Non Euclidean geometry

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Euclidean geometry is a mathematical study of geometry concerning explanations of terms, points, shape, and flat surfaces based on the assumptions of Greek mathematician Euclid. Euclid lived during the reign of Alexandria the great in the year 330 B.C. In the geometric analysis, the texts Elements of Euclid were the initial systematic analysis used. Even though most of Euclid’s work had used by the past Greek mathematicians, Euclid remains the founder of first developed and comprehensive deductive method of solving mathematical problems on paper Otal, 2014). His approach to tackling geometrical problems consisted of justifying all previous theorems based on predetermined numbers of axioms.  Therefore, Euclidean geometry primary refers to the study of plane areas. To demonstrate this, flat piece of paper is used or on the chalkboard. However, the concept of Euclidean geometry was unchallenged until the start of the ninetieth century when other forms of geometric solutions emerged. The modernized form of solving the mathematical problems termed as non-Euclidean geometry. Therefore, from that time on, the assumption that Euclidean geometry was the only means of describing physical space changed. Hence, this paper more specifically gives alternatives to Euclidean geometry and their practical applications.

Alternatives of Euclidean Geometry

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The alternatives of Euclidean geometry are grouped under non-Euclidean geometry. Non-Euclidean geometry relates to any form of geometry containing postulate the same to the negation of the Euclidean parallel axioms. Some of the examples of non-Euclidean geometry include Riemannian geometry and hyperbolic geometry. To the practical application of the geometries, this paper analyses each geometric alternative at a time. For instance, Riemannian geometry concerns the study of curved surfaces. Riemannian geometry, also called the elliptic geometry uses parallel postulates (Tromba, 2012). In practical terms, one should not consider working on a plane surface as in the case of Euclidean geometry, but on a curved area like a sphere. Today the Riemannian geometry is used in determining earth distance mathematically. The Riemannian geometry also helps in figuring out the sum of angles of a triangle which are more than eighty degrees. Since, on the sphere straight lines do not exist, so as one starts to draw a line it will automatically curve. Due to this development in mathematics, experts can navigate earth and locate any point on the globe.

The second alternative is hyperbolic geometry also termed as saddle geometry.

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The saddle or Lobachevskian geometry named after a Russian scholar called Nicholas Lobachevski. And just like Riemannian, Nicholas too studied non-Euclidean geometry (Otal, 2014). Hyperbolic geometry helps in the analysis of saddle shaped objects. For instance, instead of working on a Euclidean plane paper one works on the curved area with a shape of a saddle or Pringles potato fragments. However, in this geometrical analysis it is practically difficult to work out mathematical solutions. But it is still applicable in other scientific works like predicting orbiting bodies with powerful gradational fields, traveling in space and astronomy. Therefore, according to the words of Einstein that space is curved hence the use of hyperbolic geometry in science.

In conclusion, the two mathematical geometries were developed due to the weaknesses of Euclidean geometry. Since the Euclidean geometry could only solve mathematical geometric problems on flat paper, scientist needed a solution to work out sums even on curved areas as well as in space.

 

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