Parallelpostulaat

Parallelpostulaat, or Euclid’s fifth postulate, is a foundational axiom in plane geometry about parallels. In its standard form it states that through a point not on a given line there exists exactly one line that does not intersect the given line. This single postulate governs the behavior of lines and angles in Euclidean geometry.

Modern formulations often express the same idea as Playfair’s axiom: through a fixed point not on a line there is exactly one line parallel to the given line. The parallel postulate is also equivalent to the statement that the interior angles of a Euclidean triangle sum to 180 degrees. Together with the other four postulates, it characterizes flat (zero-curvature) geometry.

Historically, Euclid presented the postulate among his Elements as the fifth postulate, but it was long thought that it could be derived from the first four. In the 19th century, mathematicians such as Bolyai, Lobachevsky, and Gauss showed that the parallel postulate is independent of the first four, giving rise to non-Euclidean geometries. In hyperbolic geometry, through a point not on a line there are infinitely many parallel lines to the given line; in elliptic geometry, no true parallels exist.

The parallel postulate thus marks a boundary between Euclidean and non-Euclidean geometries. It underpins classical plane geometry and influences various fields, including mathematics, physics, and computer graphics, where the assumption of flat space leads to standard geometric intuitions and calculations.