Fractals are a geometric pattern that are repeat over and over again to produce irregular shapes and surfaces that cannot be classical geometry. It is also, an innovative division of geometry and art. Conceivably, this is the grounds for why most people are familiar with fractals only as attractive pictures functional as backdrop on the PC screen or unique postcard design. But what are they really? Most physical structures of nature and lots of human artifacts are not normal geometric shapes of the typical geometry resulting from Euclid. Fractal geometry proposes almost limitless ways of depicting, evaluating, and predicting these natural occurrences. But is it possible to characterize the entire world using mathematical equations? This article describes how the two most well-known fractals were fashioned and explains the most significant fractal properties, which make fractals helpful for different domains of science. Fractals are self-similarity and non-integer dimension, which are two of the most significant properties. What does self-similarity imply? If you look methodically at a fern leaf, you will become aware that every small leaf has the identical shape as the whole fern leaf. You can conclude that the fern leaf is self-similar. The same is with fractals: you can magnetize then as many times as you like and after each time you will still see the same shape. The non-integer dimension is more complicated to explain. Classical geometry involves objects of integer dimensions: points, lines and curves, plane figures, solids. However, many natural occurrences are better explained using a dimension amid two whole numbers. So while a non-curving straight line has a component of one, a fractal curve will obtain a dimension between...
... middle of paper ...
... factor of 8. Falconer (1990) explains that association between dimension D, linear scaling L and the outcome of size increasing S can be comprehensive and written as:
Rearranging this formula gives a phrase for dimension relying on how the size alters as a function of linear scaling:
In the case above, the value of D is an integer - 1, 2, or 3 - relying on the dimension of the geometry. This association holds for all Euclidean shapes glimpsing at the image of the first step in constructing the Sierpinski Triangle, we can see that if the linear dimension of the original triangle ( L) is doubled, then the area of entire fractal (blue triangles) increases by a factor of three ( S).
Using the example given above, we can compute a dimension for the Sierpinski Triangle: The outcome of this computation establishes the non-integer fractal dimension.
The behavioral dimension goes hand in hand with pragmatism. It asks how is it possible to get a child to do something effectively. So its main focus is on what the person says or thinks not what he says about what he can do or think. Which requires precise measurement. Another important question in the behavioral dimension is to ask not just was the behavior changed but whose behavior was changed. The test- retest, and the inter- rater reliability techniques are of major importance for considering the presence of this dimension.
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.
James Gleick was quoted by Yeongmahn You, where he stated that “fractal means self-similarity; self-similarity is symmetry across scale. It implies recursion, pattern inside of pattern”. In other words self-similarity is a repetition of the detail that present from the smallest to the largest scale, therefor creating a hidden pattern of order that has structure and regularity (Gleick 1987:103).
From the information presented above, it is clear that the four dimensions that Hofstede mentions, namely
if magnitudes are in the same ratio, but rather it is a condition that defines
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...
From the anatomy of a human, the social life of insects, and the way the world functions are all interconnected through complex system science. By taking fractal geometry and implementing it into larger unmanageable scales can help provide further more in depth information pertaining to not just that individual but also the system as a whole.
...are self-similar; in that case at higher and higher magnification the fractal image resembles the original.
Structural dimension is a pattern of connection and network between actors, where in general this dimension is emphasized on the pattern of relationships between individuals with networks of colleagues who are owned. Relational dimension is a relationship or interaction that occurs because of the trust between the actors, where in general measurements on this dimension is focused on the nature and quality of relationships and interactions between individuals. The cognitive dimension is the goal and the shared value of the actors, which in general is measured by focusing on the representation of objectives, norms, values, and the existence of mutual
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.
dimensional shifts. So it is safe to say it is better not to assume the latter, yet.
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
middle of paper ... ... Olckers, Gibbs, Duncan 2007:2. Knowing Dimensions - What is it The knowledge dimension focuses on gaining the relevant information and understanding of healthcare as well as the skills needed to provide it. It also encompasses the “appropriate application” of this knowledge.
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……