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Symmetry of Scale Expository Paper
, 2006
"... A checkerboard, a beehive, a brick wall and a mud flat dried in the sun all have something in common. They are all examples of tilings or tessellations. Although you may think of mosaics or other pieces of artwork when you hear these words, in actuality you should also think of mathematics and scien ..."
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A checkerboard, a beehive, a brick wall and a mud flat dried in the sun all have something in common. They are all examples of tilings or tessellations. Although you may think of mosaics or other pieces of artwork when you hear these words, in actuality you should also think of mathematics and science. I will describe in more detail the mathematics involved with tessellations and tilings, and discuss specific tilings such as the Pinwheel Tiling and the Penrose Tiles. Dr. Math, from the “Ask Dr. Math ” website, defines the word “tessellate ” as: “to form or arrange small pieces (like squares) in a checkered or mosaic pattern”. Dr. Math also explained that “the word ‘tessellate ’ is derived from the Ionic version of the Greek word ‘tesseres, ’ which in English means ‘four’ ” (Drexel University, 19942006). Although these are helpful in understanding what tessellations are, these descriptions are quite limited. Tessellations are “created when a shape is repeated over and over again covering a plane without any gaps or overlaps ” (Drexel University, 19942006). The definitions mentioned previously refer to squares which is the reason for the reference to the number four. However, tessellations can involve any shape. (See figure 1)
Taylor Polynomials Expository Paper
, 2006
"... Before the age of calculators, studying functions such as sin x, cos x, ex, and ln x was quite time consuming. The graphs of these functions are important when studying their characteristics. James Gregory, a Scottish mathematician in the 17th century, made an important discovery about these functio ..."
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and less time consuming than evaluating a function like sin x. In this paper, I will look at the background needed before one can truly understand polynomials, the definition of Taylor polynomials, and how to use Taylor polynomials to approximate the functions I mentioned above. Polynomials To work
The Game of Nim Expository Paper
, 2006
"... The game of Nim is possibly one of the most frustrating games I have ever played. Just when I started to feel that I had figured the strategy out, my brother, who is a computer programmer, blew me out of the water. I should have known better than to take on a computer wizard. Many sources claim that ..."
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that Nim possibly originated in China. It was played with pieces of paper, coins or whatever objects they could find. Charles Bouton, a professor at Harvard University, gave Nim its ’ name around 1901. He named it after an archaic English word meaning to steal or to take away. Some people have noticed
The Exponential Function Expository Paper
, 2006
"... One of the basic principles studied in mathematics is the observation of relationships between two connected quantities. A function is this connecting relationship, typically expressed in a formula that describes how one element from the domain is related to exactly one element located in the range ..."
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One of the basic principles studied in mathematics is the observation of relationships between two connected quantities. A function is this connecting relationship, typically expressed in a formula that describes how one element from the domain is related to exactly one element located in the range (Lial & Miller, 1975). An exponential function is a function with the basic form f (x) = ax, where a (a fixed base that is a real, positive number) is greater than zero and not equal to 1. The exponential function is not to be confused with the polynomial functions, such as x 2. One way to recognize the difference between the two functions is by the name of the function. Exponential functions are called so because the variable lies within the exponent of the function (Allendoerfer, Oakley, & Kerr, 1977). These functions are often recognized by the fact that their rate of growth is proportional to their value (Bogley & Robson, 1999). This concept of exponential growth has been around much longer than at the dawn of calculus. Evidence of this dates back almost 4,000 years ago on a Mesopotamian clay tablet, which is now on display at the Louvre. The question translated from the stone slab simply asks, “How long will it take for a sum of money to double if invested at 20 percent interest rate compounded
The Vigenére Cipher Expository Paper
, 2006
"... French diplomat and cryptographer Blaise de Vigenére (15231596), developed the Vigenére Cipher in 16th century France in the mid1580s. Vigenére was on the court of Henry III of France. Vigenére developed a polyalphabetic coding system in which one letter of plain text may be encrypted as different ..."
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(18531856). It was left to a Prussian officer, Friedrich Wilhelm Kasiski, to share a statistical method for breaking the Vigenére Cipher in 1863, which will be discussed later in this paper. The Vigenére Cipher involves using a table that entails 26 shifts of the alphabet and a key word. The encoding
The Art Gallery Question Expository Paper
, 2006
"... MAT question Suppose you have an arbitrary room in an art gallery with v corners, and you wanted to set up a security system consisting of cameras placed at some of the corners so that each point in the room can be seen by one of the cameras. How many cameras do we need? (See the example at right fo ..."
