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## History

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."

## 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, 1010â€“,

### Points

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 â€“ 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, 61â€“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=https://web.archive.org/web/20110717210751weblink |date=17 July 2011 }}" in Buekenhout, F., Kantor, W. eds., Handbook of incidence geometry: buildings and foundations. North-Holland: 1015â€“1031.

### Lines

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=https://web.archive.org/web/20160317230113weblink |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 â€“ definition of geodesic in English from the Oxford dictionary, OxfordDictionaries.com, 2016-01-20,weblink" title="web.archive.org/web/20160715034047weblink">weblink 15 July 2016, live,

### Planes

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.

### Angles

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.GelÊ¹fand, IzrailÊ¹ MoiseeviÄ, and Mark Saul. "Trigonometry." 'Trigonometry'. BirkhÃ¤user Boston, 2001. 1â€“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=JÃ¼rgen |title=Riemannian Geometry and Geometric Analysis |year=2002 |publisher=Springer-Verlag |location=Berlin |isbn=978-3-540-42627-1}}.

### Curves

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.

### Surfaces

(File:Sphere wireframe.svg|thumb|upright=0.85|A sphere is a surface that can be defined parametrically (by {{nowrap|x {{=}} r sin Î¸ cos Ï†,}} {{nowrap|y {{=}} r sin Î¸ sin Ï†,}} {{nowrap|z {{=}} r cos Î¸)}} or implicitly (by {{nowrap|x2 + y2 + z2 âˆ’ 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,

### Manifolds

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, 101â€“, 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 500â€“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.

## Contemporary geometry

### 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, JÃ¼rgen Jost, HÃ´ng VÃ¢n LÃª, Lorenz SchwachhÃ¶fer, Information Geometry,weblink 25 August 2017, Springer, 978-3-319-56478-4, 185,

### Topology

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 Calabiâ€“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, 489â€“, 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, JiÅ™Ã­ MatouÅ¡ek, 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, KÃ¡roly 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 LÃ¶h, 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 PoincarÃ© 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 LÃ¶h, 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.

## Applications

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

### Art

File:Fes Medersa Bou Inania Mosaique2.jpg
-

### Architecture

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, BirkhÃ¤user, 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.

### Physics

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, DmitriÄ­ Vladimirovich AlekseevskiÄ­, 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, Å»yczkowski, Karol, Karol Å»yczkowski, 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 RenÃ© 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, Ãlvaro 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,

{{reflist|40em}}

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