Parallelism of planes: condition and properties

Parallelism of planes is a concept that first appeared in Euclidean geometry more than two thousand years ago.

Basic characteristics of classical geometry

The birth of this scientific discipline is connected with the famous work of the ancient Greek thinker Euclid, who wrote in the third century BC the pamphlet of the "Beginning". Divided into thirteen books, the "Elements" were the highest achievement of all ancient mathematics and expounded the fundamental postulates connected with the properties of plane figures.

The classical condition for the parallelism of planes was formulated as follows: two planes can be called parallel if they have no common points among themselves. This was the fifth postulate of Euclidean labor.

Properties of parallel planes

In Euclidean geometry, as a rule, they are distinguished by five:

• The first property (describes the parallelism of planes and their uniqueness). Through a single point that lies outside a particular given plane, we can draw one and only one plane parallel to it

• The second property (also called the properties of three parallelisms). In the case when two planes are parallel with respect to the third, they are also parallel to each other.

• The third property (in other words, it is called the property of a straight line that intersects the parallelism of planes). If a single straight line crosses one of these parallel planes, it will intersect the other.

• The fourth property (the property of straight lines carved on planes parallel to each other). When two parallel planes intersect the third (at any angle), the lines of their intersection are also parallel

• The fifth property (a property describing segments of different parallel lines that are enclosed between planes parallel to each other). The segments of those parallel lines that are enclosed between two parallel planes are necessarily equal.

Parallelism of planes in non-Euclidean geometries

Such approaches are in particular the geometry of Lobachevsky and Riemann. If the geometry of Euclid was realized on flat spaces, then in Lobachevsky in negatively curved spaces (curved simply), and in Riemann, it finds its realization in positively curved spaces (in other words - spheres). There is a very widespread stereotyped view that Lobachevsky's parallel planes (and lines too) overlap. However, this is not true. Indeed, the birth of hyperbolic geometry was associated with the proof of the fifth postulate of Euclid and the change in views on it, but the very definition of parallel planes and lines implies that they can not intersect in neither Lobachevsky nor Riemann, in whatever spaces they are realized. A change in views and formulations was as follows. To replace the postulate that only one parallel plane can be drawn through a point that does not lie on a given plane, another formulation has come: through a point that does not lie on a given concrete plane, there can pass two, at least, straight lines that lie in a One plane from a given one and do not intersect it.