parallèles

Parallèles is the French term for parallel lines, a foundational concept in geometry and related disciplines. In a plane, two distinct lines are parallèles if they do not intersect. They share the same direction and, in analytic terms, have equal slopes; the distance between them remains constant.

Etymology: from Latin parallelus, from Greek para ("beside") and allos ("other").

In Euclidean geometry, the parallel postulate states that through any point not on a given line there exists exactly one line parallel to the given line. This unique parallel property underpins much of plane geometry. When a transversal intersects two parallels, corresponding angles are equal and alternate interior angles are equal.

In non-Euclidean geometries, the situation changes: in hyperbolic geometry, infinitely many lines can be drawn through a point and be parallel to a given line; in spherical geometry, no two distinct great circles are parallel because all great circles intersect. In projective geometry, parallel lines meet at a point at infinity, a concept formalized in the homogeneous-coordinates framework.

Notation: AB ∥ CD denotes that lines AB and CD are parallel.

Parallèles also appear in geography and cartography, where lines of latitude (parallels) run east–west and never meet, unlike meridians which converge at the poles. Outside pure math, parallelism informs design, architecture, and computer graphics where parallel lines contribute to depth and structure.