Fractals, a Mathematical Description of the World Around us
In being characterized with fractional dimensions, Fractals are considered to be a new division of math and art, which is perhaps why the common man recognizes them as nice-looking and appealing pictures that are valuable as background on computer screens and art patterns. But they are more meaningfully understood by way of the recognition that many of nature’s physical systems and a lot of human works of art are not standard geometry forms. Fractal geometry enables infinite methods of relating, evaluating and forecasting these kinds of natural phenomena. Simply said it’s a never ending pattern that repeats itself at different scales again and again. Thinking in patterns refers to fractals as “the approach to study roughness in both pure mathematics and in mathematical sciences of the real world (Novak 1).” Roughness is the idea that mathematics can measure obscure geometric patterns. This includes nature (turbulence, clusters of statistical physics, broken solids, art, finance, noises, and many more every day encounters). The issue that arises from the concept is whether it is possible to classify the entire world by making use of mathematical equations.
Benoit Mandelbrot (1982) is recognized as the father of fractals and he coined the term in describing objects, surfaces and curves having a number of extremely unusual properties. Other mathematicians; Cantor, Julia, Koch, and Peano had insight into fractal math, but it wasn’t until Mandelbrot came along that this math was widely understood.
In 1917, Gaston Julia came the closest to discovering the Mandelbrot set with his publication of Complex Numbers. Gaston Julia spent his life studying the iteratio...
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...in to understand fractals or at least appreciate them as beautiful art.
Works Cited
Batty, Michael. Fractals – Geometry between dimensions, New Scientist, 1985.
Dewy, David- Introduction to the Mandolin Set. 1996
Fractal Foundation. Fractal pack 1 an Educators guide, 2009,
Mandelbrot, Benoit B. The Fractal Geometry of Nature, W. H. Freeman, 1982.
Mosher Dave, Hidden Fractals Suggest Answer to Ancient Math Problem, Wired Science, 2011.
Mandelbrot, Benoit B., and M. M. Novak. Thinking in Patterns: Fractals and Related
Phenomena in Nature. River Edge, NJ: World Scientific, 2004.
Print.Patrzalek, Edtya. General Introduction to Fractal Geometry, 2011,
http://www.fractal.org, Accessed on 07 November, 2011.
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.
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...
The great field of mathematics stretches back in history some 8 millennia to the age of primitive man, who learned to count to ten on his fingers. This led to the development of the decimal scale, the numeric scale of base ten (Hooper 4). Mathematics has grown greatly since those primitive times, in the present day there are literally thousands of laws, theorems, and equations which govern the use of ten simple symbols representing the ten base numbers. The field of mathematics is ever changing, and therefor, there is a great demand for mathematicians to keep improving our skills in utilizing the numeric system. Many great people, both past and present, have made great contributions to the field. Among the most famous are Archimedes, Euclid, Ptolemy, and Pythagoras, all of which are men. This seems to be a common trend in mathematics, for almost all classical mathematicians were male.
Complex science provides an opportunity to break down something known to be a very large structure in order to see answers from a more individual point of view. It can become very beneficial in the long run depending on how it is applied. However, that’s just it you do not need to know how the entire global economic system works in order to benefit from it. When Benoit Mandelbrot created fractals they were used in various ways, to measure nature an...
Although these two men had substantial evidence prior to the seventeenth century their work was not recognized by other scholars of that era. It was not until Descartes, a systematist of his field, used his exper...
Born in the Netherlands, Daniel Bernoulli was one of the most well-known Bernoulli mathematicians. He contributed plenty to mathematics and advanced it, ahead of its time. His father, Johann, made him study medicine at first, as there was little money in mathematics, but eventually, Johann gave in and tutored Daniel in mathematics. Johann treated his son’s desire to lea...
