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{{Use dmy dates|date=August 2019}}{{short description|Branch of mathematics that studies the shape, size and position of objects}}{{other uses}}{{pp-semi-indef}}{{pp-move-indef}}{{General geometry}}File:Teorema de desargues.svg|thumb|right|An illustration of Desargues' theorem, an important result in Euclidean and projective geometryprojective geometryGeometry (from the ; (wikt:γῆ|geo-) "earth", (wikt:μέτρον|-metron) "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.BOOK, Vincenzo De Risi, Mathematizing Space: The Objects of Geometry from Antiquity to the Early Modern Age,weblink 31 January 2015, Birkhäuser, 978-3-319-12102-4, 1–, A mathematician who works in the field of geometry is called a geometer.Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of formal mathematical science emerging in Greek mathematics as early as the 6th century BC.{{Harv|Boyer|1991|loc="Ionia and the Pythagoreans" p. 43}} By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow.Martin J. Turner,Jonathan M. Blackledge,Patrick R. Andrews (1998). Fractal geometry in digital imaging {{Webarchive|url= |date=6 September 2015 }}. Academic Press. p. 1. {{ISBN|0-12-703970-8}} Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists preserved Greek ideas and expanded on them during the Middle Ages.{{MacTutor Biography|id=Thabit|title=Al-Sabi Thabit ibn Qurra al-Harrani}} By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.MAGAZINE, Lamb, Evelyn, 8 November 2015, By Solving the Mysteries of Shape-Shifting Spaces, Mathematician Wins $3-Million Prize,weblink Scientific American, 2016-08-29,weblink" title="">weblink 18 August 2016, live, While geometry has evolved significantly throughout the years, there are some general concepts that are fundamental to geometry. These include the concepts of point, line, plane, distance, angle, surface, and curve, as well as the more advanced notions of topology and manifold.BOOK, Tabak, John, 2014, Geometry: the language of space and form, Infobase Publishing, xiv, 978-0816049530, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics.BOOK, Walter A. Meyer, Geometry and Its Applications,weblink 21 February 2006, Elsevier, 978-0-08-047803-6,


