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Golden ratio research
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Golden ratio research
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The Golden Ratio The Golden Rectangle and Ratio The Golden Rectangle and Golden Ratio have always existed in the physical universe. Nobody knows exactly when it was first discovered and applied to mankind. Many mathematicians assume that the Golden Rectangle has been discovered and rediscovered multiple times throughout history. This would explain why it is called many different names such as the Golden Mean, divine proportion, or the Golden Section. The first person who is believed to have used the Golden Ratio is Phidias when he used it to design the statues inside of the Parthenon. This happened between 490 and 430 BC. In the early 300’s BC Plato used the Golden Ratio when he described the five platonic solids which are the tetrahedron, cube, octahedron, dodecahedron, and the icosahedron. Later in the 300’s Euclid gave the first written definition of the Golden Ratio which is an extreme and mean ratio. Then between 1170 and 1250 Fibonacci discovered a numerical series which had sequential elements that approaches the Golden Ratio asymptotically. Between 1445 and 1517 Luca Pacioli defined the Golden Ratio as the “divine proportion”. Then in between 1550 and 1631 Michael Maestlin published the first known approximation of the inverse golden ratio as a decimal fraction which is 1.61803398875. Very soon afterwards Johannes Kepler proved that the golden ratio is the limit of the ratio of consecutive Fibonacci numbers. Then in between 1842 and 1891 Edouard Lucas named the numerical sequence the Fibonacci sequence. In 1974 Roger Penrose discovered Penrose tiling which is a pattern that is related to the golden ratio both in the ratio of areas of its two rhombic tiles and in their relative frequency within the pattern. The g... ... middle of paper ... ...n whether or not they were put into the logo on purpose or if it was just by accident to make it more visually appealing. It makes me wonder if there are places in nature that also contain the golden ratio but we just haven’t discovered them yet. Works Cited "15 Uncanny Examples of the Golden Ratio in Nature." Io9. N.p., n.d. Web. 10 Mar. 2014. "Golden Ratio." Golden Ratio. N.p., n.d. Web. 12 Mar. 2014. "Golden Ratio in Geometry." Golden Ratio in Geometry. N.p., n.d. Web. 11 Mar. 2014. Livio, Mario. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway, 2002. Print. "Logo Design." Phi 1618 The Golden Number RSS. N.p., n.d. Web. 11 Mar. 2014. "Phi Formula Geometry." Phi 1618 The Golden Number RSS. N.p., n.d. Web. 12 Mar. 2014. "Theology." Phi 1618 The Golden Number RSS. N.p., n.d. Web. 12 Mar. 2014.
4 "Etude Golden Arrow II [Study Golden Arrow II]." Wolfsonian-FIU. http://www.wolfsonian.org/explore/collections/etude-golden-arrow-ii-study-golden-arrow-ii (accessed April 9, 2014).
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...
Fractions have been a around long enough for me to understand that I do not like them, but they play a significant part in simplifying, for some, division of goods or time. There is no one person who can be credited with the invention of fractions, but their use has been traced back as early as 1000 BC, in Egypt--using the formula to trade tangibles, currency, and build pyramids.
The Ancient Egyptians are commonly known as the first people to use geometry. Not only did they use it, but they were masters of it. Their work constructing the pyramids only provides evidence of their vast mathematical knowledge. The Ancient Egyptians invented many different mathematical techniques in order to make daily life easier. Luckily, there are still records from the Egyptians that have been decoded so that we may learn how they invented their version of math.
Leonardo Da Vinci is one of the few artists and mathematicians who used the Golden Ratio frequently. In the Renaissance, the Golden Ratio was often used to create balance and beauty in statues and and paintings. Da Vinci, however, called it the “Golden Section”. He used it in famous
While studying the golden mean it becomes evident just how relevant this number is in the world. Many architects and artists have used this ratio as a scale and proportion sequence. The sequence is also relevant in music, nature and even the human body. Ancient mathematicians were so fascinated in the ratio because of its frequency in geometry. The first person to provide a written definition was Euclid. He stated “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less” this has been studied thoroughly by many mathematicians but the most relevant was the studies of Leonardo Fibonacci. Fibonacci is famous for the work he put in to come up with the Fibonacci sequence.
