parallel axiom

Học thuật
Thân thiện
Definition

Noun: * Parallel Axiom: A fundamental postulate in Euclidean geometry. It states that through any given point not on a given line, there is exactly one line that can be drawn which is parallel to the given line. This axiom is also known as Euclid's fifth postulate.

Usage

The term is used exclusively in the context of geometry, specifically when discussing the foundations of Euclidean geometry and its alternatives. * The consistency of Euclidean geometry depends on the acceptance of the parallel axiom. * For centuries, mathematicians tried to prove the parallel axiom from Euclid's other four postulates. * The discovery that the parallel axiom could be replaced led to the development of non-Euclidean geometries.

Advanced Usage
  • The attempt to prove the parallel axiom from Euclid's other postulates was one of the most famous unsolved problems in mathematics until the 19th century.
  • By denying the parallel axiom, mathematicians like Gauss, Bolyai, and Lobachevsky developed consistent geometric systems where, through a point not on a line, infinitely many lines could be drawn parallel to the given line (hyperbolic geometry) or none could be drawn (elliptic geometry).
Variants and Related Words
  • Euclid's fifth postulate: A direct synonym for the parallel axiom.
  • Playfair's axiom: A modern, equivalent formulation of the parallel axiom, named after John Playfair. It is the version most commonly stated today: "In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point."
Synonyms
  • Euclid's fifth postulate
  • Playfair's axiom (equivalent formulation)
Related Concepts (Not Phrasal Verbs or Idioms)
  • Euclidean geometry: The system of geometry based on Euclid's axioms, including the parallel axiom.
  • Non-Euclidean geometry: A geometry in which the parallel axiom does not hold.
  • Axiom / Postulate: A statement that is accepted as true without proof, serving as a starting point for further reasoning.
Noun
  1. only one line can be drawn through a point parallel to another line

Từ đồng nghĩa