, ten, etc. One important and strange pattern of numbers is the set of Fibonacci numbers. This is the sequence of numbers that follow in this pattern: 1, 1, 2, 3, 5, 8, 13, 21, etc. The idea is that each number is the sum of its previous two numbers (n=[n-1]+[n-2]) (Kreith). The Fibonacci numbers appear in various topics of math, such as Pascal?s Triangle and the Golden Ratio/Section. It falls under number theory, which is the study of whole or rational numbers. Number Theory develops theories, simple equations, and uses special tools to find specific numbers. Some topic examples from number theory are the Euclidean Algorithm, Fermat?s Little Theorem, and Prime Numbers.
Strangely, the Fibonacci numbers appear in nature too. One familiar way in which the Fibonacci numbers appear in nature is the rabbit family line (and bee family line as well). Another strange way in which the Fibonacci numbers relate to nature is the plant kingdom. Because of these strange relationships, I ask the question: How and why do the Fibonacci numbers appear in nature? In this paper, I will attempt to answer this question. Pascal?s Triangle - Golden Rectangle
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The man behind the Fibonacci numbers, Leonardo Fibonacci, was born in Pisa in 1175 A.D. During his life, he was a customs officer in Africa and businessman who traveled to various places. During these trips he gained knowledge and skills which enabled him to be recognized by Emperor Fredrick II. Fredrick II noticed Fibonacci and ordered him to take part in a mathematical tournament. This place would eventuall...
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...its relation to the Golden Angle, which appears in the primordia of plants in order to give the maximum number of primordia for plants. I like to think of an idea in the book, ?Life?s Other Secret,? which says that it?s not just Fibonacci Numbers that matter; it?s also the matter in which they arise (Stewart).
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Works Cited
Adam, John. Mathematics in Nature. Princeton, New Jersey: Princeton University Press, 2003.
Knott, Ron. ?Fibonacci Numbers in Nature? 18, July 2005. 03, Aug 2005.
Kreith, Kurt. COSMOS Professor. Davis, California.
Muldrew, Lola. COSMOS Teacher Fellow. Davis, California.
Stewart, Ian. Life?s Other Secret. Canada: John Wiley & Sons, Inc.
University of Cambridge. ?The Life and Numbers of Fibonacci? Sep 1997. 03, Aug 2005.
This paper will discuss three specific instances: Le Sacrifice, Psappha, and Metastasis. The first principle that I will discuss is the Golden Section. The Golden Section can be found in art and architecture dating as far back as the Parthenon, as well as different places in nature, such as the nautilus shell. The Golden Section is essentially a proportion that is established by taking a single line and dividing that line into two separate sections of unequal lengths, one quite longer than the other.
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...
People accumulate knowledge as well as experiences on a daily basis and organize them in the so-called schemas. These schemas help people process and evaluate information more efficiently because they do not have to reassess similar instance whenever they encounter familiar experiences (Bartlett & Burt, 1933). Besides, schemas provide expectations, the extent to which information in an ad conform to some predefined knowledge structures (Lee & Mason, 1999), which affect the processing of specific situations (Goodman, 1980). For example, an individual who has only purchased liquid honey as the only form of honey would expect honey texture to be liquid. If he sees liquid honey at the supermarket, his expectation is confirmed and he does not need
Born in the Netherlands, Daniel Bernoulli was one of the most well-known Bernoulli mathematicians. He contributed plenty to mathematics and advanced it, ahead of its time. His father, Johann, made him study medicine at first, as there was little money in mathematics, but eventually, Johann gave in and tutored Daniel in mathematics. Johann treated his son’s desire to lea...
Pierre de Fermat Pierre de Fermat was born in the year 1601 in Beaumont-de-Lomages, France. Mr. Fermat's education began in 1631. He was home schooled. Mr. Fermat was a single man through his life. Pierre de Fermat, like many mathematicians of the early 17th century, found solutions to the four major problems that created a form of math called calculus. Before Sir Isaac Newton was even born, Fermat found a method for finding the tangent to a curve. He tried different ways in math to improve the system. This was his occupation. Mr. Fermat was a good scholar, and amused himself by restoring the work of Apollonius on plane loci. Mr. Fermat published only a few papers in his lifetime and gave no systematic exposition of his methods. He had a habit of scribbling notes in the margins of books or in letters rather than publishing them. He was modest because he thought if he published his theorems the people would not believe them. He did not seem to have the intention to publish his papers. It is probable that he revised his notes as the occasion required. His published works represent the final form of his research, and therefore cannot be dated earlier than 1660. Mr. Pierre de Fermat discovered many things in his lifetime. Some things that he did include: -If p is a prime and a is a prime to p then ap-1-1 is divisible by p, that is, ap-1-1=0 (mod p). The proof of this, first given by Euler, was known quite well. A more general theorem is that a0-(n)-1=0 (mod n), where a is prime...
