Instruct: Given a question, retrieve relevant passages that can help answer the question.
Query: Who is considered the father of geometry?
Euclid (/ˈjuːklɪd/; Ancient Greek: Εὐκλείδης – Eukleídēs, pronounced[eu̯.klěː.dɛːs]; fl. 300 BC), sometimes given the name Euclid of Alexandria[1] to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry"[1] or the "father of geometry". He was active in Alexandria during the reign of PtolemyI (323–283 BC). His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century.[2][3][4] In the Elements, Euclid deduced the theorems of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, and rigor.
Ambrose became an assistant professor at MIT in 1947, an associate professor in 1950 and full professor in 1957. He was several times between 1939 and 1959 a visiting scholar at the Institute for Advanced Study in Princeton, New Jersey. Ambrose is often considered one of the fathers of modern geometry. He is noted for making changes in the pure mathematics undergraduate curriculum at MIT to reflect recent findings in differential geometry. For example, less than ten years after André Weil presented the differential form, Ambrose was using it in his undergraduate differential geometry courses. In the 1950s, Ambrose (together with I. M. Singer) made MIT into the only center in geometry in the United States outside the University of Chicago. His influence continued through his students, in particular Hung-Hsi Wu and John Rhodes, both of the University of California, Berkeley. In the 1960s Ambrose was a visiting professor at the University of Buenos Aires, Argentina. He is noted for his opposition of takeover of South American countries military regimes, specifically Argentina.
In 1872, Felix Klein noted that the many branches of geometry which had been developed during the 19th century (affine geometry, projective geometry, hyperbolic geometry, etc.) could all be treated in a uniform way. He did this by considering the groups under which the geometric objects were invariant. This unification of geometry goes by the name of the Erlangen programme.
Early in the 1920s the Dantzig family moved from Baltimore to Washington. His mother became a linguist at the Library of Congress, and his father became a math tutor at the University of Maryland, College Park. Dantzig attended Powell Junior High School and Central High School; one of his friends there was Abraham Seidenberg, who also became a professional mathematician. By the time he reached high school he was already fascinated by geometry, and this interest was further nurtured by his father, challenging him with complicated problems, particularly in projective 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.
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 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.
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.
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.
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.
Special relativity places space and time on equal footing, so that one considers the geometry of a unified spacetime instead of considering space and time separately. Minkowski geometry replaces Galilean geometry (which is the three-dimensional Euclidean space with time of Galilean relativity).
Thurston's contributions to the theory allow one to also consider, in many cases, the additional structure given by a particular Thurston model geometry (of which there are eight). The most prevalent geometry is hyperbolic geometry. Using a geometry in addition to special surfaces is often fruitful.
Then, around 1830, the Hungarian mathematician János Bolyai and the Russian mathematician Nikolai Ivanovich Lobachevsky separately published treatises on hyperbolic geometry. Consequently, hyperbolic geometry is called Bolyai-Lobachevskian geometry, as both mathematicians, independent of each other, are the basic authors of non-Euclidean geometry. Gauss mentioned to Bolyai's father, when shown the younger Bolyai's work, that he had developed such a geometry several years before, though he did not publish. While Lobachevsky created a non-Euclidean geometry by negating the parallel postulate, Bolyai worked out a geometry where both the Euclidean and the hyperbolic geometry are possible depending on a parameter "k". Bolyai ends his work by mentioning that it is not possible to decide through mathematical reasoning alone if the geometry of the physical universe is Euclidean or non-Euclidean; this is a task for the physical sciences.
Descriptive geometry is the branch of geometry which allows the representation of three-dimensional objects in two dimensions by using a specific set of procedures. The resulting techniques are important for engineering, architecture, design and in art. The theoretical basis for descriptive geometry is provided by planar geometric projections. The earliest known publication on the technique was "Underweysung der Messung mit dem Zirckel und Richtscheyt", published in Linien, Nuremberg: 1525, by Albrecht Dürer. Gaspard Monge is usually considered the "father of descriptive geometry" due to his developments in geometric problem solving. His first discoveries were in 1765 while he was working as a draftsman for military fortifications, although his findings were published later on.
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.
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.
Bernhard Riemann, in a famous lecture in 1854, founded the field of Riemannian geometry, discussing in particular the ideas now called manifolds, Riemannian metric, and curvature.
He constructed an infinite family of geometries which are not Euclidean by giving a formula for a family of Riemannian metrics on the unit ball in Euclidean space. The simplest of these is called elliptic geometry and it is considered to be a non-Euclidean geometry due to its lack of parallel lines.
Digital geometry heavily overlaps with discrete geometry and may be considered as a part thereof.
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]
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.
The geometrical system described in the Elements was long known simply as geometry, and was considered to be the only geometry possible. Today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century.
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.
It is possible (but considered unlikely) that this strictly quantized understanding of geometry will be consistent with the quantum picture of geometry arising from string theory.
Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational geometry. While modern computational geometry is a recent development, it is one of the oldest fields of computing with history stretching back to antiquity.
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.