File:Westerner and Arab practicing geometry 15th century manuscript.jpg|right|thumb|A European and an ArabArabThe earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC.J. Friberg, "Methods and traditions of Babylonian mathematics. Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations", Historia Mathematica, 8, 1981, pp. 277—318.BOOK, 2, Dover Publications, Neugebauer, Otto, Otto E. Neugebauer, The Exact Sciences in Antiquity, 1957, 1969, 978-0-486-22332-2,weblink Chap. IV "Egyptian Mathematics and Astronomy", pp. 71–96. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts. The earliest known texts on geometry are the Egyptian Rhind Papyrus (2000–1800 BC) and Moscow Papyrus (c. 1890 BC), the Babylonian clay tablets such as Plimpton 322 (1900 BC). For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum.{{Harv|Boyer|1991|loc="Egypt" p. 19}} Later clay tablets (350–50 BC) demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space. These geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries.JOURNAL, Ossendrijver, Mathieu, 29 January 2016, Ancient Babylonian astronomers calculated Jupiter's position from the area under a time-velocity graph, Science, 351, 6272, 482–484, 10.1126/science.aad8085, 26823423, 2016Sci...351..482O, South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks.JOURNAL, Gnomons at Meroë and Early Trigonometry, Leo, Depuydt, 1 January 1998, The Journal of Egyptian Archaeology, 84, 171–180, 10.2307/3822211, 3822211, WEB,weblink Neolithic Skywatchers, 27 May 1998, Andrew, Slayman, Archaeology Magazine Archive, 17 April 2011,weblink" title="">weblink 5 June 2011, live, In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, which is credited with the first proof of the Pythagorean theorem,Eves, Howard, An Introduction to the History of Mathematics, Saunders, 1990, {{ISBN|0-03-029558-0}}. though the statement of the theorem has a long history.JOURNAL, The Discovery of Incommensurability by Hippasus of Metapontum, Kurt Von Fritz, The Annals of Mathematics, 1945, harv, JOURNAL, The Pentagram and the Discovery of an Irrational Number, The Two-Year College Mathematics Journal, James R. Choike, 1980, harv, Eudoxus (408–c. 355 BC) developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures,{{Harv|Boyer|1991|loc="The Age of Plato and Aristotle" p. 92}} as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements, widely considered the most successful and influential textbook of all time,{{Harv|Boyer|1991|loc="Euclid of Alexandria" p. 119}} introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework.{{Harv|Boyer|1991|loc="Euclid of Alexandria" p. 104}} The Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today.Howard Eves, An Introduction to the History of Mathematics, Saunders, 1990, {{ISBN|0-03-029558-0}} p. 141: "No work, except The Bible, has been more widely used...." Archimedes (c. 287–212 BC) of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave remarkably accurate approximations of Pi.WEB, A history of calculus, O'Connor, J.J., Robertson, E.F., University of St Andrews,weblink February 1996, 7 August 2007,weblink" title="">weblink 15 July 2007, live, He also studied the spiral bearing his name and obtained formulas for the volumes of surfaces of revolution.File:Woman teaching geometry.jpg|left|thumb|Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's ElementsEuclid's ElementsIndian mathematicians also made many important contributions in geometry. The Satapatha Brahmana (3rd century BC) contains rules for ritual geometric constructions that are similar to the Sulba Sutras.{{Citation | last=Staal | first=Frits | author-link=Frits Staal | title=Greek and Vedic Geometry | journal=Journal of Indian Philosophy | volume=27 | issue=1–2 | year=1999 | pages=105–127 | doi=10.1023/A:1004364417713 }}
According to {{Harv|Hayashi|2005|p=363}}, the Åšulba SÅ«tras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. They contain lists of Pythagorean triples,Pythagorean triples are triples of integers (a,b,c) with the property: a^2+b^2=c^2. Thus, 3^2+4^2=5^2, 8^2+15^2=17^2, 12^2+35^2=37^2 etc. which are particular cases of Diophantine equations.{{Harv|Cooke|2005|p=198}}: "The arithmetic content of the Åšulva SÅ«tras consists of rules for finding Pythagorean triples such as (3, 4, 5), (5, 12, 13), (8, 15, 17), and (12, 35, 37). It is not certain what practical use these arithmetic rules had. The best conjecture is that they were part of religious ritual. A Hindu home was required to have three fires burning at three different altars. The three altars were to be of different shapes, but all three were to have the same area. These conditions led to certain "Diophantine" problems, a particular case of which is the generation of Pythagorean triples, so as to make one square integer equal to the sum of two others."
In the Bakhshali manuscript, there is a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs a decimal place value system with a dot for zero."{{Harv|Hayashi|2005|p=371}} Aryabhata's Aryabhatiya (499) includes the computation of areas and volumes.Brahmagupta wrote his astronomical work {{IAST|Brāhma Sphuá¹­a Siddhānta}} in 628. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain).{{Harv|Hayashi|2003|pp=121–122}} In the latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral. Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of Heron's formula), as well as a complete description of rational triangles (i.e. triangles with rational sides and rational areas).In the Middle Ages, mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry.R. Rashed (1994), The development of Arabic mathematics: between arithmetic and algebra, p. 35 LondonBOOK, Boyer, Carl Benjamin Boyer, A History of Mathematics, 1991, The Arabic Hegemony, 241–242, Omar Khayyam (c. 1050–1123), the "tent-maker," wrote an Algebra that went beyond that of al-Khwarizmi to include equations of third degree. Like his Arab predecessors, Omar Khayyam provided for quadratic equations both arithmetic and geometric solutions; for general cubic equations, he believed (mistakenly, as the 16th century later showed), arithmetic solutions were impossible; hence he gave only geometric solutions. The scheme of using intersecting conics to solve cubics had been used earlier by Menaechmus, Archimedes, and Alhazan, but Omar Khayyam took the praiseworthy step of generalizing the method to cover all third-degree equations (having positive roots). .. For equations of higher degree than three, Omar Khayyam evidently did not envision similar geometric methods, for space does not contain more than three dimensions, ... One of the most fruitful contributions of Arabic eclecticism was the tendency to close the gap between numerical and geometric algebra. The decisive step in this direction came much later with Descartes, but Omar Khayyam was moving in this direction when he wrote, "Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved.", Al-Mahani (b. 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra.{{MacTutor Biography|id=Al-Mahani|title=Al-Mahani}} Thābit ibn Qurra (known as Thebit in Latin) (836–901) dealt with arithmetic operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry. Omar Khayyám (1048–1131) found geometric solutions to cubic equations.{{MacTutor Biography|id=Khayyam|title=Omar Khayyam}} The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were early results in hyperbolic geometry, and along with their alternative postulates, such as Playfair's axiom, these works had a considerable influence on the development of non-Euclidean geometry among later European geometers, including Witelo (c. 1230–c. 1314), Gersonides (1288–1344), Alfonso, John Wallis, and Giovanni Girolamo Saccheri.Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2, pp. 447–494 [470], Routledge, London and New York: In the early 17th century, there were two important developments in geometry. The first was the creation of analytic geometry, or geometry with coordinates and equations, by René Descartes (1596–1650) and Pierre de Fermat (1601–1665).BOOK, Carl B. Boyer, History of Analytic Geometry,weblink 28 June 2012, Courier Corporation, 978-0-486-15451-0, This was a necessary precursor to the development of calculus and a precise quantitative science of physics.BOOK, C.H.Jr. Edwards, The Historical Development of the Calculus,weblink 6 December 2012, Springer Science & Business Media, 978-1-4612-6230-5, 95, The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661).BOOK, Judith V. Field, Jeremy Gray, The Geometrical Work of Girard Desargues,weblink 6 December 2012, Springer Science & Business Media, 978-1-4613-8692-6, 43, Projective geometry studies properties of shapes which are unchanged under projections and sections, especially as they relate to artistic perspective.BOOK, C. R. Wylie, Introduction to Projective Geometry,weblink 12 September 2011, Courier Corporation, 978-0-486-14170-1, Two developments in geometry in the 19th century changed the way it had been studied previously.BOOK, Jeremy Gray, Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century,weblink 1 February 2011, Springer Science & Business Media, 978-0-85729-060-1, These were the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation of symmetry as the central consideration in the Erlangen Programme of Felix Klein (which generalized the Euclidean and non-Euclidean geometries). Two of the master geometers of the time were Bernhard Riemann (1826–1866), working primarily with tools from mathematical analysis, and introducing the Riemann surface, and Henri Poincaré, the founder of algebraic topology and the geometric theory of dynamical systems. As a consequence of these major changes in the conception of geometry, the concept of "space" became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics.BOOK, Eduardo Bayro-Corrochano, Geometric Algebra Applications Vol. I: Computer Vision, Graphics and Neurocomputing,weblink 20 June 2018, Springer, 978-3-319-74830-6, 4,

Important concepts in geometry

The following are some of the most important concepts in geometry.BOOK, Morris Kline, Mathematical Thought From Ancient to Modern Times: Volume 3,weblink March 1990, Oxford University Press, USA, 978-0-19-506137-6, 1010–,


