The Sierpinski Triangle
Deep within the realm of fractal math lies a fascinating triangle filled with unique properties and intriguing patterns. This is the Sierpinski Triangle, a fractal of triangles with an area of zero and an infinitely long perimeter. There are many ways to create this triangle and many areas of study in which it appears.
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.
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The Sierpinski Triangle holds many secrets that have yet to be discovered, and I am intrigued by the triangle’s apparent simplicity that hides all of its unique properties. One of the most interesting properties of the Sierpinski Triangle is the Fractal Dimension. Although it appears to be two-dimensional on paper, the triangle is actually about 1.58 dimensional. As stated on the Chaos in the Classroom website, the formula for determining dimension is:
Log(number of self...
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...t. The Chaos Game can be applied to create other fractals and shapes, and is a major part of an entirely separate area of study: chaos theory. The fact that the Sierpinski Triangle transcends the boundaries of fractal and number theory proves that it is an important part of mathematics. Perhaps the Sierpinski Triangle still holds secrets that, if discovered, will change the way we think about mathematics forever.
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Bibliography
Crilly A.J., R.A. Earnshaw, H. Jones. Fractals and Chaos. New York: Springer-Verlag, 1991.
Devaney, Robert L. “Fractal Dimension”. 2 April, 1995. Boston University. 28 June, 2005 .
"Mandelbrot Set." Wikipedia: The Free Encyclopedia. 3 Aug 2005. 10 Aug 2004 .
Tricot Claude. Curves and Fractal Dimension. New York: Springer-Verlag, 1995.
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.
In the story a stranger visits a family's home because he used to live there back in 1949 and wants to reminisce. While visiting he goes to what was room back in 1949. The son in the family is working on math work. The stranger notices this and shows him a drawing that represents infinity. The drawing consists of a square with triangles drawn within it that gradually get smaller. This is infinity that can’t be perceived due to the fact the triangles
The development of the Chaos began with a computer and mathematic problems of random data that can calculate and predict patterns that repeat themselves. For example, it picks up the pattern of a person’s heart beat and the pattern of snowflakes hitting the ground. Researchers have found that the patterns may be viewed as “unstable”, “random” and “disorderly” they tend to mimic zig-zags, lightning bolts or electrical currents. This theory has not only been used by physicist, but has also been used by astronomers, mathematicians, biologists, and computer scientists. The Chaos Theory can be applied to predict air turbulence, weather and other underlying parts of nature that is not easily understood (Fiero, p.
A triangle has certain properties such as all of the angles. add up to 180o and even if we have never thought about it before we clearly recognise these properties ‘whether we want to or not’. Cottingham. J. 1986). The 'Secondary' of the 'Se A triangle’s real meaning is independent of our mind, just as God’s existence is.
Using the book, The Greedy Triangle, by Marilyn Burns, students will learn to identify and sort two-dimensional and three-dimensional objects.
It is a legend that has terrified sailors since Columbus first sailed towards America. Its name is not on any official map, but a quick Google search turns up 10,400,000 web pages, and 101,000 books. What legend is this? It is the legend of the Bermuda triangle. A host of theories attempts to explain the supposedly abnormal events in the Bermuda triangle in a supernatural or physically impossible way. These theories attract the most attention, and are what have promoted the Bermuda triangle to the status of “Legend”. Now, let us explore some of the more prominent ones, namely the Electronic Fog theory, the Hutchinson effect , and government experiments with advanced radar at AUTEC naval base.
-The Eiffel Tower is made out of many triangles some very small and some very big.
In mathematics, Pascal’s triangle is taught everywhere throughout schools. He also started probability theory that many if not all mathematicians today use. Pascal even changed science by his experiments on atmospheric pressure and later had units of pressure named after him for his study. Pascal also, has a law in physics named after him. His inventions were just as impactful. Pascal created one of the first digital calculators. Pascal also invented the core principles of the roulette machine when study a perpetual motion theory.
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.
Statistic images and landscapes, or know as fractal landscapes, and the way that this component works is that these statistic images...
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
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.
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……
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.
The Bermuda triangle is a place that boggles many scientists even in this day and age. The Bermuda Triangle, referred to by some as the Devil's Triangle, is in a western region of the North Atlantic Ocean where countless aircraft, ships and people have inexplicably disappeared. Throughout the years of 1955 and 1975 more than 428 vessels disappeared, along with 100 ships and 1000 lives (Obringer1). Where did these people and ships disappear off too and how come no remains were found is the mysterious question people yearn to find out. Back in the 1964, the Bermuda triangle was often nicknamed as The Devil's Islands, because sea travelers could hear various different screeching noises coming off the shores (Obringer1). The Bermuda Triangle is a whirl pool of mysterious occurrences where things have magically disappeared without any remains and no matter how many theories scientists come up with to solve the mysteries of this enchanted island, none come close to having answers for any incident that occurred on this island.