Fibonacci was born in approximately 1175 AD with the birth name of Leonardo in Pisa, Italy. During his life he went by many names, but Leonardo was the one constant. Very little is known of his early life, and what is known is only found through his works. Leonardo’s history begins with his father’s reassignment to North Africa, and that is where Fibonacci’s mathematical journey begins. His father, Guilielmo, was an Italian man who worked as a secretary for the Republic of Pisa. When reassigned to Algeria in about 1192, he took his son Leonardo with him. This is where Leonardo first learned of arithmetic, and was interested in the “Hindu-Arabic” numerical style (St. Andrews, Biography). In 1200 Leonardo ended his travels around the Mediterranean and returned to Pisa. Two years later he published his first book. Liber Abaci, meaning “The Book of Calculations”.
On first thought, mathematics and art seem to be totally opposite fields of study with absolutely no connections. However, after careful consideration, the great degree of relation between these two subjects is amazing. Mathematics is the central ingredient in many artworks. Through the exploration of many artists and their works, common mathematical themes can be discovered. For instance, the art of tessellations, or tilings, relies on geometry. M.C. Escher used his knowledge of geometry, and mathematics in general, to create his tessellations, some of his most well admired works.
If you have ever heard the phrase, “I think; therefore I am.” Then you might not know who said that famous quote. The author behind those famous words is none other than Rene Descartes. He was a 17th century philosopher, mathematician, and writer. As a mathematician, he is credited with being the creator of techniques for algebraic geometry. As a philosopher, he created views of the world that is still seen as fact today. Such as how the world is made of matter and some fundamental properties for matter. Descartes is also a co-creator of the law of refraction, which is used for rainbows. In his day, Descartes was an innovative mathematician who developed many theories and properties for math and science. He was a writer who had many works that explained his ideas. His most famous work was Meditations on First Philosophy. This book was mostly about his ideas about science, but he had books about mathematics too. Descartes’ Dream: The World According to Mathematics is a collection of essays talking about his views of algebra and geometry.
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.
Burton, D. (2011). The History of Mathematics: An Introduction. (Seventh Ed.) New York, NY. McGraw-Hill Companies, Inc.
The history of math has become an important study, from ancient to modern times it has been fundamental to advances in science, engineering, and philosophy. Mathematics started with counting. In Babylonia mathematics developed from 2000B.C. A place value notation system had evolved over a lengthy time with a number base of 60. Number problems were studied from at least 1700B.C. Systems of linear equations were studied in the context of solving number problems.
Abstractions from nature are one the important element in mathematics. Mathematics is a universal subject that has connections to many different areas including nature. [IMAGE] [IMAGE] Bibliography: 1. http://users.powernet.co.uk/bearsoft/Maths.html 2. http://weblife.bangor.ac.uk/cyfrif/eng/resources/spirals.htm 3.
A rectangle is a very common shape. There are rectangles everywhere, and some of the dimensions of these rectangles are more impressive to look at then others. The reason for this, is that the rectangles that are pleasing to look at, are in the golden ratio. The Golden Ratio is one of the most mysterious and magnificent numbers/ratios in all of math. The Golden Ratio appears almost everywhere you look, yet not everyone has ever heard about it. The Golden Ratio is a special number that is equal to 1.618. An American mathematician named Mark Barr, presented the ratio using the Greek symbol “Φ”. It has been discovered in many places, such as art, architectures, humans, and plants. The Golden Ratio, also known as Phi, was used by ancient mathematicians in Egypt, about 3 thousand years ago. It is extraordinary that one simple ratio has affected and designed most of the world. In math, the golden ratio is when two quantities ratio is same as the ratio of their sum to the larger of the two quantities. The Golden Ratio is also know as the Golden Rectangle. In a Golden Rectangle, you can take out a square and then a smaller version of the same rectangle will remain. You can continue doing this, and a spiral will eventually appear. The Golden Rectangle is a very important and unique shape in math. Ancient artists, mathematicians, and architects thought that this ratio was the most pleasing ratio to look at. In the designing of buildings, sculptures or paintings, artists would make sure they used this ratio. There are so many components and interesting things about the Golden Ratio, and in the following essay it will cover the occurrences of the ratio in the world, the relationships, applications, and the construction of the ratio. (add ...