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The world of mathematics usually tends to be thought of as abstract. Complex and imaginary numbers, real numbers, logarithms, functions, some tangible and others im perceivable. 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. 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. When the output was what is now called a fractal, no one called it artificial.
Fractals suddenly broadened the realm in which understanding can be based on a plain physical basis. (McGuire, Foreword by Benoit Mandelbrot) A fractal is a geometric shape that is complex and detailed at every level of magnification, as well as self-similar. Self-similarity is something looking the same over all ranges of scale, meaning a small portion of a fractal can be viewed as a microcosm of the larger fractal. One of the simplest examples of a fractal is the snowflake. 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. This infinity appears when one tries to measure them. The resolution lies in regarding them as falling between dimensions.
The dimension of a fractal in general is not a whole number, not an integer. So a fractal curve, a one-dimensional object in a plane which has two-dimensions, has a fractal dimension that lies between 1 and 2. Likewise, a fractal surface has a dimension between 2 and 3. The value depends on how the fractal is constructed. The closer the dimension of a fractal is to its possible upper limit which is the dimension of the space in which it is embedded, the rougher, the more filling of that space it is. (McGuire, p. 14) Fractal Dimensions are an attempt to measure, or define the pattern, in fractals. A zero-dimensional universe is one point.
A one-dimensional universe is a single line, extending infinitely. A two-dimensional universe is a plane, a flat surface extending in all directions, and a three-dimensional universe, such as ours, extends in all directions. All of these dimensions are defined by a whole number. What, then, would a 2. 5 or 3. 2 dimensional universe look like? This is answered by fractal geometry, the word fractal coming from the concept of fractional dimensions. A fractal lying in a plane has a dimension between 1 and 2.
The closer the number is to 2, say 1. 9, the more space it would fill. Three-dimensional fractal mountains can be generated using a random number sequence, and those with a dimension of 2. 9 (very close to the upper limit of 3) are incredibly jagged. Fractal mountains with a dimension of 2. 5 are less jagged, and a dimension of 2. 2 presents a model of about what is found in nature. The spread in spatial frequency of a landscape is directly related to its fractal dimension. Some of the best applications of fractals in modern technology are digital image compression and virtual reality rendering. First of all, the beauty of fractals makes them a key element in computer graphics, adding flare to simple text, and texture to plain backgrounds.
In 1987 a mathematician named Michael F. Barnsley created a computer program called the Fractal Transform, which detected fractal codes in real-world images, such as pictures which have been scanned and converted into a digital format. This spawned fractal image compression, which is used in a plethora of computer applications, especially in the areas of video, virtual reality, and graphics. The basic nature of fractals is what makes them so useful.
If someone was Rendering a virtual reality environment, each leaf on every tree and every rock on every mountain would have to be stored. Instead, a simple equation can be used to generate any level of detail needed. A complex landscape can be stored in the form of a few equations in less than 1 kilobyte, 1 / 1440 of a 3. 25 disk, as opposed to the same landscape being stored as 2. 5 megabytes of image data (almost 2 full 3. 25 disks). Fractal image compression is a major factor for making the multimedia revolution of the 1990 s take place.
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Research essay sample on 1 And 2 Virtual Reality
