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Fractal geometry applied to physical
Mandelbrot father of fractals
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Fractal Geometry
In the past, mathematics has been concerned largely with sets and functions to which the methods of classical calculus could be applied. Sets or functions that are not sufficiently smooth or regular have tended to be named as " pathological" and not worthy of study. They were regarded as individual curiosities and only rarely were thought of as a class to which a general theory might be applicable. However, in recent years this attitude has changed. Irregular sets provide a much better representation of many natural phenomena than do the figures of classical geometry. Fractal geometry provides a general framework for the study of such irregular sets. (Falconer) The word ‘fractal’ was coined in 1975 by mathematician Benoit B. Mandelbrot to describe an intricate-looking set of curves, many of which were not yet seen before the creation of the computer. Fractals often exhibit self-similarity, which means that various copies of an object can be found in the original object at smaller size scales. This continues for many magnifications – like an endless nesting of Russian dolls within dolls. (Pickover) Fractals appear everywhere in nature, in galaxies and landscapes, in earthquakes and geological cracks, in aggregates and colloids, and even in the human body. Fractal geometry is an important tool in the analysis of phenomena, ranging from rhythms in music melodies to the human heartbeat and DNA sequences. Many professions including, mathematics, astronomy, physics, chemistry, engineering, and biology use fractal geometry. (Bunde)
Waclaw Sierpinski was born on March 14, 1882, in Warsaw, Poland. Sierpinski attended the University of Warsaw in 1899, when all classes were taught in Russian. He graduated in 1904 and went on to teach mathematics and physics at a girl's school in Warsaw. He left teaching in 1905 to get his doctorate at the Jagiellonian University in Cracow. After receiving his doctorate in 1908, Sierpinski went on to teach at the University of Lvov. During his years at Lvov, he wrote three books and many research papers. These books were The Theory of Irrational numbers (1910), Outline of Set Theory (1912), and The Theory of Numbers (1912). In 1919, Sierpinski accepted a job as a professor at the University of Warsaw, and this is where
Waclaw Sierpinski (The Mactutor)
he would spend the rest of his life. Throughout his career, Sierpinski wrote 724 papers and an amazing 50 books. Sierpinski studied many areas of mathematics, including, irrational numbers, set theory, fractal geometry, and theory of numbers.
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.
to many to list, in total 23. Some books that he has written are The
First, even from birth this mathematical and theoretical genius has an irregular story. Hawking had been born on the three hundredth anniversary of Galileo’s death, this proving later to be a great point of inspiration and motivation for him and his research. The date being January 8, 1942, a later recognized genius was born
Music effect on stress relief is due in part to it’s effect on mood. This may seem like a sentence within a sentence, but it is much more than that. When listening to classical music, one is brought into a state of calm. When listening to rap or faster tempo music, one is brought into a state of action. These effects all have to do with the power of music on our mood. Classical music, is the most effective at calming someone down. In terms of numbers, listening to classical music, as used in this study, was associated with a significant (5-5.6%, p<0.05) lowering of the resting heart rate and a consistent improvement of oxygen saturation (by 1-1.4%).(V K Paul1 #4.) These may seem like strange numbers, but they have a great deal to do with the topic at hand. Later on in the paper, it discusses more vividly into the terms of oxygen saturation and heart rate...
We don’t hear too much about Euler, but he is one of the most important and influential mathematicians ever, along with Archimedes and Newton. He created more published works than any other mathematician and wrote in a very understandable way. There is a fundamental part of geometry that all other mathematicians before him missed, but Euler discovered it and made the polyhedron formula: V-E+F=2
As Jackie Joyner Kersee once said, “The rewards are going to come, but my happiness is just loving the sport and having fun performing” (“Sports Quotes,” 2013). Sports play a major role in entertainment all across the world. Sports can range from driving in NASCAR to playing in the NBA (Careers in Pro Sports, 1999). Sports usually involve teamwork and is fun for everyone. Although sports are supposed to be fun, there is natural competition. Being a feminine art, dance is considered to be just about anything other than a sport. Dancers are trained to the height of their ability, just like other athletes in other sports. Even though the dancer might not perform with their whole dance studio on stage at the same time, the dancers are all apart of the same team. Dancing is not an easy sport to participate in, so if you are looking for a sport that does not require practice then you might want to look into a different sport. There are hour-long practices, healthy diets, and several routines to remember. Normally, if someone is convinced that dance is not a sport, they have not fully engaged or experienced dance. In order to better understand dance, it is important to understand the history of dance, the competition of dance, and the comparison of dance to other sports.
During early times in history dance was seen as a way to heal sickness, a way to break a spell, or perhaps to put a spell on someone.
This first section is based on what dance is. Dance is defined differently by different people. Some people define it as a sport and some people define it as an art. Some people do dance for different reasons. Some people do dance as a sport, others do it as therapy, and others just do it recreationally. According to an article named When will Dancers Be Considered Atheletes, Dance can be done competitively. Dance is extremely beneficial for several reasons. According to When will Dancers be considered athletes, football players found that they had strengthened their ankles and feet, had increased agility, and were less likely to get hurt because of muscle strength after practicing a form of dance named ballet ( Dancers: Artists or Athletes?).
Stephen Hawking has been hailed as one of the most brilliant theoretical physicists since Albert Einstein. Hawking was born on January 8, 1942, which as he likes to point out is the 300th anniversary of Galileo's death. Hawking originally studied at Oxford University in England studying physics even though he would have preferred math. He moved onto Cambridge University to work on his PhD in cosmology. Hawking's career has focused upon the cosmic entities known as black holes, and has extended to specialized areas such as quantum gravity, particle physics, and supersymmetry.
Leonardo created five mathematical works during his lifetime, and four of these became popular books about his discoveries. It has later been discovered that during his lifetime
Many people will insist that dance is not a sport, but they are wrong. Dance, though also an art, is an intense sport, especially ballet. Sports are physically demanding, require a commitment, involve being fit and active, and are competitive and enjoyable. Ballet fits every single criterion above. A professional ballerina attends class every day, including tests of flexibility, gracefulness, and strength. Ballerinas are also continually rehearsing for their performances, which involve stressful costume changes. Nevertheless, dancers love it. Indeed, ballet should be considered a sport by all. Ballet is an old art and intense discipline that has endured for hundreds of years with a long, complex, and remarkable history including famous men and women remembered to this day.
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...
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.
This means that math work with numbers, symbols, geometric shapes, etc. One could say that nearly all human activities have some sort of relationship with mathematics. These links may be evident, as in the case of engineering, or be less noticeable, as in medicine or music. You can divide mathematics in different areas or fields of study. In this sense we can speak of arithmetic (the study of numbers), algebra (the study of structures), geometry (the study of the segments and figures) and statistics (data analysis collected), between
There is scientific evidence that shows that listening to soothing classical music is effective in eliminating pain. This is a break-through treatment for achieving control over pain in patients. Test results show that the neurons in the amygdala and hypothalamus slow down their action potential while the patient is listening to relaxing music. In decreasing the sympathetic nervous system the parasympathetic takes over to decrease heart rate, respiration, and muscle tension. This calming effect on the immune system allows the body to relax, thereby allowing the medical staff to focus on medical intervention (Stuckey). For example, one patient who had Parkinson’s disease, it was reported that the tremors had been random and at times uncontrollable. After receiving music therapy, the tremors became mild and the patient was...