Different Interpretations Of Geometry According To Different Mathematicians (Essay Sample)

The Beauty of Mathematics:
Alternatives to Euclidean Geometry
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Introduction
Euclidean geometry is derived from a set of statements known as "postulates." Axioms are visualized as basic truths which require no proof. They were regarded to as "self-evident truths" in the olden times. In current times, postulates are seen as arbitrary rules which can differ. They are similar to game rules which are needed to start playing but does not necessarily signify the reality. One of these ancient axioms is the Euclidean "parallel line" postulate which states that parallel lines do not meet at any point. This axiom solely led to the rise of non-Euclidean geometries whereby it was replaced with statements like "Parallel lines meet at least once" in Riemannian geometry and "There are no parallel lines" in hyperbolic geometry. This paper seeks to provide alternatives to Euclidean geometry, which are based on different "parallel lines" postulates, and also to prove that these alternatives are as practical as Euclidean geometry.
Riemann Geometry
This geometry is also known as elliptical or spherical geometry named after Bernhard Riemann, who was a great German mathematician. It is a non-Euclidean geometry which substitutes the Euclidean "parallel postulate" with an alternative postulate that every pair of parallel lines will meet at some point. When working with Spherical geometry, the following statement holds: If A is any line and B is a random point which is not in A, then there are no lines passing through B that can be parallel to A (Daniels, 2014).
This geometry, unlike Euclidean geometry, deals with spherical surfaces. It has a one-on-one connection to our daily lives since we live in a spherical universe. This change of surface has several effects on Euclid's geometrical truths including the angle sum of a triangle is greater than 180 degrees, there are no straight lines on a sphere as all lines curve around the sphere, and the shortest distance between two points on a sphere is no longer unlike in Euclidean geometry (Daniels, 2014).
Hyperbolic Geometry
This geometry is also known as Lobachevskian or saddle geometry which is named after Russian mathematician Nicholas Lobachevsky. It is a non-Euclidean geometry in which the following parallel postulate holds: If A is any line and B is a random point which is not in A, then there is a minimum of two lines passing through B that can be parallel to A (Henderson, 2013).
This geometry, unlike Riemannian geometry, deals with hyperbolic surfaces which are curved in such a way the angle sum of a triangle is less than 180 degrees. This can be realized when dealing with saddle surfaces or the hyperbolic plane. This geometry is applied in certain areas of science such as astronomy, space tra...