Definition:Parallel (Geometry)/Lines
Definition
In the words of Euclid:
- Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in either direction, do not meet one another in either direction.
(The Elements: Book $\text{I}$: Definition $23$)
The contemporary interpretation of the concept of parallelism declares that a straight line is parallel to itself.
Comment
Expand each of these statements into well-defined pages in their own right
Until this has been finished, please leave {{refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
Different geometries allow different conditions for the existence of parallel lines.
- Euclidean geometry allows that exactly one line through a given point can be constructed parallel to a given line;
- Hyperbolic geometry allows for an infinite number of such;
- Elliptical geometry does not allow construction of such lines.
Reflexivity
Expand this and put into its own page
Until this has been finished, please leave {{refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
An attempt can be made to define parallelism by suggesting that the perpendiculars dropped from one (line or plane) to another (line or plane) are the same length everywhere along the line or plane, but this interpretation does not work in the context of non-Euclidean geometries, and is in fact no more than a derivable consequence of the definition of parallel as given here.
Also see
- Results about parallel lines can be found here.
Sources
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: parallel