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

THIS ESSAY DISCUSSES DIFFERENT INTERPRETATIONS OF GEOMETRY ACCORDING TO DIFFERENT MATHEMATICIANS.

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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,

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