The Geometry of Euclides

Euclidean geometry is often called parabolic geometries, ie geometries which follow a set of propositions that are based on five postulates of Euclid has defined in his book The Elements.More specifically, Euclidean geometry is different from other types of geometry in the fifth postulate, often referred dengaan parallel postulate. Non-Euclidean geometry of the fifth postulate is replaced by one of two alternative postulates, and leads to the geometry of hyperbolic or elliptic geometry. There are two types of Euclidean geometry: the geometry of the field, which is a two-dimensional Euclidean geometry and solid geometry, which is a three-dimensional Euclidean geometry.
The five postulates of Euclid can be expressed as follows:1) It is possible to draw a straight line segment joining two points.2) It is possible to permanently extend the straight line segment himpunaniap continuously in a straight line.3) Given himpunaniap straight line segments, it is possible to have menggambarlingkaran segment as radius and one endpoint as center.4) All right angles with each other or congruent.5) If two lines are drawn so that they intersect a third in such a way that the sum of interior angles on one side less than two right angles, then those two lines, if extended far enough, must intersect each other on a particular side.
The fifth postulate is known as the parallel postulate. This parallel postulate states that given the set of straight line segments and each point is not that the line segment, there is one and only one straight line that passes through it and never cut the first row, no matter how far extended line segments. Although Euclid's fifth postulate can not be proved as a theorem, for years many claimed evidence was published. Many efforts are aimed to formulate the theorem for this postulate because it is necessary to prove an important result and it does not seem as intuitive as the other arguments. More than two thousand years of research the fifth postulate was found to be independent of the other four.This is the fifth postulate is to be continued to Euclidean geometry to be considered.
In 1826 Nikolay Lobachevsky and János Bolyai in 1832, developed independennon-Euclidean geometries are fully self-consistent in which the fifth postulate does not hold. Johann Carl Friedrich Gauss had found this but the results are not published.Euclid tried to avoid using the fifth postulate and succeeded in the first 28 propositions of The Elements, but for the proposition 29 he needed it. Part of geometry that can be derived using only the first four postulates of Euclid became known as absolute geometry. As stated above, the fifth postulate and hence describe the parallel postulate of Euclidean geometry. If part of the parallel postulate is replaced with "no line through the" geometry of the elliptical or round dijelaskan.Jika part of the parallel postulate is replaced by "at least two lines there came to pass that through the point that" the hyperbolic geometry are described.
As stated above, two types of Euclidean geometry, solid geometry and geometry of the field, which is clearly different.Geometry of the field is part of the geometry in two-dimensional space associated with an overview of the field, such as lines, circles and polygons. Solid Geometry is a part of the geometry of three-dimensional space associated with solid foods, such as polyhedra, spheres, and lines and fields. In both types of the fifth postulate of Euclidean geometry according to Euclid, but each figure illustrates the various types of space. Euclidean space is the space of all n-tuples of real numbers and is denoted as R ^ n.This space is a vector space and has a topological dimension (Lebesgue covering dimension) n (see topology). Contravariant and the number of covariance that himpunanara in Euclidean space. R ^ 1 is the real line, namely line with the scale fixed in accordance with the number of unique points on the line.Generalization real line in two-dimensional space is called Euclidean field and is denoted R ^ 2.
Sources:http://www.bookrags.com/research/euclidean-geometries-wom/