# Non-Euclidean Geometry Essays

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 non-Euclidean geometry, and the angles of triangles do not have a sum of 180 degrees.

Overall, non-Euclidean geometry follows almost all of the same postulates as Euclidean geometry. The main difference is non-Euclidean involves the study of curved surfaces, while Euclidean geometry involves the study of flat space. Around 1830, the Hungarian mathematician Janos Bolyai and a Russian mathematician named Nikolai Ivanovich Lobachevsky separately published studies on hyperbolic geometry. Both mathematicians spent years working with the fifth postulate.

Neither of them gained public recognition for the work they put into their geometric discoveries. Hyperbolic geometry is a type of non-Euclidean geometry that uses the statement “If l is any line and P is any point not on l, then there exists at least two lines through P that are parallel to l” or any statement equivalent to this statement as its parallel postulate. It is the study of saddle shaped space. It applies to areas of science such as determining the orbit of objects within intense gradational fields, space travel, and astronomy.

Einstein’s theory of relativity also involves hyperbolic geometry. There are many major differences between hyperbolic geometry and Euclidean geometry. In hyperbolic geometry, the sum of the angles of a triangle is less than 180 degrees and triangles with the same angles have the same area, but there are no similar triangles. In 1854, Bernhard Riemann founded Riemannian geometry, which elliptic geometry was a part of. Elliptic geometry is considered to be a non-Euclidean geometry due to its lack of parallel lines.

It has a parallel postulate is or is equivalent to “If l is any line and P is any point not on l, then there are no lines through P that are parallel to l”. Elliptic geometry deals with the study of curved surfaces such as a sphere. Like in hyperbolic geometry, there are many major differences between elliptic geometry and Euclidean geometry. The sum of the angles of any triangle is now always greater than 180 degrees. Also, there are no straight lines, as they will always curve. Non-Euclidean geometry is an example of a paradigm shift in the history of geometry.

Before the subjects of non-Euclidean geometry were brought up, Euclidean geometry stood unchallenged as the mathematical model of space. The Euclidean point of view was how people viewed the world. Without non-Euclidean geometry, the world would not have many of the everyday luxuries to which we have become accustomed. Without non-Euclidean geometry, we would not have many of the methods of digital entertainment and communication we use today because we would not be able to find a way to get satellites to orbit the Earth.

We also would have never been able to send astronauts into space because we would never have been able to design our rockets to get out of the gravitational pull the Earth holds on them. Non-Euclidean geometry has played a major role in allowing for technological developments that have completely transformed our world. Sources: http://regentsprep. org/Regents/math/geometry/GG1/Euclidean. htm http://noneuclidean. tripod. com/history. html