Parallelism of planes is a concept that first appeared in Euclidean geometry over two thousand years ago.
Main characteristics of classical geometry
The birth of this scientific discipline is associated 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 Elements was the highest achievement of all ancient mathematics and set out the fundamental postulates associated 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 do not have common points with each other. This was the fifth postulate of Euclidean labor.
Properties of parallel planes
In Euclidean geometry, there are usually five of them:
Property one (describes the parallelism of planes and their uniqueness). Through one point that lies outside a particular given plane, we can draw one and only one plane parallel to it
- Second property (also called the property of three parallels). When two planes areparallel 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 intersecting the parallelism of the planes). If a single straight line intersects one of these parallel planes, then it will intersect the other
Fourth property (property of straight lines cut on planes parallel to each other). When two parallel planes intersect with a third (at any angle), their intersection lines are also parallel
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 Euclid's geometry was realized on flat spaces, then Lobachevsky's geometry was realized in negatively curved spaces (simply curved), and in Riemann's it finds its realization in positively curved spaces (in other words, spheres). There is a very common stereotypical opinion that Lobachevsky's parallel planes (and lines too) intersect.
However, this is not true. Indeed, the birth of hyperbolic geometry was associated with the proof of Euclid's fifth postulate and the changeviews on it, however, the very definition of parallel planes and lines implies that they cannot intersect either in Lobachevsky or Riemann, no matter in what spaces they are realized. And the change in views and formulations 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 a given particular plane, two, at least, lines that lie in the same plane as the given one and do not intersect it.
