Parallel (geometry)
Parallel lines are coplanar lines that never meet.
In geometry, two straight lines that lie in the same plane and extend infinitely without ever crossing are called parallel. Similarly, two flat planes in three-dimensional space that never meet are parallel. A line and a plane in three-dimensional Euclidean space are also considered parallel if they share no point. However, lines that are not in the same plane are referred to as skew, not parallel. For line segments and Euclidean vectors, parallelism is defined by having the same direction or exactly opposite directions, regardless of length.
The symbol for parallel is ∥, as in AB ∥ CD, meaning line AB is parallel to line CD. In Unicode, the parallel sign is U+2225, the not-parallel sign is U+2226, and the sign for "equal and parallel to" is U+22D5.
Parallelism is central to Euclid's parallel postulate and is primarily a concept in affine geometries, with Euclidean geometry being a specific case. In other geometries, like hyperbolic geometry, lines have analogous properties also called parallelism. The idea extends to non-straight curves and non-flat surfaces that maintain a fixed minimum distance without touching or intersecting.
In Euclidean geometry, for two parallel lines l and m, several equivalent properties hold: every point on m is the same minimum distance from l; m lies in the same plane as l but never meets it; and when a third line (a transversal) crosses both, the corresponding angles formed are equal. Although any of these could serve as a definition, the second property—non-intersection in the same plane—is usually chosen because it does not involve measurement. The other properties then follow from Euclid's parallel postulate.
Historically, the definition of parallel lines as straight lines in a plane that do not meet appears as Definition 23 in Book I of Euclid's *Elements*. Other Greek thinkers proposed alternatives, often while trying to prove the parallel postulate. Proclus credited Posidonius with defining parallel lines as equidistant lines, and Simplicius mentioned a modification by the philosopher Aganis. By the late 19th century in England, Euclid's *Elements* was still the standard school text, but new developments in projective and non-Euclidean geometry prompted reform textbooks. These texts varied in their treatment of parallels. Charles Dodgson (Lewis Carroll) criticized them in his play *Euclid and His Modern Rivals*. One r
- field
- Geometry
- key_concept
- Parallelism in Euclidean and non-Euclidean geometries
- symbol
- ∥ (U+2225)
- not_parallel_symbol
- ∦ (U+2226)
- equal_and_parallel_symbol
- ⋕ (U+22D5)
- defining_property
- Two lines in a plane that do not intersect
Lore & Background
The definition of parallel lines as a pair of straight lines in a plane which do not meet appears as Definition 23 in Book I of Euclid's Elements. Alternative definitions were discussed by other Greeks, often as part of an attempt to prove the parallel postulate. Proclus attributes a definition of parallel lines as equidistant lines to Posidonius and quotes Geminus in a similar vein. Simplicius also mentions Posidonius' definition as well as its modification by the philosopher Aganis.
At the end of the nineteenth century, in England, Euclid's Elements was still the standard textbook in secondary schools. The traditional treatment of geometry was pressured to change by new developments in projective geometry and non-Euclidean geometry, leading to several new textbooks. A major difference between these reform texts, both between themselves and between them and Euclid, is the treatment of parallel lines. One of the early reform textbooks was James Maurice Wilson's Elementary Geometry of 1868, which based its definition of parallel lines on the primitive notion of direction. Augustus De Morgan reviewed this text and declared it a failure, primarily on the basis of this definition. Charles Dodgson (Lewis Carroll) also devoted a large section of his play Euclid and His Modern Rivals to denouncing Wilson's treatment of parallels.
Other properties proposed as replacements for the definition of parallel lines did not fare much better. The equidistant line definition of Posidonius, expounded by Francis Cuthbertson in his 1874 text Euclidean Geometry, suffers from the problem that the points at a fixed given distance on one side of a straight line must be shown to form a straight line, which cannot be proved and must be assumed. The corresponding angles formed by a transversal property, used by W. D. Cooley in his 1860 text, requires a proof that if one transversal meets a pair of lines in congruent corresponding angles then all transversals must do so, again requiring a new axiom.
Reader's Guide
Parallelism is primarily a property of affine geometries, and Euclidean geometry is a special instance of this type. In some other geometries, such as hyperbolic geometry, lines can have analogous properties referred to as parallelism. The concept can also be generalized to non-straight parallel curves and non-flat parallel surfaces, which keep a fixed minimum distance and do not touch each other or intersect.
Given parallel straight lines l and m in Euclidean space, several properties are equivalent: every point on line m is at exactly the same minimum distance from line l (equidistant lines); line m is in the same plane as line l but does not intersect l; when lines m and l are both intersected by a transversal, the corresponding angles of intersection are congruent. The second property is usually chosen as the defining property of parallel lines in Euclidean geometry, with the other properties being consequences of Euclid's Parallel Postulate.
The distance between two parallel lines in a Euclidean plane can be found by locating two points (one on each line) that lie on a common perpendicular and calculating the distance between them. For non-vertical, non-horizontal parallel lines given by equations y = mx + b₁ and y = mx + b₂, the distance can be computed using the common perpendicular line with slope −1/m.
Did You Know?
- The parallel symbol ∥ has Unicode codepoint U+2225, and the not parallel symbol ∦ has codepoint U+2226.
- In three-dimensional Euclidean space, a line and a plane that do not share a point are also said to be parallel.
- Two noncoplanar lines are called skew lines, not parallel.
- The definition of parallel lines as equidistant lines is attributed to Posidonius by Proclus.
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