Differences in Geometry
Geometry is the branch of mathematics that deals with the properties of space. Geometry is classified between two separate branches, Euclidean and Non-Euclidean Geometry. Being based off different postulates, theorems, and proofs, Euclidean Geometry deals mostly with two-dimensional figures, while Demonstrative, Analytic, Descriptive, Conic, Spherical, Hyperbolic, are Non-Euclidean, dealing with figures containing more than two-dimensions. The main difference between Euclidean, and Non-Euclidean Geometry is the assumption of how many lines are parallel to another. In Euclidean Geometry it is stated that there is one unique parallel line to a point not on that line.
Euclidean Geometry has been around for over thousands of years, and is studied the most in high school as well as college courses. In it's simplest form, Euclidean geometry, is concerned with problems such as determining the areas and diameters of two-dimensional figures and the surface areas and volumes of solids. Euclidean Geometry is based off of the parallel postulate, Postulate V in Euclid's elements, which states that, "If a straight line meets two other straight lines so as to make the two interior angles on one side of it together less than two right angles, the other straight lines, if extended indefinitely, will meet on that side on which the angles are less than two right angles."
For centuries, mathematicians tried to contradict Euclid's Postulate V, and determine that there was more than one line parallel to that of another. It was declared impossible until the 19th century when Non-Euclidean Geometry was developed. Non-Euclidean geometry was classified as any geometry that differed from the standards of Euclidean geo...
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...is a theorem in Euclidean Geometry, yet in Hyperbolic Geometry it is proved false by the above counter example (Both BA and BC are parallel to DE, yet BA is not parallel to BC). However, you may not be convinced that BA and DE are parallel.
Bibliography:
Bibliography:
Books:
O'Reilly, Geometry in a Nutshell, O'Reilly & Associates, Inc. California, ©1996.
Textbooks:
Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen, Geometry, Houghton Mifflin Company. Boston, ©1988.
John C. Peterson, Technical Mathematics 2nd Edition, Delmar Publishers, Inc. Washington, ©1997.
Reference:
Leon L. Bram, Funk & Wagnalls New Encyclopedia, Funk & Wagnalls, Inc. ©1990.
URL Reference:
NonEuclid: http://math.rice.edu/~joel/NonEuclid/
The Geometry of a Sphere: http://math.rice.edu/~pcmi/sphere/
Euclid propositions can be called theorems in common language. In the Book I Euclid main considerations was on geometry. He began with a long list of definitions which followed by the small number of basic statements to take the essential properties of points, lines, angles etc. then he obtained the remaining geometry from these basic statements with proofs. (Berlinghoff, 2015, p.158).
According to Roland Shearer (1992) the release of non-Euclidean geometries at the end of the 19th Century copied the announcement of art movements occurring at that time, which included Cubism, Constructivism, Orphism, De Stijl, Futurism, Suprematism and Kinetic art. Most of the artists who were involved in these beginnings of Modern art were directly working with the new ideas from non-Euclidean geometry or were at least exposed to it – artists such as Picasso, Braque, Malevich, Mondrian and Duchamp. To explain human-created geometries (Euclidean, non-Euclidean), it is a representation of human-made objects and technology (Shearer
The Mayans used tons and tons of geometry throughout their creations. Which is obviously figured out just by thinking of the one thing that most of everybody knows and relates to the Mayans, the calendar, and the Aztec’s then took the Mayan calendar and adapted it to come up with their own calendar. They probably used trial and error, I’m sure of. They created many drawings that all involved geometry in one form or another.
The construction phase would not be possible without the knowledge of basic geometry. Points, lines, measurements and angles are often used to lay out the building in accordance to the architect drawings.
Most of geometry is based on two main constructions, circles and straight lines. In geometry, there are many different tools used for construction such as the compass, the straightedge, carpenter’s square, and mirrors. (Princeton) A compass is an instrument that is used to help draw circles. The two most well-known compasses are the modern compass and the collapsible compass. The straightedge is a tool that has no curves. It is used to draw straight line when knowing two points. (Princeton) The only difference between a straightedge and a ruler is that a ruler has measurements while a straightedge does not. A carpenter’s square is two straight edges connected to look like an “L”. This tool allows others to construct right angles. (Princeton) Finally mirrors are used to see an object’s reflection. These tools have been used for years to make the construction of geometric shapes easier. (Princeton)
In this essay the conic sections in taxicab geometry will be researched. The area of mathematics used is geometry. I have chosen this topic because it seemed interesting to me. I have never heard for this topic before, but then our math teacher presented us mathematic web page and taxicab geometry was one of the topics discussed there. I looked at the topic before and it encounter problems, which seemed interesting to explore. I started with a basic example, just to compare Euclidean and taxicab distance and after that I went further into the world of taxicab geometry. I explored the conic sections (circle, ellipse, parabola and hyperbola) of taxicab geometry. All pictures, except figure 12, were drawn by me in the program called Geogebra.