File:Parallel postulate en.svg|thumb|right|An illustration of Euclid's parallel postulateparallel postulate{{See also|Euclidean geometry|Axiom}}Euclid took an abstract approach to geometry in his Elements,BOOK, Victor J. Katz, Using History to Teach Mathematics: An International Perspective,weblink 21 September 2000, Cambridge University Press, 978-0-88385-163-0, 45–, one of the most influential books ever written.BOOK, David Berlinski, The King of Infinite Space: Euclid and His Elements,weblink 8 April 2014, Basic Books, 978-0-465-03863-3, Euclid introduced certain axioms, or postulates, expressing primary or self-evident properties of points, lines, and planes.BOOK, Robin Hartshorne, Geometry: Euclid and Beyond,weblink 11 November 2013, Springer Science & Business Media, 978-0-387-22676-7, 29–, He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as axiomatic or synthetic geometry.BOOK, Pat Herbst, Taro Fujita, Stefan Halverscheid, Michael Weiss, The Learning and Teaching of Geometry in Secondary Schools: A Modeling Perspective,weblink 16 March 2017, Taylor & Francis, 978-1-351-97353-3, 20–, At the start of the 19th century, the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky (1792–1856), János Bolyai (1802–1860), Carl Friedrich Gauss (1777–1855) and othersBOOK, I.M. Yaglom, A Simple Non-Euclidean Geometry and Its Physical Basis: An Elementary Account of Galilean Geometry and the Galilean Principle of Relativity,weblink 6 December 2012, Springer Science & Business Media, 978-1-4612-6135-3, 6–, led to a revival of interest in this discipline, and in the 20th century, David Hilbert (1862–1943) employed axiomatic reasoning in an attempt to provide a modern foundation of geometry.BOOK, Audun Holme, Geometry: Our Cultural Heritage,weblink 23 September 2010, Springer Science & Business Media, 978-3-642-14441-7, 254–,


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'Euclid's Elements – All thirteen books in one volume, Based on Heath's translation, Green Lion Press {{ISBN|1-888009-18-7}}. and through the use of algebra or nested sets.JOURNAL, Clark, Bowman L., Jan 1985, Individuals and Points, Notre Dame Journal of Formal Logic, 26, 1, 61–75, 10.1305/ndjfl/1093870761, 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.Gerla, G., 1995, "Pointless Geometries {{Webarchive|url= |date=17 July 2011 }}" in Buekenhout, F., Kantor, W. eds., Handbook of incidence geometry: buildings and foundations. North-Holland: 1015–1031.


Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself". In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry, a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation,John Casey (1885) Analytic Geometry of the Point, Line, Circle, and Conic Sections {{Webarchive|url= |date=17 March 2016 }}, link from Internet Archive. but in a more abstract setting, such as incidence geometry, a line may be an independent object, distinct from the set of points which lie on it.Buekenhout, Francis (1995), Handbook of Incidence Geometry: Buildings and Foundations, Elsevier B.V. In differential geometry, a geodesic is a generalization of the notion of a line to curved spaces.WEB,weblink geodesic – definition of geodesic in English from the Oxford dictionary,, 2016-01-20,weblink" title="">weblink 15 July 2016, live,


A plane is a flat, two-dimensional surface that extends infinitely far. Planes are used in every area of geometry. For instance, planes can be studied as a topological surface without reference to distances or angles;Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000. it can be studied as an affine space, where collinearity and ratios can be studied but not distances;Szmielew, Wanda. 'From affine to Euclidean geometry: An axiomatic approach.' Springer, 1983. it can be studied as the complex plane using techniques of complex analysis;Ahlfors, Lars V. Complex analysis: an introduction to the theory of analytic functions of one complex variable. New York, London (1953). and so on.


Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. In modern terms, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.{{SpringerEOM|id=Angle&oldid=13323|title=Angle|year=2001|last=Sidorov|first=L.A.}}(File:Angle obtuse acute straight.svg|thumb|Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.)In Euclidean geometry, angles are used to study polygons and triangles, as well as forming an object of study in their own right. The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry.Gelʹfand, Izrailʹ Moiseevič, and Mark Saul. "Trigonometry." 'Trigonometry'. Birkhäuser Boston, 2001. 1–20.In differential geometry and calculus, the angles between plane curves or space curves or surfaces can be calculated using the derivative.Stewart, James (2012). Calculus: Early Transcendentals, 7th ed., Brooks Cole Cengage Learning. {{ISBN|978-0-538-49790-9}}{{citation |last=Jost |first=Jürgen |title=Riemannian Geometry and Geometric Analysis |year=2002 |publisher=Springer-Verlag |location=Berlin |isbn=978-3-540-42627-1}}.


A curve is a 1-dimensional object that may be straight (like a line) or not; curves in 2-dimensional space are called plane curves and those in 3-dimensional space are called space curves.Baker, Henry Frederick. Principles of geometry. Vol. 2. CUP Archive, 1954.In topology, a curve is defined by a function from an interval of the real numbers to another space. In differential geometry, the same definition is used, but the defining function is required to be differentiable Algebraic geometry studies algebraic curves, which are defined as algebraic varieties of dimension one.


