Instruct: Given a question, retrieve relevant documents that answer the question.
Query: Who is considered the founder of geometry?
Classic geometry was focused in compass and straightedge constructions. Geometry was revolutionized by Euclid, who introduced mathematical rigor and the axiomatic method still in use today. His book, The Elements is widely considered the most influential textbook of all time, and was known to all educated people in the West until the middle of the 20th century.[1]
Beniamino Segre (16 February 1903 – 2 October 1977) was an Italian mathematician who is remembered today as a major contributor to algebraic geometry and one of the founders of finite geometry.
Projective geometry is an elementary non-metrical form of geometry, meaning that it is not based on a concept of distance. In two dimensions it begins with the study of configurations of points and lines. That there is indeed some geometric interest in this sparse setting was first established by Desargues and others in their exploration of the principles of perspective art. In higher dimensional spaces there are considered hyperplanes (that always meet), and other linear subspaces, which exhibit the principle of duality. The simplest illustration of duality is in the projective plane, where the statements "two distinct points determine a unique line" (i.e. the line through them) and "two distinct lines determine a unique point" (i.e. their point of intersection) show the same structure as propositions. Projective geometry can also be seen as a geometry of constructions with a straight-edge alone. Since projective geometry excludes compass constructions, there are no circles, no angles, no measurements, no parallels, and no concept of intermediacy. It was realised that the theorems that do apply to projective geometry are simpler statements. For example, the different conic sections are all equivalent in (complex) projective geometry, and some theorems about circles can be considered as special cases of these general theorems.
In traditional geometry, affine geometry is considered to be a study between Euclidean geometry and projective geometry. On the one hand, affine geometry is Euclidean geometry with congruence left out; on the other hand, affine geometry may be obtained from projective geometry by the designation of a particular line or plane to represent the points at infinity. In affine geometry, there is no metric structure but the parallel postulate does hold. Affine geometry provides the basis for Euclidean structure when perpendicular lines are defined, or the basis for Minkowski geometry through the notion of hyperbolic orthogonality. In this viewpoint, an affine transformation is a projective transformation that does not permute finite points with points at infinity, and affine transformation geometry is the study of geometrical properties through the action of the group of affine transformations.
Because of the essential role of the circle (considered as the non-degenerate conic in a projective plane) and the plane description of the original models the three types of geometries are subsumed to planar circle geometries or in honor of Walter Benz, who considered these geometric structures from a common point of view, Benz planes.
Nothing of his life is known about the ten years after his graduation. In unknown date he went to Moscow where he taught in the German "Gymnasium Peter and Paul" of this city from 1865. Peterson never had an academic position at university level, but he was one of the founders of the Moscow Mathematical Society with Nikolai Brashman and August Davidov. Peterson was a notable collaborator in the journal of the Society and he is considered the founder of the Moscow school of geometry.
In relativity, rather than considering Euclidean, elliptic and hyperbolic geometries, the appropriate geometries to consider are Minkowski space, de Sitter space and anti-de Sitter space, corresponding to zero, positive and negative curvature respectively.
According to Panofsky, who wrote about the print three times between 1923 and 1964, "Melencolia I" combines the traditional iconographies of melancholy and geometry, both governed by Saturn. Geometry was one of the Seven Liberal Arts and its mastery was considered vital to the creation of high art, which had been revolutionised by new understandings of perspective. In the engraving, symbols of geometry, measurement, and trades are numerous: the compass, the scale, the hammer and nails, the plane and saw, the sphere and the unusual polyhedron. Panofsky examined earlier personifications of geometry and found much similarity between Dürer's engraving and an allegory of geometry from Gregor Reisch's "Margarita philosophica", a popular encyclopedia.
Digital geometry heavily overlaps with discrete geometry and may be considered as a part thereof.
In the nearly two thousand years since Euclid, while the range of geometrical questions asked and answered inevitably expanded, the basic understanding of space remained essentially the same. Immanuel Kant argued that there is only one, "absolute", geometry, which is known to be true "a priori" by an inner faculty of mind: Euclidean geometry was synthetic a priori. This dominant view was overturned by the revolutionary discovery of non-Euclidean geometry in the works of Bolyai, Lobachevsky, and Gauss (who never published his theory). They demonstrated that ordinary Euclidean space is only one possibility for development of geometry. A broad vision of the subject of geometry was then expressed by Riemann in his 1867 inauguration lecture "Über die Hypothesen, welche der Geometrie zu Grunde liegen" ("On the hypotheses on which geometry is based"), published only after his death. Riemann's new idea of space proved crucial in Einstein's general relativity theory, and Riemannian geometry, that considers very general spaces in which the notion of length is defined, is a mainstay of modern geometry.
Those known as the father, mother, or considered a founder in a Humanities field are those who have made important contributions to that field. In some fields several people are considered the founders, while in others the title of being the "father" is debatable. Some of the people who have humanity are given in REFERENCES.
With Arthur Cayley, Menger is considered one of the founders of distance geometry; especially by having formalized definitions to the notions of "angle" and of "curvature" in terms of directly measurable physical quantities, namely ratios of "distance" values.
Ernest Julius Wilczynski (November 13, 1876 – September 14, 1932) was an American mathematician considered the founder of projective differential geometry.
A finite geometry is any geometric system that has only a finite number of points.
The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are many systems that could be called finite geometries, attention is mostly paid to the finite projective and affine spaces because of their regularity and simplicity. Other significant types of finite geometry are finite Möbius or inversive planes and Laguerre planes, which are examples of a general type called Benz planes, and their higher-dimensional analogs such as higher finite inversive geometries.