Carl Friedrich Gauss was born April 30, 1777 in Brunswick, Germany to a stern father and a loving mother. At a young age, his mother sensed how intelligent her son was and insisted on sending him to school to develop even though his dad displayed much resistance to the idea. The first test of Gauss’ brilliance was at age ten in his arithmetic class when the teacher asked the students to find the sum of all whole numbers 1 to 100. In his mind, Gauss was able to connect that 1+100=101, 2+99=101, and so on, deducing that all 50 pairs of numbers would equal 101. By this logic all Gauss had to do was multiply 50 by 101 and get his answer of 5,050. Gauss was bound to the mathematics field when at the age of 14, Gauss met the Duke of Brunswick. The duke was so astounded by Gauss’ photographic memory that he financially supported him through his studies at Caroline College and other universities afterwards. A major feat that Gauss had while he was enrolled college helped him decide that he wanted to focus on studying mathematics as opposed to languages. Besides his life of math, Gauss also had six children, three with Johanna Osthoff and three with his first deceased wife’s best fri...
‘Nature abounds with example of mathematical concepts’ (Pappas, 2011, .107). It is interesting how much we see this now we know, regarding the Fibonacci Sequence, which is number pattern where the first number added to itself creates a new number, then adding that previous number to the new number and so on. You will notice how in nature this sequence always adds up to a Fibonacci number, but alas this is no coincidence it is a way in which plants can pack in the most seeds in a small space creating the most efficient way to receive sunlight and catches the most
Named after the Polish mathematician, Waclaw Sierpinski, the Sierpinski Triangle has been the topic of much study since Sierpinski first discovered it in the early twentieth century. Although it appears simple, the Sierpinski Triangle is actually a complex and intriguing fractal. Fractals have been studied since 1905, when the Mandelbrot Set was discovered, and since then have been used in many ways. One important aspect of fractals is their self-similarity, the idea that if you zoom in on any patch of the fractal, you will see an image that is similar to the original. Because of this, fractals are infinitely detailed and have many interesting properties. Fractals also have a practical use: they can be used to measure the length of coastlines. Because fractals are broken into infinitely small, similar pieces, they prove useful when measuring the length of irregularly shaped objects. Fractals also make beautiful art.
Fractal Geometry The world of mathematics usually tends to be thought of as abstract. Complex and imaginary numbers, real numbers, logarithms, functions, some tangible and others imperceivable. But these abstract numbers, simply symbols that conjure an image, a quantity, in our mind, and complex equations, take on a new meaning with fractals - a concrete one. Fractals go from being very simple equations on a piece of paper to colorful, extraordinary images, and most of all, offer an explanation to things. The importance of fractal geometry is that it provides an answer, a comprehension, to nature, the world, and the universe.
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).
Irrational numbers are real numbers that cannot be written as a simple fraction or a whole number. For example, irrational numbers can be included in the category of √2, e, Π, Φ, and many more. The √2 is equal to 1.4142. e is equal to 2.718. Π is equal to 3.1415. Φ is equal to 1.6180. None of these numbers are “pretty” numbers. Their decimal places keep going and do not end. There is no pattern to the numbers of the decimal places. They are all random numbers that make up the one irrational number. The concept of irrational numbers took many years and many people to discover and prove (I.P., 1997).
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.
The history of math has become an important study, from ancient to modern times it has been fundamental to advances in science, engineering, and philosophy. Mathematics started with counting. In Babylonia mathematics developed from 2000B.C. A place value notation system had evolved over a lengthy time with a number base of 60. Number problems were studied from at least 1700B.C. Systems of linear equations were studied in the context of solving number problems.
The Golden Ratio is also known as the golden rectangle. The Golden Rectangle has the property that when a square is removed a smaller rectangle of the same shape remains, a smaller square can be removed and so on, resulting in a spiral pattern.
Abstractions from nature are one the important element in mathematics. Mathematics is a universal subject that has connections to many different areas including nature. [IMAGE] [IMAGE] Bibliography: 1. http://users.powernet.co.uk/bearsoft/Maths.html 2. http://weblife.bangor.ac.uk/cyfrif/eng/resources/spirals.htm 3.