(File:Sphere wireframe.svg|thumb|upright=0.85|A sphere is a surface that can be defined parametrically (by {{nowrap|x {{=}} r sin θ cos φ,}} {{nowrap|y {{=}} r sin θ sin φ,}} {{nowrap|z {{=}} r cos θ)}} or implicitly (by {{nowrap|x2 + y2 + z2 − r2 {{=}} 0}}.))A surface is a two-dimensional object, such as a sphere or paraboloid.Briggs, William L., and Lyle Cochran Calculus. "Early Transcendentals." {{ISBN|978-0321570567}}. In differential geometryDo Carmo, Manfredo Perdigao, and Manfredo Perdigao Do Carmo. Differential geometry of curves and surfaces. Vol. 2. Englewood Cliffs: Prentice-hall, 1976. and topology, surfaces are described by two-dimensional 'patches' (or neighborhoods) that are assembled by diffeomorphisms or homeomorphisms, respectively. In algebraic geometry, surfaces are described by polynomial equations.BOOK, Mumford, David, David Mumford, The Red Book of Varieties and Schemes Includes the Michigan Lectures on Curves and Their Jacobians, 2nd, 1999, Springer Science+Business Media, Springer-Verlag, 978-3-540-63293-1, 0945.14001,


A manifold is a generalization of the concepts of curve and surface. In topology, a manifold is a topological space where every point has a neighborhood that is homeomorphic to Euclidean space. In differential geometry, a differentiable manifold is a space where each neighborhood is diffeomorphic to Euclidean space.Manifolds are used extensively in physics, including in general relativity and string theory.Yau, Shing-Tung; Nadis, Steve (2010). The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions. Basic Books. {{ISBN|978-0-465-02023-2}}.

Length, area, and volume

{{See also|Area#List of formulas|Volume#Volume formulas}}Length, area, and volume describe the size or extant of an object in one dimension, two dimension, and three dimensions respectively.BOOK, Steven A. Treese, History and Measurement of the Base and Derived Units,weblink 17 May 2018, Springer International Publishing, 978-3-319-77577-7, 101–, In Euclidean geometry and analytic geometry, the length of a line segment can often be calculated by the Pythagorean theorem.BOOK, James W. Cannon, Geometry of Lengths, Areas, and Volumes,weblink 16 November 2017, American Mathematical Soc., 978-1-4704-3714-5, 11, Area and volume can be defined as fundamental quantities separate from length, or they can be described and calculated in terms of lengths in a plane or 3-dimensional space. Mathematicians have found many explicit formulas for area and formulas for volume of various geometric objects. In calculus, area and volume can be defined in terms of integrals, such as the Riemann integralBOOK, Gilbert Strang, Calculus,weblink 1 January 1991, SIAM, 978-0-9614088-2-4, or the Lebesgue integral.BOOK, H. S. Bear, A Primer of Lebesgue Integration,weblink 2002, Academic Press, 978-0-12-083971-1,

Metrics and measures

File:Chinese pythagoras.jpg|thumb|right|Visual checking of the Pythagorean theorem for the (3, 4, 5) triangle as in the Zhoubi Suanjing 500–200 BC. The Pythagorean theorem is a consequence of the Euclidean metricEuclidean metricThe concept of length or distance can be generalized, leading to the idea of metrics.Dmitri Burago, Yu D Burago, Sergei Ivanov, A Course in Metric Geometry, American Mathematical Society, 2001, {{ISBN|0-8218-2129-6}}. For instance, the Euclidean metric measures the distance between points in the Euclidean plane, while the hyperbolic metric measures the distance in the hyperbolic plane. Other important examples of metrics include the Lorentz metric of special relativity and the semi-Riemannian metrics of general relativity.{{Citation|last=Wald|first=Robert M.|authorlink=Robert Wald|title=General Relativity|publisher=University of Chicago Press|date=1984|isbn=978-0-226-87033-5|title-link=General Relativity (book)}}In a different direction, the concepts of length, area and volume are extended by measure theory, which studies methods of assigning a size or measure to sets, where the measures follow rules similar to those of classical area and volume.BOOK, Terence Tao, An Introduction to Measure Theory,weblink 14 September 2011, American Mathematical Soc., 978-0-8218-6919-2,

Congruence and similarity

Congruence and similarity are concepts that describe when two shapes have similar characteristics.BOOK, Shlomo Libeskind, Euclidean and Transformational Geometry: A Deductive Inquiry,weblink 12 February 2008, Jones & Bartlett Learning, 978-0-7637-4366-6, 255, In Euclidean geometry, similarity is used to describe objects that have the same shape, while congruence is used to describe objects that are the same in both size and shape.BOOK, Mark A. Freitag, Mathematics for Elementary School Teachers: A Process Approach,weblink 1 January 2013, Cengage Learning, 978-0-618-61008-2, 614, Hilbert, in his work on creating a more rigorous foundation for geometry, treated congruence as an undefined term whose properties are defined by axioms.Congruence and similarity are generalized in transformation geometry, which studies the properties of geometric objects that are preserved by different kinds of transformations.BOOK, George E. Martin, Transformation Geometry: An Introduction to Symmetry,weblink 6 December 2012, Springer Science & Business Media, 978-1-4612-5680-9,

Compass and straightedge constructions

Classical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically, the only instruments allowed in geometric constructions are the compass and straightedge. Also, every construction had to be complete in a finite number of steps. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using parabolas and other curves, as well as mechanical devices, were found.


