Alternatives to Euclidean Geometry
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Alternatives to Euclidean Geometry
According to Johnson (2013) Euclidean Geometry , commonly known as high school geometry, is a mathematical study of geometry based on undefined terms such as points, lines and or planes; definitions and other theories of a mathematician known as Euclid (330 B.C.) While a number of Euclid’s research findings had been earlier stated by Greek Mathematicians, Euclid has received a lot of recognition for developing the very first comprehensive deductive systems. Euclid’s approach to mathematical geometry involved providing all the theorems from a finite number of axioms (postulates).
Euclidean Geometry is basically a study of flat surfaces. Geometrical concepts can easily be illustrated by drawings on a chalkboard or a piece of paper. A number of concepts are known in a flat surface. These concepts include, the shortest distance between points, which is known to be one unique straight line, the angle sum of a triangle, which adds up to 180 degrees and the concept of perpendicular to any line.( Johnson, 2013, p.45)
In his text, Mr. Euclid detailed his fifth axiom, the famous parallel axiom, in this manner:
If a straight line traversing any two straight lines forms interior angles on one side less than two right angles, the two straight lines, if indefinitely extrapolated, will meet on that same side where the angles smaller than the two right angles.
In today’s mathematics, the parallel axiom is simply stated as: through a point outside a line, there is only one line parallel to that particular line. Euclid’s geometrical concepts remained unchallenged until around early nineteenth century when other concepts in geometry started to emerge. The new geometrical concepts are majorly referred to as non-Euclidean geometries and are used as the alternatives to Euclid’s geometry. (Johnson, 2013).
Since early the periods of the nineteenth century, it is no longer an assumption that Euclid’s concepts are useful in describing all the physical space.
Non Euclidean geometry is a form of geometry that contains an axiom equivalent to that of Euclidean parallel axiom (postulate). There exist a number of non-Euclidean geometry. Some of the examples are discussed below: 1. Riemannian Geometry
Riemannian geometry is also known as spherical or elliptical geometry. This type of geometry is named after the German Mathematician by the name Bernhard Riemann. In 1889, Riemann discovered some shortcomings of Euclidean Geometry. He discovered the work of Girolamo Sacceri,an Italian mathematician, which was challenging the Euclidean geometry. Riemann geometry states that if there is a line l and a point p outside the line l, then there are no parallel lines to l passing through point p. (Wendling, 2001, p.103)
Riemann geometry majorly deals with the study of curved surfaces. It can be said that it is an improvement of Euclidean concept. Euclidean geometry cannot be used to analyze curved surfaces. This form of geometry is directly connected to our daily existence because we live on the planet earth, whose surface is curved.
A number of concepts on a curved surface have been brought forward by the Riemann Geometry. These concepts include, the angles sum of any triangle on a curved surface, which is known to be greater than 180 degrees; the fact that there are no lines on a spherical surface; in spherical surfaces, the shortest distance between any given two points, also known as ageodestic is not unique. For instance, there are several geodesics between the south and north poles on the earth’s surface that are not parallel. These lines intersect at the poles. 2. Hyperbolic geometry
Hyperbolic geometry is also known as Lobachevskian or saddle geometry. It states that if there is a line l and a point p outside the line l, then there are at least two parallel lines to line p. This geometry is named for a Russian Mathematician by the name Nicholas Lobachevsky. He, like Riemann, advanced on the non-Euclidean geometrical concepts. Hyperbolic geometry has a number of applications in the areas of science. These areas include the orbit prediction, astronomy and space travel. For instance Einstein suggested that the space is spherical through his theory of relativity, which uses the concepts of hyperbolic geometry.
According to Wendling (2001), hyperbolic geometry has the following concepts: a) The surface areas of any set of triangles having the same angle are equal, b) the angles sum of a triangle is less than 180 degrees, c) it is possible to draw parallel lines on an hyperbolic space and d) that there are no similar triangles on an hyperbolic space.
Conclusion
Due to advanced studies in the field of mathematics, it is necessary to replace the Euclidean geometrical concepts with non-geometries. Euclidean geometry is so limited in that it is only useful when analyzing a point, line or a flat surface. Non- Euclidean geometries can be used to analyze any form of surface. |

References
Johnson, R. A. (2013). Advanced Euclidean geometry. , : Courier Corporation.
Wendling, J. (2001). Advanced geometry. , : Lorenz Educational Press.