ESSAY ON ALTERNATIVES TO EUCLIDEAN GEOMETRY

  • INTRODUCTION

    The Euclidean geometry is named after Euclid, who was a Greek mathematician in 300 BC. Euclid’s geometry studies of flat space. The concepts are illustrated as; shortest distance between two places on a particular straight line, the sum of angles triangle adding to 180 degrees and the perpendicular line which cuts the original line by 90 degrees. The concepts are of great importance were used from ancient Greek to the modern times in the design of buildings, land surveys and predicting the location of mloving objects. However, over time, there have been other geometry concepts that been developed. They are commonly referred as non-Euclidean geometry. They are spherical geometry and hyperbolic geometry. The essay will discuss the alternatives to Euclidean geometry and their possible applications and areas of use.

    Alternatives to Euclidean geometry are as follows

    Euclidean Geometry is constructive. Postulates 1, 2, 3, and 5 assert the existence and uniqueness of certain geometric figures, and these assertions are of a constructive nature: that is, we are not only told that certain things exist, but are also given methods for creating them with no more than a straightedge. In this sense, Euclidean geometry is more concrete than many modern axiomatic systems such as set theory, which often assert the existence of objects without saying how to construct them, or even assert the existence of objects that cannot be constructed within the theory. Sincerely speaking, the lines on paper are models of the objects defined within the formal system, rather than instances of those objects. In addition to this we have an example in Euclidean where straight line has no width, but any real drawn line will. Nevertheless  nearly all modern mathematicians consider non constructive methods  just as sound as constructive ones, Euclid's constructive proofs often supplanted fallacious no constructive one. Euclid often used proof by contradiction. Euclidean geometry also allows the method of superposition, in which a figure is transferred to another point in space. For example, proposition I.4, side-angle-side congruence of triangles, is proved by moving one of the two triangles so that one of its sides coincides with the other triangle's equal side, and then proving that the other sides coincide as well. Some modern treatments add a sixth postulate, the rigidity of the triangle, which can be used as an alternative to superposition. To conclude Euclidean geometry is a diversified field of mathematics where mathematicians have explained technically and professionally, there new form of mathematics are used in real life like in the field of architecture.

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