File:Von Koch curve.gif|thumb|The Koch snowflake, with fractal dimension=log4/log3 and topological dimensiontopological dimensionWhere the traditional geometry allowed dimensions 1 (a line), 2 (a plane) and 3 (our ambient world conceived of as three-dimensional space), mathematicians and physicists have used higher dimensions for nearly two centuries.BOOK, Mark Blacklock, The Emergence of the Fourth Dimension: Higher Spatial Thinking in the Fin de Siècle,weblink 2018, Oxford University Press, 978-0-19-875548-7, One example of a mathematical use for higher dimensions is the configuration space of a physical system, which has a dimension equal to the system's degrees of freedom. For instance, the configuration of a screw can be described by five coordinates.BOOK, Charles Jasper Joly, Papers,weblink 1895, The Academy, 62–, In general topology, the concept of dimension has been extended from natural numbers, to infinite dimension (Hilbert spaces, for example) and positive real numbers (in fractal geometry).BOOK, Roger Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,weblink 11 December 2013, Springer Science & Business Media, 978-1-4612-0645-3, 367, In algebraic geometry, the dimension of an algebraic variety has received a number of apparently different definitions, which are all equivalent in the most common cases.BOOK, Bill Jacob, Tsit-Yuen Lam, Recent Advances in Real Algebraic Geometry and Quadratic Forms: Proceedings of the RAGSQUAD Year, Berkeley, 1990-1991,weblink 1994, American Mathematical Soc., 978-0-8218-5154-8, 111,


File:Order-3 heptakis heptagonal tiling.png|right|thumb|A tiling of the hyperbolic plane ]]The theme of symmetry in geometry is nearly as old as the science of geometry itself.BOOK, Ian Stewart, Why Beauty Is Truth: A History of Symmetry,weblink 29 April 2008, Basic Books, 978-0-465-08237-7, 14, Symmetric shapes such as the circle, regular polygons and platonic solids held deep significance for many ancient philosophersBOOK, Stakhov Alexey, Mathematics Of Harmony: From Euclid To Contemporary Mathematics And Computer Science,weblink 11 September 2009, World Scientific, 978-981-4472-57-9, 144, and were investigated in detail before the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in a multitude of forms, including the graphics of da Vinci, M.C. Escher, and others.BOOK, Werner Hahn, Symmetry as a Developmental Principle in Nature and Art,weblink 1998, World Scientific, 978-981-02-2363-2, In the second half of 19th century, the relationship between symmetry and geometry came under intense scrutiny. Felix Klein's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group, determines what geometry is.BOOK, Brian J. Cantwell, Introduction to Symmetry Analysis,weblink 23 September 2002, Cambridge University Press, 978-1-139-43171-2, 34, Symmetry in classical Euclidean geometry is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations, geometric transformations that take straight lines into straight lines.BOOK, B. Rosenfeld, Bill Wiebe, Geometry of Lie Groups,weblink 9 March 2013, Springer Science & Business Media, 978-1-4757-5325-7, 158ff, However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group' found its inspiration. Both discrete and continuous symmetries play prominent roles in geometry, the former in topology and geometric group theory,BOOK, Michio Kaku, Strings, Conformal Fields, and Topology: An Introduction,weblink 6 December 2012, Springer Science & Business Media, 978-1-4684-0397-8, 151, BOOK, Mladen Bestvina, Michah Sageev, Karen Vogtmann, Geometric Group Theory,weblink 24 December 2014, American Mathematical Soc., 978-1-4704-1227-2, 132, the latter in Lie theory and Riemannian geometry.BOOK, W-H Steeb, Continuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra,weblink 30 September 1996, World Scientific Publishing Company, 978-981-310-503-4, BOOK, Charles W. Misner, Directions in General Relativity: Volume 1: Proceedings of the 1993 International Symposium, Maryland: Papers in Honor of Charles Misner,weblink 20 October 2005, Cambridge University Press, 978-0-521-02139-5, 272, A different type of symmetry is the principle of duality in projective geometry, among other fields. This meta-phenomenon can roughly be described as follows: in any theorem, exchange point with plane, join with meet, lies in with contains, and the result is an equally true theorem.BOOK, Linnaeus Wayland Dowling, Projective Geometry,weblink 1917, McGraw-Hill book Company, Incorporated, 10, A similar and closely related form of duality exists between a vector space and its dual space.BOOK, G. Gierz, Bundles of Topological Vector Spaces and Their Duality,weblink 15 November 2006, Springer, 978-3-540-39437-2, 252,

Contemporary geometry

Euclidean geometry

Euclidean geometry is geometry in its classical sense.BOOK, Robert E. Butts, J.R. Brown, Constructivism and Science: Essays in Recent German Philosophy,weblink 6 December 2012, Springer Science & Business Media, 978-94-009-0959-5, 127–, As its models the space of the physical world, it is used in many scientific areas, such as mechanics, astronomy, crystallography,BOOK, Science,weblink 1886, Moses King, 181–, and many technical fields, such as engineering,BOOK, W. Abbot, Practical Geometry and Engineering Graphics: A Textbook for Engineering and Other Students,weblink 11 November 2013, Springer Science & Business Media, 978-94-017-2742-6, 6–, architecture,BOOK, George L. Hersey, Professor George L Hersey, Architecture and Geometry in the Age of the Baroque,weblink March 2001, University of Chicago Press, 978-0-226-32783-9, geodesy,BOOK, P. Vanícek, E.J. Krakiwsky, Geodesy: The Concepts,weblink 3 June 2015, Elsevier, 978-1-4832-9079-9, 23, aerodynamics,BOOK, Russell M. Cummings, Scott A. Morton, William H. Mason, David R. McDaniel, Applied Computational Aerodynamics,weblink 27 April 2015, Cambridge University Press, 978-1-107-05374-8, 449, and navigation.BOOK, Roy Williams, Geometry of Navigation,weblink 1998, Horwood Pub., 978-1-898563-46-4, The mandatory educational curriculum of the majority of nations includes the study of Euclidean concepts such as points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, and analytic geometry.Schmidt, W., Houang, R., & Cogan, L. (2002). "A coherent curriculum". American Educator, 26(2), 1–18.

