Parallel to the plane: the condition and properties

parallel plane is a concept first appeared in the Euclidean geometry of more than two thousand years ago.

main characteristics of classical geometry

birth of this scientific discipline related to the well-known works of ancient Greek philosopher Euclid, wrote in the third century BC, the pamphlet "Elements".Divided into thirteen books, "Elements" is the supreme achievement of the entire ancient mathematics and outlines the fundamental tenets associated with the properties of plane figures.

classic condition of parallelism of planes was formulated as follows: the two planes may be called parallel to each other if they have no common points.This read Euclidean fifth postulate labor.

properties of parallel planes

In Euclidean geometry, they are isolated, usually five:

  • property first (describes the parallel planes and uniqueness).Through a single point, which lies outside of this particular plane, we can make one and only one parallel plane
  • second property (also known as the properties of the three parallel).In the case where the two planes are parallel with respect to the third, and between them they are parallel.
  • property third (in other words, it is called a property line intersecting parallel to the plane).If taken separately straight line intersects one of these parallel planes, it will cross and another.
  • fourth property (property of the straight lines carved on planes parallel to each other).When two parallel planes intersect the third (at any angle), the line of intersection are also parallel
  • property fifth (property describing the different segments of parallel lines that lie between planes parallel to each other).The segments of the parallel lines that lie between two parallel planes necessarily equal.

parallel planes in non-Euclidean geometry

Such an approach is particularly geometry of Lobachevsky and Riemann.If the geometry of Euclid implemented on flat spaces, then Lobachevsky negatively curved spaces (curved simply put), while Riemann it finds its realization in positively curved spaces (in other words - areas).There is a very common stereotypical view that Lobachevsky plane parallel (and also line) intersect.However, this is not true.Indeed the birth of hyperbolic geometry was associated with proof of Euclid's fifth postulate and changing views on it, but the very definition of parallel planes and straight lines means that they can not cross nor Lobachevsky, nor Riemann, in whatever spaces they are implemented.A change of heart and the language is as follows.In place of the postulate that only one plane parallel can be drawn through a point not on a given plane came another formulation: through a point that is not on this particular plane can take two, at least directly, that lieCurrent coplanar with and do not cross it.