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A plane midway between the two given planes and parallel to them.
Given a line, there are an infinite number of different planes that it lies in.
The given description fits that of a cylinder
Infinitely many planes contain any two given points- it takes three (non-collinear) points to determine a plane.
If 2 points determine a line, then a line contains infinitely many planes.
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There is a subtle distinction between Euclidean, Hilbert and Non-Euclidean planes. Euclidean planes are those that satisfy the 5 axioms, while Non-Euclidean planes do not satisfy the fifth postulate. This means that in Non-Euclidean planes, given a line and a point not on that line, then there are two (or more) lines that contain that point and are parallel to the original line. There are geometries where there must be exactly one line through that point and parallel to the original line and then there are also geometries where no such line contains that point and is parallel to the original line.Basically, the fifth postulate can be satisfied by multiple geometries.
In hyperbolic geometry, lines are typically represented by arcs of circles that intersect the boundary of the hyperbolic plane orthogonally or by straight lines that extend infinitely in both directions. Unlike Euclidean geometry, where two parallel lines never intersect, hyperbolic planes can contain multiple lines that do not intersect a given line, leading to unique properties of parallelism. This results in a richer structure where the concepts of distance and angle differ significantly from those in Euclidean space.
There's no limit to the number of them. A more cool and sexy way to say the same thing might be: "An infinite number."
The Playfair Axiom (or "Parallel Postulate")
Euclid's parallel postulate.