Parallels
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| Parallels | |
|---|---|
| Name | Parallels |
| Field | Mathematics |
| Introduced | Antiquity |
| Notable figures | Euclid, Pythagoras, Plato, Aristotle, Archimedes, Alhazen, Omar Khayyam, Ibn al-Haytham, Girard Desargues, René Descartes, Blaise Pascal, Jean-Victor Poncelet, Carl Friedrich Gauss, Bernhard Riemann, Nikolai Lobachevsky, János Bolyai, David Hilbert, Felix Klein, Élie Cartan, Henri Poincaré, Felix Klein, André Weil, Paul Erdős, John Conway, Donald Knuth, Alan Turing, Ada Lovelace, George Boole, Augustin-Louis Cauchy, Joseph-Louis Lagrange, Sofia Kovalevskaya, Emmy Noether, David Hilbert, Bernhard Riemann |
Parallels are relationships and configurations in which entities maintain constant separation, directional equivalence, or structural correspondence across contexts. In mathematical geometry, parallels typically denote lines or planes that never meet and preserve equidistance properties; in algebra and computing, the term extends to operations, structures, or processes that run alongside each other without interference. Parallels permeate the work of classical figures such as Euclid, Pythagoras, René Descartes, and modern researchers like Bernhard Riemann and David Hilbert, appearing across disciplines including Euclidean geometry, Projective geometry, Analytic geometry, Non-Euclidean geometry, and Parallel computing.
Definition and basic properties
In classical Euclidean geometry a parallel relation identifies two distinct straight lines in a plane that do not intersect, often formalized by the parallel postulate in Euclid's Elements and rephrased in Playfair's axiom. Parallel lines share directional equivalence and yield corresponding angle congruences in transversal configurations exemplified in proofs by Euclid and later treatments by René Descartes in Analytic geometry. In three-dimensional Euclidean space parallels generalize to planes and lines: a line can be parallel to a plane, two planes can be parallel, and skew lines describe non-intersecting, non-parallel spatial relationships studied by Girard Desargues and Jean-Victor Poncelet. In metric settings influenced by Bernhard Riemann and Nikolai Lobachevsky, parallelism can fail or take multiple forms, producing the multiple parallels of Hyperbolic geometry or the absence of parallels in Elliptic geometry. Algebraic analogues include parallelism of vectors in Linear algebra (scalar multiples), parallel morphisms in Category theory, and concurrency constraints in Ring theory and module homomorphisms.
Types and contexts (geometry, algebra, computing)
Geometry: In Euclidean geometry parallels follow Playfair's axiom, while Projective geometry treats parallels via points at infinity, embedding them into the Projective plane studied by Desargues and Blaise Pascal. In Non-Euclidean geometry, János Bolyai and Nikolai Lobachevsky developed hyperbolic models with infinitely many parallels through a point; Bernhard Riemann's elliptic perspective removes parallels entirely. Analytic approaches by René Descartes and Carl Friedrich Gauss use vector coordinates and inner products to detect parallelism via proportional components.
Algebra: In Linear algebra vectors are parallel when linearly dependent; bases and subspaces introduce concepts of parallel subspaces and complementarity investigated by Élie Cartan and David Hilbert. In Algebraic geometry parallelisms can appear in tangent directions on varieties explored by André Weil and Henri Cartan; in Group theory commuting subgroups sometimes model parallel processes in symmetry contexts examined by Élie Cartan.
Computing: In Parallel computing hardware and software execute tasks concurrently, relying on paradigms from Alan Turing's computation models and implementations by figures like Donald Knuth and John von Neumann. Parallel algorithms, distributed systems, and concurrency control use synchronization primitives, scheduling theories shaped by Edsger Dijkstra and Leslie Lamport, and architectures from companies such as Intel Corporation and IBM. Models like the PRAM and concepts in Graph theory map parallelism in dataflow, while parallels in programming languages draw on work by Ada Lovelace and Alonzo Church.
Historical development and notable contributors
Ancient foundations: Euclid codified parallelism in Elements via the parallel postulate; predecessors such as Pythagoras, Plato, Aristotle, and Archimedes developed geometric insight underlying parallel constructions. Medieval and Islamic scholars including Omar Khayyam and Ibn al-Haytham advanced parallels through conic and optical studies.
Renaissance to Enlightenment: Girard Desargues and Blaise Pascal investigated projective notions that reinterpret parallels as points at infinity; René Descartes introduced coordinates linking algebra to geometric parallels, while Carl Friedrich Gauss and Augustin-Louis Cauchy refined analytic and differential perspectives.
19th–20th centuries: Breakthroughs by Nikolai Lobachevsky, János Bolyai, and Bernhard Riemann reshaped the understanding of parallels into non-Euclidean frameworks; Felix Klein and Henri Poincaré popularized models linking group theory and geometry. In modern algebra and topology, Élie Cartan, Emmy Noether, David Hilbert, and André Weil extended parallel-like structures into symmetry, bundle theory, and sheaf theory. In computing, foundational work by Alan Turing, John von Neumann, and later by Donald Knuth and Edsger Dijkstra translated the metaphor of parallelism into practical concurrent computation.
Applications and examples
Architecture and engineering employ parallel lines and planes in drafting traditions from Vitruvius to modern firms; civil and mechanical designs use coordinate methods of René Descartes and Carl Friedrich Gauss for alignment. Cartography and navigation exploit parallels of latitude in Geography and celestial parallels in Astronomy following conventions of Claudius Ptolemy and instruments refined by Johannes Kepler.
In physics, parallel transport and geodesic parallelism arise in General relativity and differential geometry developed by Bernhard Riemann and applied by Albert Einstein; electromagnetic field lines and optical rays reference parallel approximations in models by James Clerk Maxwell and Thomas Young. In computer science, parallel algorithms accelerate computations in Graph theory, Numerical analysis, and Machine learning implementations on hardware from Intel Corporation and NVIDIA Corporation; distributed databases and fault-tolerant systems employ concurrency theories by Leslie Lamport and Edsger Dijkstra.
Educational curricula in Mathematics emphasize parallels in secondary and tertiary instruction from classical texts to modern treatises by Serge Lang and Ian Stewart.
Related concepts and generalizations
Related ideas include affine structures in Affine geometry, where parallels are preserved under affine transformations studied by Jean-Victor Poncelet; projective completion via points at infinity in Projective geometry; and curvature-driven departures from parallelism in Riemannian geometry by Bernhard Riemann. Algebraic generalizations involve linear dependence in Linear algebra, parallel transport in Differential geometry, and concurrency models in Theoretical computer science influenced by Alan Turing and Alonzo Church. Concepts such as geodesics, foliations, and bundle connections link parallelism across Topology, Differential topology, and Algebraic topology explored by Henri Poincaré and Élie Cartan.