Topology
Topology is the study of those properties of geometric figures that are unchanged when the shape of the figure is twisted, stretched, shrunk, or otherwise distorted without breaking. It is sometimes referred to as "rubber sheet geometry" (West 577). Topology is a basic and essential part of any post school mathematics curriculum. Johann Benedict Listing introduced this subject, while Euler is regarded as the founder of topology. Mathematicians such as August Ferdinand Möbius, Felix Christian Klein, Camille Marie Ennemond Jordan and others have contributed to this field of mathematics. The Möbius band, Klein bottle, and Jordan curve are all examples of objects commonly studied. These and other topics prove to be intricate and fascinating mathematical themes.
Topologists are mathematicians who study qualitative questions about geometrical structures. They ask questions like does the structure have any holes in it? Is it all connected, or can it be separated into parts? Topologists are not concerned with size, straightness, distance, angle, or other such properties. An often-cited example is the London Underground map. This will not reliably tell you how far it is from Kings Cross to Picadilly, or even the compass direction from one to the other. However, it will tell you how the lines connect between them, using topological rather than geometric information (What 1).
Furthermore, if one figure can be distorted into another figure without breaking, then the two figures are described as being topologically equivalent to each other. Two examples of topologically equivalent figures are a coffee cup and doughnut, and groups of the letters of the alphabet. First, an object shaped like a doughnut is a torus. A torus can...
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The streetlamp outside paints shapes across the wall next to my bed. I can see them in the darkness, dull orange lines that have become familiar in my many restless nights. At the heart of their canvas, they intersect to form a rectangle. A rectangle? For months I believed in this reality of form with the inborn certainty that accompanies that which is obvious. I didn’t have to think about it. Nightly, I would study the shape in a sleep haze, unconsciously harboring knowledge of its regularity. Except that it is not a rectangle.
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.
Abstract geometric shapes are easily seen in “Always.” The subject of geometry is apparent from any angle. This sculpture has a unitary form of a long and large rectangular shape that bends several times in different directions and angles before springing into space. Mass controls the composition of this sculpture; it is a large sculpture with lots of volume. The mass creates a line of movement into space. There is a dynamic action of the geometric shapes extending into space. The sculpture appears unstable and off-balanced as if it is going to tip over.
In the science-fiction short story “And He Built a Crooked House” by Robert A. Heinlein, a mathematically inclined architect named Quintus Teal constructs a house based on the unfolded net of a tesseract in order to save on real estate costs. However, to Teal’s dismay, an earthquake occurs the night before he shows a friend the house, and the house had fallen through a section of space and seemingly had been shaken into an actual tesseract. Despite its mathematical basis, “And He Built a Crooked House” is a quality example of science-fiction.
The Columbian exchange was the widespread transfer of various products such as animals, plants, and culture between the Americas and Europe. Though most likely unintentional, the byproduct that had the largest impact from this exchange between the old and new world was communicable diseases. Europeans and other immigrants brought a host of diseases with them to America, which killed as much as ninety percent of the native population. Epidemics ravaged both native and nonnative populations of the new world destroying civilizations. The source of these epidemics were due to low resistance, poor sanitation, and inadequate medical knowledge- “more die of the practitioner than of the natural course of the disease (Duffy).” These diseases of the new world posed a serious
The Columbian Exchange is used to describe the transfer of peoples, animals, plants, and diseases between the New World and the Old World. There were many social, economic, political, environmental, and cultural changes that ensued; however, the Columbian Exchange evoked more of an economic change than anything else in both the New and Old Worlds. The diseases brought to the New World, slavery, Triangular Trade, and widespread use of sugar were major game changers that indicate the economic change was greatest.
Study of Geometry gives students the tools to logical reasoning and deductive thinking to solve abstract equations. Geometry is an important mathematical concept to grasp as we use it in our life every day. Geometry is the study of shape- and there are shapes all around us. Examples of geometry in everyday life are- in sport, nature, games and architecture. The game Jenga involves geometry as it is important to keep the stack of tiles at a 90 degrees angle, otherwise the stack of tiles will fall over. Architects use geometry everyday- it is essential when designing buildings- shape, angles and area and perimeter are some of the geometry concepts architects
Geometry, a cornerstone in modern civilization, also had its beginnings in Ancient Greece. Euclid, a mathematician, formed many geometric proofs and theories [Document 5]. He also came to one of the most significant discoveries of math, Pi. This number showed the ratio between the diameter and circumference of a circle.
"A shape is that which limits a solid; in a word, a shape is the limit of a solid."
Furthermore, the Möbius Strip composes of various mathematical properties. If a strip of paper is taken and it's ends taped together, it would most likely end up being a belt. It would be a loop with both an inside surface and an outside surface. But what if you took that same strip of paper and gave it a half twist before taping the ends together? The result would be this fascinating geometric complexity, which is known as the Möbius Strip. If a Möbius Strip is cut length-wise (all the way around) it will end up with a loop ...
Shapes are two- dimensional surfaces such as circles or squares, and forms are three-dimensional shapes like spheres or cubes. A concave form has a pushed-in surface like the inside of a bowl and a convex form has a raised surface like the outside of a bowl. When you are looking at shapes and forms, the shape that you see first is called a figure or positive shape and the area around it is called the ground or the negative shape. The natural curves in different objects, such as trees or clouds are called organic shapes. Geometric shapes and forms are precise and regular such as cubes, pyramids, and circles. A free-form is an irregular invented shape or form that has qualities of a geometric form or an organic form.
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.
There are six diagonal lines. At one end there are circles on them giving the impression of three circular prongs. At the other end the same size lines have cross connecting lines consistent with two square prongs. These perceptions can violate our expectations for what is possible often to a delightful effect.
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.
Many artists say that they were born to do art, that it was always in their blood and that they cannot remember a time that art was not a part of their lives. For me, this was exactly the opposite. I was always trying to do my best in science and mathematics and art was not even on my radar until I was a freshman in high school where I met my first inspiration for art, Zack Smithey. I was lucky enough to have Mr. Smithey as a guide for the start of my art career all four years in high school and he really pushed me to develop my portrait work. He helped me develop the foundation of my artwork, but at that point I was merely duplicating what I was seen and really had not developed an aesthetic of my own. For me, art was a challenge for me to