Geometry
Geometry  

Geometers  
The earliest recorded beginnings of geometry can be traced to ancient motion within timevelocity space. These geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries.^{[7]} South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks.^{[8]}^{[9]}
In the 7th century BC, the .Diophantine equations.^{[21]}
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."^{[22]} Aryabhata's Aryabhatiya (499) includes the computation of areas and volumes.
rational triangles (i.e. triangles with rational sides and rational areas).^{[23]}
In the 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).^{[30]} This was a necessary precursor to the development of calculus and a precise quantitative science of physics.^{[31]} The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661).^{[32]} Projective geometry studies properties of shapes which are unchanged under projections and sections, especially as they relate to artistic perspective.^{[33]} Two developments in geometry in the 19th century changed the way it had been studied previously.^{} Erlangen Programme of Felix Klein (which generalized the Euclidean and nonEuclidean 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.^{[35]}
Main conceptsThe following are some of the most important concepts in geometry.^{}[2]^{[36]}^{[37]} AxiomsNikolai Ivanovich Lobachevsky (1792–1856), János Bolyai (1802–1860), Carl Friedrich Gauss (1777–1855) and others^{[42]} 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.^{[43]}
ObjectsPointsPoints are generally considered fundamental objects for building geometry. They may be defined by the properties that they must have, as in Euclid's definition as "that which has no part",^{} axiomatically defined.
With these modern definitions, every geometric shape is defined as a set of points; this is not the case in synthetic geometry, where a line is another fundamental object that is not viewed as the set of the points through which it passes. However, there are modern geometries in which points are not primitive objects, or even without points.^{[45]}^{[46]} One of the oldest such geometries is Whitehead's pointfree geometry, formulated by Alfred North Whitehead in 1919–1920. LinesEuclid described a line as "breadthless length" which "lies equally with respect to the points on itself".^{[44]} 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,^{[47]} 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.^{[48]} In differential geometry, a geodesic is a generalization of the notion of a line to curved spaces.^{[49]} PlanesIn Euclidean geometry a plane is a flat, twodimensional surface that extends infinitely;^{[44]} the definitions for other types of geometries are generalizations of that. Planes are used in many areas of geometry. For instance, planes can be studied as a topological surface without reference to distances or angles;^{[50]} it can be studied as an affine space, where collinearity and ratios can be studied but not distances;^{[51]} it can be studied as the complex plane using techniques of complex analysis;^{[52]} and so on. Anglesrays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.^{[53]}
In Euclidean geometry, angles are used to study polygons and triangles, as well as forming an object of study in their own right.^{[44]} The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry.^{[54]} In derivative.^{[55]}^{[56]}
CurvesA space curves.^{[57]}
In topology, a curve is defined by a function from an interval of the real numbers to another space.^{} algebraic varieties of dimension one.^{[59]}
SurfacesA polynomial equations.^{[59]}
ManifoldsA neighborhood that is homeomorphic to Euclidean space.^{[50]} In differential geometry, a differentiable manifold is a space where each neighborhood is diffeomorphic to Euclidean space.^{[58]}
Manifolds are used extensively in physics, including in general relativity and string theory.^{[61]} Length, area, and volumeLength, area, and volume describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively.^{[62]} In Euclidean geometry and analytic geometry, the length of a line segment can often be calculated by the Pythagorean theorem.^{[63]} 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 3dimensional space.^{} Lebesgue integral.^{[65]}
Metrics and measuresEuclidean metric .The concept of length or distance can be generalized, leading to the idea of Riemannian metrics of general relativity.^{[67]}
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.^{[68]}
Congruence and similarityHilbert, 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.^{[71]} Compass and straightedge constructionsClassical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically, the only instruments used in most geometric constructions are the compass and straightedge.^{[b]} 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 neusis , parabolas and other curves, or mechanical devices, were found.