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.
Believe it or not, Geometry is actually useful! All our lives we have been told that we would use this in our lives and we have thought “no we won't.” but we do use it in life. Geometry is used for home decorating. Also architects use it for home building.
Janos Bolyai was born in December 1802 in Kolozsvar, Hungary. Janos’ father, Farkas Bolyai, was also a mathematician. This most likely where Janos attained his touch in mathematics. He taught Janos much about mathematics and other skills. Janos proved to be a sponge soaking up every bit of knowledge given to him. Farkas Bolyai was a student of mathematical genius Carl Friedrich Gauss, a German mathematician who had made many mathematical discoveries. He tried to persuade Gauss to take Janos and give him the education that Farkas himself had gotten, but Gauss turned him down. This didn’t slow down Janos in his education. He had an amazing learning ability and was able to comprehend complex mathematics at a young age as well as quickly learning new languages. Farkas claimed that Janos had learned everything that Farkas could teach him by the time he was fifteen. Janos could speak many languages, and was very knowledgeable in calculus, trigonometry, algebra, and geometry. He was also a student at the Academy of Military Engineering in Vienna at the young age of 16. He studied for 4 years completing his degree in a little over half the time it took most students. Janos became interested in the problem of the axiom of parallelism or Euclid’s 5th postulate which states, “if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” This was a theory that many mathematicians had tried to prove or disprove using the other postulates since it was created. He was determined to solve the problem despite the attempted dissuasion of his father as his father had also studied the subject extensively with little result. Janos continued to study this subject for sometime even though the college he attended did not have much to teach him in the mathematics field as he already knew most all of it. There is evidence that while still in college, Janos had created a new concept of the axiom of parallelism and a new system of non-Euclidian Geometry. Janos found that it was possible to have consistent geometries that did not fall under the rule of the parallel postulate. Janos’ conclusion was this “The geometry of curved spaces on a saddle-shaped plane, where the angles of a triangle did not add up to 180° and apparently parallel lines were NOT actually parallel.
Physical-mathematical knowledge was the first to understand the conventional character that is typical of axiomatic reasoning: ".. which firstly, and in the most rigorous manner, became conscious of the symbolic character of its fundamental instruments" [Cassirer, 1929]. The attempt to render Euclid's works without contradictions has caused a review of the form in which scientific work is carried out [Saccheri, 1733]. The verification of the existence of many types of points and lineshas sanctioned the distinction, even in the field of knowledge, between common language and technical language, clarifying once and for all that it is the the type of link established between the symbol and the meaning that provides the symbol with its significance.
Russell was, early on in life, fascinated by geometry -- in fact, he found an inherent beauty in it. He approached everything in life analytically, and of course mathematics ...
"The Foundations of Geometry: From Thales to Euclid." Science and Its Times. Ed. Neil Schlager and Josh Lauer. Vol. 1. Detroit: Gale, 2001. Gale Power Search. Web. 20 Dec. 2013.
Analytic geometry combines algebra and geometry in a way that allows for the visualization of algebraic functions. Rene Descartes, a French philosopher, and Pierre de Fermat, a French lawyer, independently founded analytic geometry in the early 1600s. Analytic geometry subsequently paved the way for calculus and physics.
Euclid also came up with a number of axioms and proofs, which he called “postulates.” Some of these postulates relate to all sciences, while other postulates relate only to geometry. An example of a Euclidean postulate that relates to all sciences is “The whole is greater than the part.” An example of a Euclidean postulate relating only to geometry is “You can draw a straight line between any two points.” Although these postulates seem extremely simple and obvious to us, Euclid was the first person to state them, as well as prove them to be true without question. These simple postulates really help with more complicated math and sciences, such as advanced geometry. For example, when doing advanced geometry involving a lot of lines and shapes, it is extremely helpful to know for sure that any single line can never contain more than one parallel line.
Then in Euclid II, 7, it goes farther to explain that “if a straight line be cut at random, the square on the whole and that on one of the segments both together, are equal to twice the rectangle contained by the whole and said segm...