Alternatives to Euclidean Geometry (Essay Sample)

TITLE: ALTERNATIVES TO EUCLIDEAN GEOMETRY
NAME:
INSTITUTION: TOP WRITERS LIST
TEST ESSAY
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 moving 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.
Hyperbolic geometry
It is also called as the saddle geometry or Lobachevskian geometry (Roberts, 2014). It is named Lobachevskian after Nicholas Lobachevsky, a Russian mathematician, who furthered the non-Euclidean Geometry. Hyperbolic geometry studies saddle-shaped space, like the outer surface of the horse saddle. In hyperbolic geometry, the circle of fixed radius has more surface area than the flat surfaces.
In the hyperbolic geometry, the following concepts hold;
* The angles in a triangle do not sum to 180 degrees.
* There are no similar triangles.
* Triangles that have same angles have the same area.
* Lines that are drawn in the hyperbolic space are a parallel and should not intersect.
* The perpendicular lines in hyperbolic geometry are from tangents, as illustrated below.
(Source: Roberts, 2014)
It has applications to areas of science that include orbit prediction of objects in intense gradational fields, astronomy and space travel. Moreover, the geometry is used in the research of the part of curvature in molecular materials; the role of a hyperbolic surface in describing the properties of crystalline materials (People.physics.anu.edu.au, 2014).
Elliptic Geometry
It also called the Riemannian geometry or the spherical geometry. It was named after Bernhard Riemann. It involves the study of curved surfaces and the figures on the sphere’s surface. The sphere is a three-dimensional surface that is made up of the set of points in space, at a length from the center. The antipodal points are formed by the intersection of the sphere and the line passing through the sphere’s center.
In the Elliptic geometry, the following concepts are followed:
* The spherical triangle is a 3-sided region that is enclosed by arcs of the great circle
The sum of angles in triangles is greater than 180 degrees. The triangle...