Differential geometry

File:Hyperbolic triangle.svg|thumb|150px|right|Differential geometry uses tools from calculuscalculusDifferential geometry uses techniques of calculus and linear algebra to study problems in geometry.BOOK, Gerard Walschap, Multivariable Calculus and Differential Geometry,weblink 1 July 2015, De Gruyter, 978-3-11-036954-0, It has applications in physics,BOOK, Harley Flanders, Differential Forms with Applications to the Physical Sciences,weblink 26 April 2012, Courier Corporation, 978-0-486-13961-6, , econometrics,BOOK, Paul Marriott, Mark Salmon, Applications of Differential Geometry to Econometrics,weblink 31 August 2000, Cambridge University Press, 978-0-521-65116-5, , and bioinformatics,BOOK, Matthew He, Sergey Petoukhov, Mathematics of Bioinformatics: Theory, Methods and Applications,weblink 16 March 2011, John Wiley & Sons, 978-1-118-09952-0, 106, , among others.In particular, differential geometry is of importance to mathematical physics due to Einstein's general relativity postulation that the universe is curved.BOOK, P. A.M. Dirac, General Theory of Relativity,weblink 10 August 2016, Princeton University Press, 978-1-4008-8419-3, Differential geometry can either be intrinsic (meaning that the spaces it considers are smooth manifolds whose geometric structure is governed by a Riemannian metric, which determines how distances are measured near each point) or extrinsic (where the object under study is a part of some ambient flat Euclidean space).BOOK, Nihat Ay, Jürgen Jost, Hông Vân Lê, Lorenz Schwachhöfer, Information Geometry,weblink 25 August 2017, Springer, 978-3-319-56478-4, 185,

Non-Euclidean geometry

Euclidean geometry was not the only historical form of geometry studied. Spherical geometry has long been used by astronomers, astrologers, and navigators.BOOK, Boris A. Rosenfeld, A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space,weblink 8 September 2012, Springer Science & Business Media, 978-1-4419-8680-1, 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.Kline (1972) "Mathematical thought from ancient to modern times", Oxford University Press, p. 1032. Kant did not reject the logical (analytic a priori) possibility of non-Euclidean geometry, see Jeremy Gray, "Ideas of Space Euclidean, Non-Euclidean, and Relativistic", Oxford, 1989; p. 85. Some have implied that, in light of this, Kant had in fact predicted the development of non-Euclidean geometry, cf. Leonard Nelson, "Philosophy and Axiomatics," Socratic Method and Critical Philosophy, Dover, 1965, p. 164. This view was at first somewhat challenged by thinkers such as Saccheri, then finally overturned by the revolutionary discovery of non-Euclidean geometry in the works of Bolyai, Lobachevsky, and Gauss (who never published his theory).BOOK, Duncan M'Laren Young Sommerville, Elements of Non-Euclidean Geometry ...,weblink 1919, Open Court, 15ff, 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 Ãœber die Hypothesen, welche der Geometrie zu Grunde liegen (On the hypotheses on which geometry is based),WEB,weblink Ueber die Hypothesen, welche der Geometrie zu Grunde liegen, dead,weblink" title="">weblink 18 March 2016, published only after his death. Riemann's new idea of space proved crucial in Einstein's general relativity theory. Riemannian geometry, which considers very general spaces in which the notion of length is defined, is a mainstay of modern geometry.BOOK, Peter Pesic, Beyond Geometry: Classic Papers from Riemann to Einstein,weblink 1 January 2007, Courier Corporation, 978-0-486-45350-7,


File:Trefoil knot arb.png|thumb|150px|right|A thickening of the trefoil knottrefoil knotTopology is the field concerned with the properties of continuous mappings,BOOK, Martin D. Crossley, Essential Topology,weblink 11 February 2011, Springer Science & Business Media, 978-1-85233-782-7, and can be considered a generalization of Euclidean geometry.BOOK, Charles Nash, Siddhartha Sen, Topology and Geometry for Physicists,weblink 4 January 1988, Elsevier, 978-0-08-057085-3, 1, In practice, topology often means dealing with large-scale properties of spaces, such as connectedness and compactness.The field of topology, which saw massive development in the 20th century, is in a technical sense a type of transformation geometry, in which transformations are homeomorphisms.BOOK, George E. Martin, Transformation Geometry: An Introduction to Symmetry,weblink 20 December 1996, Springer Science & Business Media, 978-0-387-90636-2, This has often been expressed in the form of the saying 'topology is rubber-sheet geometry'. Subfields of topology include geometric topology, differential topology, algebraic topology and general topology.BOOK, J. P. May, A Concise Course in Algebraic Topology,weblink September 1999, University of Chicago Press, 978-0-226-51183-2,