In fact, a rectangle with side lengths φ is said to be a golden rectangle, which is a result of the assumption that h=1 [2]. The number phi has even gripped theologians to ...
It is constructed by taking an equilateral triangle, and after many iterations of adding smaller triangles to increasingly smaller sizes, resulting in a "snowflake" pattern, sometimes called the von Koch snowflake. The theoretical result of multiple iterations is the creation of a finite area with an infinite perimeter, meaning the dimension is incomprehensible. Fractals, before that word was coined, were simply considered above mathematical understanding, until experiments were done in the 1970's by Benoit Mandelbrot, the "father of fractal geometry". Mandelbrot developed a method that treated fractals as a part of standard Euclidean geometry, with the dimension of a fractal being an exponent. Fractals pack an infinity into "a grain of sand".
It is not possible to precisely tell who first became conscious of this number. There are writings from 35000 years ago that reveal the knowledge of a concept closely related to pi. According to Beckmann in his book A History of Pi, in order to understand how in 2000 B.C. the concept of pi and its significance more or less clearly arrived to human minds “we must return into the stone age and beyond, and into realm of speculation” (Beckmann, 1971). Pi is the circumference of a circle divided by its diameter.
The recursive sequence of numbers that bear his name is a discovery for which Fibonacci is popularly known. While it brought him little recognition during the course of his life, is was nearly 100 years later when the majority of the mathematicians recognized and appreciated his invention. This fascinating and unique study of recursive numbers possess all sorts of intriguing properties that can be discovered by applying different mathematical procedures to a set of numbers in a given sequence. The recursive / Fibonacci numbers are present in everyday life and they are manifested in the everyday life in which we live. The formed patterns perplex and astonish the minds in real world perspectives. The recursive sequences are beautiful to study and much of their beauty falls in nature. They highlight the mathematical complexity and the incredible order of the world that we live in and this gives a clear view of the algorithm that God used to create some of these organisms and plants. Such patterns seem not have been evolved by accident but rather, they seem to have evolved by the work of God who created both heaven and
Historically speaking, ancient inventors of Greek origin, mathematicians such as Archimedes of Syracuse, and Antiphon the Sophist, were the first to discover the basic elements that translated into what we now understand and have formed into the mathematical branch called calculus. Archimedes used infinite sequences of triangular areas to calculate the area of a parabolic segment, as an example of summation of an infinite series. He also used the Method of Exhaustion, invented by Antiphon, to approximate the area of a circle, as an example of early integration.
My research demonstrates how a desire to calculate the area of circular regions for constructions purposes led ancient mathematicians to develop methods from which pi could be derived and why pi is still a major subject today even though the number of digits approximated has long since passed the point of practicality. I separated the research into two parts, one dedicated to the history of pi and another to analyzing a few of the prominent methods of approximating pi, by reading scholarly articles, books, and other resources. The analysis shows that the hunt for pi has persisted because of the challenge it presents to mathematicians and the hope they hold of uncovering other secrets of mathematics in the process. It appears that, while nothing
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
There are many people that contributed to the discovery of irrational numbers. Some of these people include Hippasus of Metapontum, Leonard Euler, Archimedes, and Phidias. Hippasus found the √2. Leonard Euler found the number e. Archimedes found Π. Phidias found the golden ratio. Hippasus found the first irrational number of √2. In the 5th century, he was trying to find the length of the sides of a pentagon. He successfully found the irrational number when he found the hypotenuse of an isosceles right triangle. He is thought to have found this magnificent finding at sea. However, his work is often discounted or not recognized because he was supposedly thrown overboard by fellow shipmates. His work contradicted the Pythagorean mathematics that was already in place. The fundamentals of the Pythagorean mathematics was that number and geometry were not able to be separated (Irrational Number, 2014).
The Golden Rectangle is a unique and important shape in mathematics. The Golden Rectangle appears in nature, music, and is often used in art and architecture. Some thing special about the golden rectangle is that the length to the width equals approximately 1.618……