Euclidean Geometries
Euclidean geometry, sometimes called parabolic geometry, is a geometry that follows a set of propositions that are based on Euclid's five postulates (see Euclid's axioms), as defined in his book The Elements. More specifically, Euclidean geometry is different from other types of geometry in that the fifth postulate, sometimes called the parallel postulate, holds to be true. Non-Euclidean geometry replaces this fifth postulate with one of two alternative postulates and leads to hyperbolic geometry or elliptic geometry. There are two types of Euclidean geometry: plane geometry, which is two-dimensional Euclidean geometry, and solid geometry, which is three-dimensional Euclidean geometry.
Euclid's five postulates can be stated as follows:
1. It is possible to draw a straight line segment joining any two points.
2. It is possible to indefinitely extend any straight line segment continuously in a straight line.
3. Given any straight line segment, it is possible to draw a circle having the segment as a radius and one endpoint as its center.
4. All right angles are equal to each other or congruent.
5. If two lines are drawn so that they intersect a third in such a way that the sum of the interior angles on one side is less than two right angles, then those two lines, if extended far enough, must intersect each other on that particular side.
The fifth postulate is equivalent to what is known as the parallel postulate. The parallel postulate states that given any straight line segment and a point not on that line segment, there exists one and only one straight line which passes through that point and never intersects the first line, no matter how far the line segments are extended. Although Euclid's fifth postulate cannot be proven as a theorem, over the years many purported proofs were published. Much effort was devoted to formulating a theorem to this postulate since it was needed to prove important results and it did not seem as intuitive as the other postulates. After more than two millennia of study the fifth postulate was found to be independent of the other four. It is this fifth postulate that must hold for geometry to be considered Euclidean. In 1826 Nikolay Lobachevsky, and, in 1832, János Bolyai, independently developed entirely self-consistent non-Euclidean geometries in which the fifth postulate did not hold. Johann Carl Friedrich Gauss had previously discovered this but had not published his results. Euclid tried to avoid using this fifth postulate and was successful in the first 28 propositions of The Elements, but for the 29th proposition he needed it. That part of geometry that can be derived using only the first four of Euclid's postulates came to be known as absolute geometry. As stated above, the fifth postulate and therefore the parallel postulate describe Euclidean geometry. If part of the parallel postulate is replaced by "no line exists which passes through that point" then elliptic or spherical geometry is described. If part of the parallel postulate is replaced by "at least two lines exist that pass through that point" then hyperbolic geometry is described.
As was stated above, the two types of Euclidean geometry, plane geometry and solid geometry, are distinctly different. Plane geometry is that portion of geometry in two-dimensional space that deals with figures in a plane, such as the circle, line and polygon. Solid geometry is that portion of geometry in three-dimensional space that deals with solids, such as polyhedra, spheres, and lines and planes. In both types of Euclidean geometry Euclid's fifth postulate holds--but each describes figures in different types of space. Euclidean space is the space of all n-tuples of real numbers and is denoted as Rn. This space is a vector space and has topological dimension (Lebesgue covering dimension) n (see topology). Contravariant and covariant quantities are equivalent in Euclidean space. R1 is the real line, that is a line with a fixed scale on which numbers correspond to unique points on the line. The generalization of the real line in two-dimensional space is called the Euclidean plane and denoted R2.
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