During 1871, while at Göttingen, Klein made major discoveries in geometry. He published two papers "On the So-called Non-Euclidean Geometry" showing that Euclidean and non-Euclidean geometries could be considered metric spaces determined by a Cayley-Klein metric. This insight had the corollary that non-Euclidean geometry was consistent if and only if Euclidean geometry was, giving the same status to geometries Euclidean and non-Euclidean, and ending all controversy about non-Euclidean geometry. Arthur Cayley never accepted Klein's argument, believing it to be circular.
In May 1996, the annual Geometry and Topology conference which was held at Harvard University was dedicated to commemorating the 30th anniversary of the journal and the 80th birthday of its founder. Similarly, in May 2008 Harvard held a conference dedicated to the 40th anniversary of the "Journal of Differential Geometry".
Thābit ibn Qurrah made important discoveries in algebra, geometry, and astronomy. In astronomy, Thābit is considered one of the first reformers of the Ptolemaic system, and in mechanics he was a founder of statics.
According to Guerraggio & Nastasi (page 9, 2005) Luigi Cremona is "considered the founder of the Italian school of algebraic geometry". Later they explain that in Turin the collaboration of Enrico D'Ovidio and Corrado Segre "would bring, either by their own efforts or those of their students, Italian algebraic geometry to full maturity". A one-time student of Segre, H.F. Baker wrote (1926, page 269), [Corrado Segre] "may probably be said to be the father of that wonderful Italian school which has achieved so much in the birational theory of algebraical loci." On this topic, Brigaglia & Ciliberto (2004) say "Segre had headed and maintained the school of geometry that Luigi Cremona had established in 1860." Reference to the Mathematics Genealogy Project shows that, in terms of "Italian doctorates", the real productivity of the school began with Guido Castelnuovo and Federigo Enriques. In the USA Oscar Zariski inspired many Ph.D.s.
It was Gauss who coined the term "non-Euclidean geometry". He was referring to his own, unpublished work, which today we call "hyperbolic geometry". Several authors still consider "non-Euclidean geometry" and "hyperbolic geometry" to be synonyms. In 1871, Felix Klein, by adapting a metric discussed by Arthur Cayley in 1852, was able to bring metric properties into a projective setting and was thus able to unify the treatments of hyperbolic, euclidean and elliptic geometry under the umbrella of projective geometry. Klein is responsible for the terms "hyperbolic" and "elliptic" (in his system he called Euclidean geometry "parabolic", a term which has not survived the test of time and is used today only in a few disciplines.) His influence has led to the common usage of the term "non-Euclidean geometry" to mean either "hyperbolic" or "elliptic" geometry.
Antonio Maria Bordoni (19 July 1789 – 26 March 1860) was an Italian mathematician who did research on mathematical analysis, geometry, and mechanics. Joining the faculty of the University of Pavia in 1817, Bordoni is generally considered to be the founder of the mathematical school of Pavia. He was a member of various learned academies, notably the Accademia dei XL. Bordoni's famous students were Francesco Brioschi, Luigi Cremona, Eugenio Beltrami and Felice Casorati.
János Bolyai (; 15 December 1802 – 27 January 1860) or Johann Bolyai, was a Hungarian mathematician, one of the founders of non-Euclidean geometry — a geometry that differs from Euclidean geometry in its definition of parallel lines. The discovery of a consistent alternative geometry that might correspond to the structure of the universe helped to free mathematicians to study abstract concepts irrespective of any possible connection with the physical world.
Points are considered fundamental objects in Euclidean geometry. They have been defined in a variety of ways, including Euclid's definition as 'that which has no part' and through the use of algebra or nested sets. In many areas of geometry, such as analytic geometry, differential geometry, and topology, all objects are considered to be built up from points. However, there has been some study of geometry without reference to points.
Girard Desargues (; 21 February 1591 – September 1661) was a French mathematician and engineer, who is considered one of the founders of projective geometry. Desargues' theorem, the Desargues graph, and the crater Desargues on the Moon are named in his honour.
Born in Lyon, Desargues came from a family devoted to service to the French crown. His father was a royal notary, an investigating commissioner of the Seneschal's court in Lyon (1574), the collector of the tithes on ecclesiastical revenues for the city of Lyon (1583) and for the diocese of Lyon.
It was Gauss who coined the term "non-Euclidean geometry". He was referring to his own work which today we call "hyperbolic geometry". Several modern authors still consider "non-Euclidean geometry" and "hyperbolic geometry" to be synonyms.
In the 19th century, hyperbolic geometry was explored extensively by János Bolyai, Nikolai Ivanovich Lobachevsky, Carl Friedrich Gauss and Franz Taurinus. Unlike their predecessors, who just wanted to eliminate the parallel postulate from the axioms of Euclidean geometry, these authors realized they had discovered a new geometry.
Gauss wrote in an 1824 letter to Franz Taurinus that he had constructed it, but Gauss did not publish his work. Gauss called it "non-Euclidean geometry" causing several modern authors to continue to consider "non-Euclidean geometry" and "hyperbolic geometry" to be synonyms. Taurinus published results on hyperbolic trigonometry in 1826, argued that hyperbolic geometry is self consistent, but still believed in the special role of Euclidean geometry. The complete system of hyperbolic geometry was published by Lobachevsky in 1829/1830, while Bolyai discovered it independently and published in 1832.