Algebraic geometry

File:Calabi yau.jpg|thumb|150px|Quintic Calabi–Yau threefold ]]The field of algebraic geometry developed from the Cartesian geometry of co-ordinates.BOOK, Scientific American, inc, The Encyclopedia Americana: A Universal Reference Library Comprising the Arts and Sciences, Literature, History, Biography, Geography, Commerce, Etc., of the World,weblink 1905, Scientific American Compiling Department, 489–, It underwent periodic periods of growth, accompanied by the creation and study of projective geometry, birational geometry, algebraic varieties, and commutative algebra, among other topics.BOOK, Suzanne C. Dieudonne, History Algebraic Geometry,weblink 30 May 1985, CRC Press, 978-0-412-99371-8, From late 1950s through mid-1970s it had undergone major foundational development, largely due to work of Jean-Pierre Serre and Alexander Grothendieck. This led to the introduction of schemes and greater emphasis on topological methods, including various cohomology theories. One of seven Millennium Prize problems, the Hodge conjecture, is a question in algebraic geometry.BOOK, James Carlson, James A. Carlson, Arthur Jaffe, Andrew Wiles, Clay Mathematics Institute, American Mathematical Society, The Millennium Prize Problems,weblink 2006, American Mathematical Soc., 978-0-8218-3679-8, Wiles' proof of Fermat's Last Theorem uses advanced methods of algebraic geometry for solving a long standing problem of number theory.In general, Algebraic geometry studies geometry through the use of concepts in commutative algebra such as multivariate polynomials.BOOK, Robin Hartshorne, Algebraic Geometry,weblink 29 June 2013, Springer Science & Business Media, 978-1-4757-3849-0, It has applications in many areas, including cryptographyBOOK, Everett W. Howe, Kristin E. Lauter, Judy L. Walker, Algebraic Geometry for Coding Theory and Cryptography: IPAM, Los Angeles, CA, February 2016,weblink 15 November 2017, Springer, 978-3-319-63931-4, and string theory.BOOK, Marcos Marino, Michael Thaddeus, Ravi Vakil, Enumerative Invariants in Algebraic Geometry and String Theory: Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 6-11, 2005,weblink 15 August 2008, Springer, 978-3-540-79814-9,

Discrete geometry

File:Closepacking.svg|thumb|150px|Discrete geometry includes the study of various sphere packingsphere packingDiscrete geometry is a subject that has close connections with convex geometry.BOOK, Jiří Matoušek, Lectures on Discrete Geometry,weblink 1 December 2013, Springer Science & Business Media, 978-1-4613-0039-7, BOOK, Chuanming Zong, The Cube-A Window to Convex and Discrete Geometry,weblink 2 February 2006, Cambridge University Press, 978-0-521-85535-8, BOOK, Peter M. Gruber, Convex and Discrete Geometry,weblink 17 May 2007, Springer Science & Business Media, 978-3-540-71133-9, It is concerned mainly with questions of relative position of simple geometric objects, such as points, lines and circles. Examples include the study of sphere packings, triangulations, the Kneser-Poulsen conjecture, etc.BOOK, Satyan L. Devadoss, Joseph O'Rourke, Discrete and Computational Geometry,weblink 11 April 2011, Princeton University Press, 978-1-4008-3898-1, BOOK, Károly Bezdek, Classical Topics in Discrete Geometry,weblink 23 June 2010, Springer Science & Business Media, 978-1-4419-0600-7, It shares many methods and principles with combinatorics.

Computational geometry

Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Important problems historically have included the travelling salesman problem, minimum spanning trees, hidden-line removal, and linear programming.BOOK, Franco P. Preparata, Michael I. Shamos, Computational Geometry: An Introduction,weblink 6 December 2012, Springer Science & Business Media, 978-1-4612-1098-6, Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc.BOOK, Xianfeng David Gu, Shing-Tung Yau, Computational Conformal Geometry,weblink 2008, International Press, 978-1-57146-171-1,

Geometric group theory

missing image!
- Cayley graph of F2.svg|right|thumb|150px|The Cayley graph of the free groupfree groupGeometric group theory uses large-scale geometric techniques to study finitely generated groups.BOOK, Clara Löh, Geometric Group Theory: An Introduction,weblink 19 December 2017, Springer, 978-3-319-72254-2, It is closely connected to low-dimensional topology, such as in Grigori Perelman's proof of the Geometrization conjecture, which included the proof of the Poincaré conjecture, a Millenium Prize Problem.BOOK, John Morgan, Gang Tian, The Geometrization Conjecture,weblink 21 May 2014, American Mathematical Soc., 978-0-8218-5201-9, Geometric group theory often revolves around the Cayley graph, which is a geometric representation of a group. Other important topics include quasi-isometries, Gromov-hyperbolic groups, and right angled Artin groups.BOOK, Clara Löh, Geometric Group Theory: An Introduction,weblink 19 December 2017, Springer, 978-3-319-72254-2, BOOK, Daniel T. Wise, From Riches to Raags: 3-Manifolds, Right-Angled Artin Groups, and Cubical Geometry: 3-manifolds, Right-angled Artin Groups, and Cubical Geometry,weblink 2012, American Mathematical Soc., 978-0-8218-8800-1,

Convex geometry

Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis and discrete mathematics.BOOK, Gerard Meurant, Handbook of Convex Geometry,weblink 28 June 2014, Elsevier Science, 978-0-08-093439-6, It has close connections to convex analysis, optimization and functional analysis and important applications in number theory.Convex geometry dates back to antiquity. Archimedes gave the first known precise definition of convexity. The isoperimetric problem, a recurring concept in convex geometry, was studied by the Greeks as well, including Zenodorus. Archimedes, Plato, Euclid, and later Kepler and Coxeter all studied convex polytopes and their properties. From the 19th century on, mathematicians have studied other areas of convex mathematics, including higher-dimensional polytopes, volume and surface area of convex bodies, Gaussian curvature, algorithms, tilings and lattices.


Geometry has found applications in many fields, some of which are described below.


