Maddux Willingham
Mr. Warfle
Geometry Honors
26 September 2011
Euclidean & Non-Euclidean Geometry Paper
Isn’t it amazing that we still study the same geometry as people did back nearly twenty-three centuries ago? Euclidean and Non-Euclidean geometry communicates to us through mathematical equations immense amounts of significant information. Without the study of geometry, many people would be unemployed. Euclidean and Non-Euclidean geometry have several similarities, however they also have numerous differences, as well as their historical aspects.

To begin with, these mathematical concepts have many similarities. For example, both studies of geometry include perpendicular lines, the drawings for these lines may be different, but they still make 90 degree angles. And both geometries are used in physics by thousands of people daily. Euclidean geometry and Non-Euclidean geometries are both hard to grasp and have a lot of theorems and postulates that say a lot about each individual mathematical study. Both have quite a few similarities, but they have even more differences.

Next, there are several differences between Euclidean and Non-Euclidean geometries, such as those in Euclidean geometry the measure of all angles of a triangle add up to 180 degrees. Euclidean Geometry is the study of flat space, and there is only one form on Euclidean geometry. But there are several types of Non-Euclidean geometry, such as Riemmanian Geometry (or spherical geometry), which is the study of curved spaces. Riemmanian geometry says that in curved space, the sum of the angles of any triangle is now always greater than 180 degrees. It also says that there are no straight lines; once you start drawing a line it immediately curves due to the sphere on which it is drawn on. Another form of Non-Euclidean geometry is Hyperbolic geometry, which is the study of saddle shaped space. In Hyperbolic geometry the measurements of all angles of a triangle is less than 180 degrees...

...Geometry is simply the study of space. There are Euclidean and Non-Euclidean Geometries. Euclidean geometry is the most common and is the basis for other Non-Euclidean types of geometry. Euclidean geometry is based on five main rules, or postulates. Differences in these rules are what make new kinds of geometries. There is Euclidean, Elliptic, and Hyperbolic Geometry....

...Non-Euclidean geometry is any form of geometry that is based on axioms, or postulates, different from those of Euclidean geometry. These geometries were developed by mathematicians to find a way to prove Euclid’s fifth postulate as a theorem using his other four postulates. They were not accepted until around the nineteenth century. These geometries are based on a curved plane, whether it is elliptic or hyperbolic. There are no parallel lines in...

...Introduction
segment PQ:
In Euclidean geometry the perpendicular distance between the rays
remains equal to the distance from P to Q as we move to the right.
However, in the early nineteenth century two alternative geometries
were proposed. In hyperbolic geometry (from the Greek hyperballein,
"to exceed") the distance between the rays increases. In elliptic
geometry (from the Greek elleipein, "to fall short") the distance decreases and the rays eventually meet. These...

...Euclidean algorithm
In mathematics, the Euclidean algorithm, or Euclid's algorithm, is a method for computing the greatest common divisor (GCD) of two (usually positive) integers, also known as the greatest common factor (GCF) or highest common factor (HCF). It is named after the Greek mathematician Euclid, who described it in Books VII and X of his Elements.
The GCD of two positive integers is the largest integer that divides both of them without leaving...

...When it comes to Euclidean Geometry, Spherical Geometry and Hyperbolic Geometry there are many similarities and differences among them. For example, what may be true for Euclidean Geometry may not be true for Spherical or Hyperbolic Geometry. Many instances exist where something is true for one or two geometries but not the other geometry. However, sometimes a property is true for all three geometries. These points bring us to the purpose of this...

...Euclidean Geometry
Geometry was thoroughly organized in about 300 BC, when the Greek
mathematician Euclid gathered what was known at the time, added original work of
his own, and arranged 465 propositions into 13 books, called 'Elements'. The
books covered not only plane and solid geometry but also much of what is now
known as algebra, trigonometry, and advanced arithmetic.
Through the ages, the propositions have been rearranged, and many of the
proofs are different, but...

...that he didn’t really eat Wheaties for breakfast I hope that was just a joke question! Though not many facts are known about Euclid’s life, we all know that he was a major contribution to mathematics, thus having a math subject named after him, Euclidean Geometry....

...—————————————————————————————————————————
Question 1
The diagram represents a large cone of height 30 cm and base diameter 58 cm.
The large cone is made by placing a small cone A of height 10 cm and base diameter
5 cm on top of a frustum B.
(a) Calculate the volume of the frustum B.
Give your answer correct to 3 significant figures.
........................ cm3
(3 marks)
The diagram shows a frustum.
The diameter of the base is 3d cm and the diameter of the...

1337 Words |
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