Math Report Fractal Geometry Fractal Geometry Is Not Just A Chapter O

Math Report Fractal Geometry ?Fractal Geometry is not just a chapter of mathematics, but one that helps everyman to see the same old world differently". - Benoit Mandelbrot 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. 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. 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 exampl es 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 frac tal 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. 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

100 Fantastic Personal Essay Topics for Winning Essays

100 Fantastic Personal Essay Topics for Winning Essays 100 Great Personal Essay Topics: Write Killer Essays! High school and college students are frequently given personal essays to write. Regardless of whether these essays are for admission purposes or for a specific class, they are usually challenging since they need to be written in both a descriptive and narrative style and in the correct tense. These essays are not just a case of telling a particular story or describing a particular person or event. The writer needs to connect a situation or event to a bigger idea. Below is a list 100 wonderful personal essay topics to get your writing project off to a smooth start. All these topics are interesting and have the potential to create the most captivating essays. Let us get started!The thing that motivates you the most. What animal would you like to be if you had the choice? What you do to relax. A time you had to make a difficult choice and how you did it. How your parents chose your name. The television show you liked most when you were young. The period in history you most like. What cartoon do you most like? How you got through a very difficult period of your life. A very daring thing you did. The karaoke song you like best. What historical figure do you admire most? Describe a stressful event or situation you recently experienced. Explain what money means to you in terms of where you are presently at in your life. Which story from your familys history do you find most interesting? A massive change that happened in yours or your familys life. The most significant conflict that has ever occurred between you and one of your siblings or cousins. The most perilous situation you have ever found yourself in. Describe something of great value you accidently found. The work or item of art you like most. An occasion when an important decision you made turned out to be the wrong one. An item in your classroom you find distracting. The place in the world you find most pleasant and beautiful. Try comparing yourself to a close member of your family i.e. your mother or father. What you do to find inner tranquility. What public personality would you ask to give a talk at your school or college if you had the chance? The time you entered a competition. The thing that annoys you most. An item of technology you could not imagine being without. The home you dream of having. The time you chose to leave your comfort zone. The most recent good cause you supported. The local place you most like to hang out in. An occasion where you did not think before you acted or reacted. Your most interesting adventure ever. Which celebrity would you most like to hold an interview with? The thing you fear most. A time you needed a very big hug. What would be the job of your dreams? The thing you find most irresistible. How you have chosen to decorate your bedroom. A heroic thing you did. Your last social media post. Behavior you find annoying. How the life you now live might be different if you were born a century ago. The poem that stands out most in your memory. The individual who has given you the most help with your school assignments. What you would be most likely to buy if someone gave you a million dollars. The approach you take to writing an essay. A memorable exploration of the outdoors. The place in your home you like best. Your most disappointing school experience ever. The time you told a well-intentioned lie. A band or artist from whom you derive the most inspiration. The last time you did a kind act. Describe how you overcome an almost-impossible obstacle to achieve something you really wanted. A local place you do your best to stay away from. A particular song that evokes happy memories for you. The reactions of your parents when they see your report cards. A special event that served to bring you and your parents closer together. How a teacher or college professor gave you the motivation to do much better. An occasion where you reacted before thinking. A movie or book character that has many of the same characteristics as you. A tradition your family continues to uphold. What celebrity would you like to come to your graduation? An occasion when you felt really good about being alone. A building that you are really impressed by. The proudest moment of your life. The most expensive item you have ever bought. An occasion when a failure taught you a valuable lesson. Do you have a special technique that helps you learn new languages? What mobile application do you use most frequently? A character from a television show you would very much like to form a friendship with. An occasion when you felt betrayed by someone close to you. An item from your childhood that you have always kept. Someone you would like to send a message of thanks to. An occasion when you were extremely angry. If you could have a superpower, what would you choose? The best moment in the history of movies. What would you invent if you had the opportunity? The best trip you ever had from school. The best neighbor you have known. How you contribute to teamwork. Which book has had the most impact on you thus far in your life? An occasion when a friend really helped you. A fake news item you found online. Which item of modern technology most excites you? Someone you cannot envisage your life without. The member of your family who most influences you. The most recent online conversation you deleted. An occasion when you were left disappointed by a friend. A real fun occasion. An event from history you would like to have taken part in or witnessed. An occasion when you felt life running away but did not. The most recent important subject you and a friend discussed. If you could become President for just one day, what is it you would most like to do? The most recent online video you saw. A kid from a previous school or class whom you thought was really cool. What thing or event has made the greatest impression on you today? An occasion when the decision you made turned out to be the wrong one.