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/
5. Collected Papers, Charles Hartshorne and Paul Weiss, (edd.) (Cambridge: The Belknap Press of Harvard University Press, 1960). Volume and page number, respectively, noted in the text.
"Geometry has two great treasures; one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold, the second we may name a precious jewel."
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.
Geometry “[is] displaced, on the border between different worlds…” (Baym 2810). It is not only a common high school math class, but also an allegory for the mind. Dove took a simple childhood event and related it to the minds expansion as it becomes aware of the world around it. She uses her famous technique of moving through space to show the house disappearing and an open world appearing.
Chudnovky (2014) wonders if it might allude to the ancient problem of expressing pi in algebraic form. Terrance Lynch (1982) speculates that the polyhedron poses the mathematical problem of squaring a circle. Remarkably, Hideko (2009) believes that the solution to the Delian Problem of ancient Greece is hidden in the form of the polyhedron. The Delian Problem is, as described by Dürer: Using only a compass and straightedge, how does one double the volume of a cube? To date, it remains
"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.
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...
Euclid, who lived from about 330 B.C.E. to 260 B.C.E., is often referred to as the Father of Geometry. Very little is known about his life or exact place of birth, other than the fact that he taught mathematics at the Alexandria library in Alexandria, Egypt during the reign of Ptolemy I. He also wrote many books based on mathematical knowledge, such as Elements, which is regarded as one of the greatest mathematical/geometrical encyclopedias of all time, only being outsold by the Bible.
“I.M. Pei and the Geometry of the NGA.” National Gallery of Art. National Gallery of Art, 2013. 6 Feb, 2014. Web.
First, non-Euclidean geometry starts with two geometry methods. These are based off of axioms related to the ones that are sorting the concept of Euclidean geometry. Non-Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is set aside (Non-Euclidi...
books covered not only plane and solid geometry but also much of what is now
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.
In order for two lines to be parallel, they must be drawn in the same plane. We created lots of parallel lines in our drawing. These parallel lines formed planes. The parallel planes formed walls and rooms of our house.
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.
The 17th Century saw Napier, Briggs and others greatly extend the power of mathematics as a calculator science with his discovery of logarithms. Cavalieri made progress towards the calculus with his infinitesimal methods and Descartes added the power of algebraic methods to geometry. Euclid, who lived around 300 BC in Alexandria, first stated his five postulates in his book The Elements that forms the base for all of his later Abu Abd-Allah ibn Musa al’Khwarizmi, was born abo...