Dimensiontopological dimension =1Where the traditional geometry allowed dimensions 1 (a higher dimensions for nearly two centuries.^{[72]} 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.^{[73]}
In fractal geometry).^{[74]} 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.^{[75]}
Symmetrytiling of the hyperbolic plane The theme of symmetry in geometry is nearly as old as the science of geometry itself.^{[76]} Symmetric shapes such as the circle, regular polygons and platonic solids held deep significance for many ancient philosophers^{[77]} and were investigated in detail before the time of Euclid.^{[40]} Symmetric patterns occur in nature and were artistically rendered in a multitude of forms, including the graphics of Leonardo da Vinci, M. C. Escher, and others.^{[78]} In the second half of the 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.^{[79]} 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.^{[80]} 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.^{[81]} Both discrete and continuous symmetries play prominent roles in geometry, the former in topology and geometric group theory,^{[82]}^{[83]} the latter in Lie theory and Riemannian geometry.^{[84]}^{[85]} A different type of symmetry is the principle of duality in projective geometry, among other fields. This metaphenomenon 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.^{[86]} A similar and closely related form of duality exists between a vector space and its dual space.^{[87]} Contemporary geometryEuclidean geometrysolid figures, circles, and analytic geometry.^{[36]}
Differential geometryDifferential geometry uses techniques of calculus and linear algebra to study problems in geometry.^{[95]} It has applications in physics,^{[96]} econometrics,^{[97]} and bioinformatics,^{[98]} among others. In particular, differential geometry is of importance to 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).^{[100]}
NonEuclidean geometryEuclidean geometry was not the only historical form of geometry studied. Spherical geometry has long been used by astronomers, astrologers, and navigators.^{[101]} general relativity theory. Riemannian geometry, which considers very general spaces in which the notion of length is defined, is a mainstay of modern geometry.^{[81]}
Topologycompactness.^{[50]}
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.^{[107]} This has often been expressed in the form of the saying 'topology is rubbersheet geometry'. Subfields of topology include geometric topology, differential topology, algebraic topology and general topology.^{[108]} Algebraic geometryThe field of Wiles' proof of Fermat's Last Theorem uses advanced methods of algebraic geometry for solving a longstanding problem of number theory .
In general, algebraic geometry studies geometry through the use of concepts in multivariate polynomials.^{[112]} It has applications in many areas, including cryptography^{[113]} and string theory.^{[114]}
Complex geometryseveral complex variables, and has found applications to string theory and mirror symmetry.^{[118]}
Complex geometry first appeared as a distinct area of study in the work of Bernhard Riemann in his study of Riemann surfaces.^{[119]}^{[120]}^{[121]} Work in the spirit of Riemann was carried out by the Italian school of algebraic geometry in the early 1900s. Contemporary treatment of complex geometry began with the work of JeanPierre Serre, who introduced the concept of sheaves to the subject, and illuminated the relations between complex geometry and algebraic geometry.^{[122]}^{[123]} The primary objects of study in complex geometry are coherent sheaves over these spaces. Special examples of spaces studied in complex geometry include Riemann surfaces, and Calabi–Yau manifolds, and these spaces find uses in string theory. In particular, worldsheets of strings are modelled by Riemann surfaces, and superstring theory predicts that the extra 6 dimensions of 10 dimensional spacetime may be modelled by Calabi–Yau manifolds.
Discrete geometryDiscrete geometry is a subject that has close connections with convex geometry.^{[124]}^{[125]}^{[126]} 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 KneserPoulsen conjecture, etc.^{[127]}^{[128]} It shares many methods and principles with combinatorics. Computational geometryimplementations for manipulating geometrical objects. Important problems historically have included the travelling salesman problem, minimum spanning trees, hiddenline removal, and linear programming.^{[129]}
Although being a young area of geometry, it has many applications in Geometric group theoryGeometric group theory uses largescale geometric techniques to study finitely generated groups.^{[131]} It is closely connected to lowdimensional topology, such as in Grigori Perelman's proof of the Geometrization conjecture, which included the proof of the Poincaré conjecture, a Millennium Prize Problem.^{[132]} Geometric group theory often revolves around the right angled Artin groups.^{[131]}^{[133]}
Convex geometryoptimization and functional analysis and important applications in number theory .
Convex geometry dates back to antiquity.^{} .ApplicationsGeometry has found applications in many fields, some of which are described below. ArtMathematics 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.^{[135]} Artists have long used concepts of proportion in design. Vitruvius developed a complicated theory of ideal proportions for the human figure.^{[136]} These concepts have been used and adapted by artists from Michelangelo to modern comic book artists.^{[137]} 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.^{[138]} Tilings, or tessellations, have been used in art throughout history. Islamic art makes frequent use of tessellations, as did the art of M. C. Escher.^{[139]} 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.^{[140]}^{[141]}
ArchitectureGeometry has many applications in architecture. In fact, it has been said that geometry lies at the core of architectural design.^{} tessellations,^{[91]} and the use of symmetry.^{[91]}
PhysicsThe 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.^{[145]} quantum information theory.^{[148]}
Other fields of mathematicsinfinitesimal calculus in the 17th century. Analytic geometry continues to be a mainstay of precalculus and calculus curriculum.^{[149]}^{[150]}
Another important area of application is scheme theory, which is used in Wiles's proof of Fermat's Last Theorem.^{[153]}
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Further readingExternal linksWikibooks has more on the topic of: Geometry
Encyclopædia Britannica. Vol. 11 (11th ed.). 1911. pp. 675–736. .