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MAT question Suppose you have an arbitrary room in an art gallery with v corners, and you wanted to set up a security system consisting of cameras placed at some of the corners so that each point in the room can be seen by one of the cameras. How many cameras do we need? (See the example at right for a possible room with an interesting shape.) In the early 1900’s a graphic artist by the name of Paul Klee was studying art, most of which involved geometry. He spent his career drawing and etching unique combinations of angles and colors. By the end of his career he had over 10,000 paintings and pieces of art that expressed his love of color and geometry. He often wondered how the angles and color were connected. Victor Klee, not a relation to Paul Klee, was just starting his life as a mathematician as Paul Klee was coming to the end of his career. Victor Klee also liked geometry and art, and was drawn to Paul Klee’s work. Victor studied Paul Klee’s work and found that Paul was not only interested in art and geometry but also interested in infinite sums. Both of Paul’s interests drew Victor further into the studies of geometry and the idea of ‘infinite sums’. As Victor continued to study mathematics his questions regarding geometry and infinite sums led to the branch of mathematics known as computational geometry. During his long career as a mathematician, Victor Klee has made contributions to a wide variety of mathematics, such as discrete and computational geometry, convexity, combinatorics, graph theory, functional analysis, mathematical programming and optimization, and theoretical computer science. In the beginning of Victor Klee’s career when he was exploring geometry he became friends with M. C. Escher. Escher is another famous graphic artist. His art was known for geometric graphics and many were infinite in manner. In my seventh grade classes we discuss tessellations which are an important part of Escher’s work and many
Comparing Infinite Sets Expository Paper
, 2006
"... I have been assigned to explore the theorem stating that there is no largest (infinite) set as established and proven by Georg Cantor. To do this I need to start by defining what it means to say that a set is infinite. This can be quite difficult because the tendency might be to say that a set is in ..."
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I have been assigned to explore the theorem stating that there is no largest (infinite) set as established and proven by Georg Cantor. To do this I need to start by defining what it means to say that a set is infinite. This can be quite difficult because the tendency might be to say that a set is infinite if it is not finite, and I don’t believe that grants us the clarity of definition we are looking for. When trying to understand the size of a given set, the number of objects (elements) in the set, we may not be able to count them as the total might be quite large. So we look to pair them evenly with objects of other sets or proper subsets of themselves: this is known as finding a onetoone correspondence. A set A is finite if it is impossible to have a onetoone correspondence between the set A and a proper subset of the set A. (This is essentially the Pigeonhole Principle.) For example, the set {A, B, C, …Y, Z} is finite because we cannot pair every element in the alphabet with the proper subset consisting of the alphabet not including Z, the set = {A, B, C,…Y}. We run out of elements to pair with the final letter. Building on this definition, a set A is infinite if it IS possible to have a onetoone correspondence between the set A and a proper subset of A. For example, the natural
Just What Do You “Mean”? Expository Paper
, 2006
"... Part IB In Ancient Greece the Pythagoreans were interested in three means. The means were the arithmetic, geometric, and harmonic. The arithmetic mean played an important role in the observations of Galileo. Along with the arithmetic mean, the geometric and harmonic mean (formerly known as the subco ..."
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Part IB In Ancient Greece the Pythagoreans were interested in three means. The means were the arithmetic, geometric, and harmonic. The arithmetic mean played an important role in the observations of Galileo. Along with the arithmetic mean, the geometric and harmonic mean (formerly known as the subcontrary mean) are said to be instrumental in the development of the musical scale. As we explore the three Pythagorean Means we will discover their unique qualities and mathematical uses for helping us solve problems.
MultiParagraph Segmentation of Expository Text
, 1994
"... This paper describes TextTiling, an algorithm for partitioning expository texts into coherent multiparagraph discourse units which reflect the subtopic structure of the texts. The algorithm uses domainindependent lexical frequency and distribution information to recognize the interactions of multi ..."
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This paper describes TextTiling, an algorithm for partitioning expository texts into coherent multiparagraph discourse units which reflect the subtopic structure of the texts. The algorithm uses domainindependent lexical frequency and distribution information to recognize the interactions
Farey Sequences, Ford Circles and Pick's Theorem Expository Paper
, 2006
"... One of the ongoing themes through the Math in the Middle coursework has been the idea of identifying patterns. From our first course, Math as a Second Language, patterns have been useful to explain phenomena and determine future values. Some patterns are numerical but can be described using algebra. ..."
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One of the ongoing themes through the Math in the Middle coursework has been the idea of identifying patterns. From our first course, Math as a Second Language, patterns have been useful to explain phenomena and determine future values. Some patterns are numerical but can be described using algebra. Some are visual or geometric and also can be described using numbers and symbols. Many of these patterns have resurfaced in different forms and at different times in new and interesting ways. It has been a humbling experience to see the interconnectedness of seemingly unconnected ideas. Pick’s Theorem, Farey Sequences and Ford Circles are concepts quite different on the surface but linked in interesting ways.
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