Parallelism of planes: condition and properties

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

The main characteristics of classical geometry

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

The classical condition of parallel planes was formulated as follows: two planes can be called parallel if they do not have common points with each other. This was stated by the fifth postulate of Euclidean labor.

Properties of parallel planes

In Euclidean geometry, they are usually distinguished by five:

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

• The second property (also called the property of three parallelities). 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 intersects one of these parallel planes, then it intersects 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), their intersection lines are also parallel

• The fifth property (a property that describes 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 Euclidean geometry was realized on flat spaces, then for Lobachevsky in negatively curved spaces (curved simply put), and in Riemann it finds its realization in positively curved spaces (in other words, spheres). There is a very common stereotypical opinion that at Lobachevsky parallel planes (and lines too) intersect.

However, this is not true. Indeed, the birth of hyperbolic geometry was connected with the proof of the fifth postulate of Euclid and a change of views on him, however, the very definition of parallel planes and lines implies that they cannot intersect at either Lobachevsky or Riemann, in whatever spaces they are realized. And the change in views and wording was as follows. The postulate that only one parallel plane can be drawn through a point that does not lie on a given plane has been replaced by another formulation: through a point that does not lie on this particular plane, at least two straight lines that lie in one plane with a given one and do not cross it.