File:Fes Medersa Bou Inania Mosaique2.jpg
Mathematics and art are related in a variety of ways. For instance, the theory of perspective showed that there is more to geometry than just the metric properties of figures: perspective is the origin of projective geometry.BOOK, Jürgen Richter-Gebert, Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry,weblink 4 February 2011, Springer Science & Business Media, 978-3-642-17286-1, Artists have long used concepts of proportion in design. Vitruvius developed a complicated theory of ideal proportions for the human figure.BOOK, Kimberly Elam, Geometry of Design: Studies in Proportion and Composition,weblink 2001, Princeton Architectural Press, 978-1-56898-249-6, These concepts have been used and adapted by artists from Michelangelo to modern comic book artists.BOOK, Brad J Guigar, The Everything Cartooning Book: Create Unique And Inspired Cartoons For Fun And Profit,weblink 4 November 2004, Adams Media, 978-1-4405-2305-2, 82–, The golden ratio is a particular proportion that has had a controversial role in art. Often claimed to be the most aesthetically pleasing ratio of lengths, it is frequently stated to be incorporated into famous works of art, though the most reliable and unambiguous examples were made deliberately by artists aware of this legend.BOOK, Mario Livio, The Golden Ratio: The Story of PHI, the World's Most Astonishing Number,weblink 12 November 2008, Crown/Archetype, 978-0-307-48552-6, 166, Tilings, or tessellations, have been used in art throughout history. Islamic art makes frequent use of tessellations, as did the art of Escher.BOOK, Michele Emmer, Doris Schattschneider, M.C. Escher's Legacy: A Centennial Celebration,weblink 8 May 2007, Springer, 978-3-540-28849-7, 107, Escher's work also made use of hyperbolic geometry.Cézanne advanced the theory that all images can be built up from the sphere, the cone, and the cylinder. This is still used in art theory today, although the exact list of shapes varies from author to author.BOOK, Robert Capitolo, Ken Schwab, Drawing Course 101,weblink 2004, Sterling Publishing Company, Inc., 978-1-4027-0383-6, 22, BOOK, Phyllis Gelineau, Integrating the Arts Across the Elementary School Curriculum,weblink 1 January 2011, Cengage Learning, 978-1-111-30126-2, 55,


Geometry has many applications in architecture. In fact, it has been said that geometry lies at the core of architectural design.BOOK, Cristiano Ceccato, Lars Hesselgren, Mark Pauly, Helmut Pottmann, Johannes Wallner, Advances in Architectural Geometry 2010,weblink 5 December 2016, Birkhäuser, 978-3-99043-371-3, 6, BOOK, Helmut Pottmann, Architectural geometry,weblink 2007, Bentley Institute Press, Applications of geometry to architecture include the use of projective geometry to create forced perspective,BOOK, Marian Moffett, Michael W. Fazio, Lawrence Wodehouse, A World History of Architecture,weblink 2003, Laurence King Publishing, 978-1-85669-371-4, 371, the use of conic sections in constructing domes and similar objects, the use of tessellations, and the use of symmetry.


The field of astronomy, especially as it relates to mapping the positions of stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, have served as an important source of geometric problems throughout history.BOOK, Robin M. Green, Robin Michael Green, Spherical Astronomy,weblink 31 October 1985, Cambridge University Press, 978-0-521-31779-5, 1, Riemannian geometry and pseudo-Riemannian geometry are used in general relativity.BOOK, Dmitriĭ Vladimirovich Alekseevskiĭ, Recent Developments in Pseudo-Riemannian Geometry,weblink 2008, European Mathematical Society, 978-3-03719-051-7, String theory makes use of several variants of geometry,BOOK, Shing-Tung Yau, Steve Nadis, The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions,weblink 7 September 2010, Basic Books, 978-0-465-02266-3, as does quantum information theory.BOOK, Bengtsson, Ingemar, Życzkowski, Karol, Karol Życzkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement, Cambridge University Press, 2nd, 2017, 9781107026254, 1004572791,

Other fields of mathematics

File:Square root of 2 triangle.svg|thumb|right|The Pythagoreans discovered that the sides of a triangle could have incommensurable lengths.]]Calculus was strongly influenced by geometry.BOOK, Carl B. Boyer, History of Analytic Geometry,weblink 28 June 2012, Courier Corporation, 978-0-486-15451-0, For instance, the introduction of coordinates by René Descartes and the concurrent developments of algebra marked a new stage for geometry, since geometric figures such as plane curves could now be represented analytically in the form of functions and equations. This played a key role in the emergence of infinitesimal calculus in the 17th century. Analytic geometry continues to be a mainstay of pre-calculus and calculus curriculum.BOOK, Harley Flanders, Justin J. Price, Calculus with Analytic Geometry,weblink 10 May 2014, Elsevier Science, 978-1-4832-6240-6, BOOK, Jon Rogawski, Colin Adams, Calculus,weblink 30 January 2015, W. H. Freeman, 978-1-4641-7499-5, Another important area of application is number theory.BOOK, Álvaro Lozano-Robledo, Number Theory and Geometry: An Introduction to Arithmetic Geometry,weblink 21 March 2019, American Mathematical Soc., 978-1-4704-5016-8, In ancient Greece the Pythagoreans considered the role of numbers in geometry. However, the discovery of incommensurable lengths contradicted their philosophical views.BOOK, Arturo Sangalli, Pythagoras' Revenge: A Mathematical Mystery,weblink 10 May 2009, Princeton University Press, 978-0-691-04955-7, 57, Since the 19th century, geometry has been used for solving problems in number theory, for example through the geometry of numbers or, more recently, scheme theory, which is used in Wiles's proof of Fermat's Last Theorem.BOOK, Gary Cornell, Joseph H. Silverman, Glenn Stevens, Modular Forms and Fermat's Last Theorem,weblink 1 December 2013, Springer Science & Business Media, 978-1-4612-1974-3,

See also


Related topics

Other fields




  • BOOK, Boyer, C.B., Carl Benjamin Boyer, A History of Mathematics, Second edition, revised by Uta Merzbach, Uta C. Merzbach, New York, Wiley, 1991, 1989, 978-0-471-54397-8, harv, registration,weblink
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Further reading

External links

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