# Geometry

**Geometry**(Greek "γεωμετρία"; geo = earth, metria = measure) is a part ofmathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerninglength s,area s, andvolume s, in the third century B.C., geometry was put into an axiomatic form byEuclid , whose treatment -Euclidean geometry - set a standard for many centuries to follow. The field ofastronomy , especially mapping the positions of the stars and planets on the celestial sphere, served as an important source of geometric problems during the next one and a half millennia.Introduction of

coordinates byRené Descartes and the concurrent development ofalgebra marked a new stage for geometry, since geometric figures, such asplane curve s, could now be represented analytically, i.e., with functions and equations. This played a key role in the emergence ofcalculus in the seventeenth century. Furthermore, the theory of perspective showed that there is more to geometry than just the metric properties of figures. The subject of geometry was further enriched by the study of intrinsic structure of geometric objects that originated withEuler and Gauss and led to the creation oftopology anddifferential geometry .Since the nineteenth century discovery of

non-Euclidean geometry , the concept ofspace has undergone a radical transformation. Contemporary geometry considersmanifold s, spaces that are considerably more abstract than the familiarEuclidean space , which they only approximately resemble at small scales. These spaces may be endowed with additional structure, allowing one to speak about length. Modern geometry has multiple strong bonds withphysics , exemplified by the ties betweenRiemannian geometry andgeneral relativity . One of the youngest physical theories,string theory , is also very geometric in flavour.The visual nature of geometry makes it initially more accessible than other parts of mathematics, such as

algebra ornumber theory . However, the geometric language is also used in contexts that are far removed from its traditional, Euclidean provenance, for example, infractal geometry , and especially inalgebraic geometry . [*It is quite common in algebraic geometry to speak about "geometry of algebraic varieties over*]finite field s", possibly singular. From a naïve perspective, these objects are just finite sets of points, but by invoking powerful geometric imagery and using well developed geometric techniques, it is possible to find structure and establish properties that make them somewhat analogous to the ordinarysphere s or cones.**History**The earliest recorded beginnings of geometry can be traced to ancient

Mesopotamia , Egypt, and the Indus Valley from around3000 BCE . Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need insurveying ,construction ,astronomy , and various crafts. The earliest known texts on geometry are the Egyptian "Rhind Papyrus" and "Moscow Papyrus", the Babylonian clay tablets, and the Indian "Shulba Sutras ", while the Chinese had the work ofMozi ,Zhang Heng , and the "Nine Chapters on the Mathematical Art ", edited byLiu Hui .Euclid's "The Elements of Geometry" (c.

300 BCE ) was one of the most important early texts on geometry, in which he presented geometry in an idealaxiom atic form, which came to be known asEuclidean geometry . The treatise is not, as is sometimes thought, a compendium of all thatHellenistic mathematicians knew about geometry at that time; rather, it is an elementary introduction to it; [*cite book|last=Boyer|authorlink=Carl Benjamin Boyer|title=|year=1991|chapter=Euclid of Alexandria|pages=104|quote=The "Elements" was not, as is sometimes thought, a compendium of all geometric knowledge; it was instead an introductory textbook covering all "elementary" mathematics-*] Euclid himself wrote eight more advanced books on geometry. We know from other references that Euclid’s was not the first elementary geometry textbook, but the others fell into disuse and were lost.Fact|date=July 2007In the

Middle Ages , Muslim mathematicians contributed to the development of geometry, especiallyalgebraic geometry andgeometric algebra .Al-Mahani (b. 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems inalgebra .Thābit ibn Qurra (known as Thebit inLatin ) (836-901) dealt witharithmetic al operations applied toratio s of geometrical quantities, and contributed to the development ofanalytic geometry .Omar Khayyám (1048-1131) found geometric solutions tocubic equation s, and his extensive studies of theparallel postulate contributed to the development ofNon-Euclidian geometry .Fact|date=July 2007In the early 17th century, there were two important developments in geometry. The first, and most important, was the creation of

analytic geometry , or geometry with coordinates andequations , byRené Descartes (1596–1650) andPierre de Fermat (1601–1665). This was a necessary precursor to the development ofcalculus and a precise quantitative science ofphysics . The second geometric development of this period was the systematic study ofprojective geometry byGirard Desargues (1591–1661). Projective geometry is the study of geometry without measurement, just the study of how points align with each other.Two developments in geometry in the nineteenth century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Lobachevsky, Bolyai and Gauss and of the formulation of

symmetry as the central consideration in theErlangen Programme ofFelix Klein (which generalized the Euclidean and non Euclidean geometries). Two of the master geometers of the time wereBernhard Riemann , working primarily with tools frommathematical analysis , and introducing theRiemann surface , andHenri Poincaré , the founder ofalgebraic topology and the geometric theory ofdynamical system s.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 andclassical mechanics . The traditional type of geometry was recognized as that ofhomogeneous space s, those spaces which have a sufficient supply of symmetry, so that from point to point they look just the same.**What is geometry?**Recorded development of geometry spans more than two

millennia . It is hardly surprising that perceptions of what constituted geometry evolved throughout the ages. The geometric paradigms presented below should be viewed as 'Pictures at an exhibition ' of a sort: they do not exhaust the subject of geometry but rather reflect some of its defining themes.**Practical geometry**There is little doubt that geometry originated as a "practical" science, concerned with surveying, measurements, areas, and volumes. Among the notable accomplishments one finds formulas for

length s,area s andvolume s, such asPythagorean theorem ,circumference and area of a circle, area of atriangle , volume of a cylinder,sphere , and a pyramid. Development ofastronomy led to emergence oftrigonometry andspherical trigonometry , together with the attendant computational techniques.**Axiomatic geometry**A method of computing certain inaccessible distances or heights based on similarity of geometric figures and attributed to

Thales presaged more abstract approach to geometry taken byEuclid in his Elements, one of the most influential books ever written. Euclid introduced certainaxiom s, orpostulate s, expressing primary or self-evident properties of points, lines, and planes. He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor. In the twentieth century,David Hilbert employed axiomatic reasoning in his attempt to update Euclid and provide modern foundations of geometry.**Geometric constructions**Ancient scientists paid special attention to constructing geometric objects that had been described in some other way. Classical instruments allowed in geometric constructions are the

compass and straightedge . 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. The approach to geometric problems with geometric or mechanical means is known assynthetic geometry .**Numbers in geometry**Already

Pythagoreans considered the role of numbers in geometry. However, the discovery of incommensurable lengths, which contradicted their philosophical views, made them abandon (abstract) numbers in favour of (concrete) geometric quantities, such as length and area of figures. Numbers were reintroduced into geometry in the form ofcoordinate s byDescartes , who realized that the study of geometric shapes can be facilitated by their algebraic representation.Analytic geometry applies methods of algebra to geometric questions, typically by relating geometriccurve s and algebraicequation s. These ideas played a key role in the development ofcalculus in the seventeenth century and led to discovery of many new properties of plane curves. Modernalgebraic geometry considers similar questions on a vastly more abstract level.**Geometry of position**Even in ancient times, geometers considered questions of relative position or spatial relationship of geometric figures and shapes. Some examples are given by inscribed and circumscribed circles of

polygon s, lines intersecting and tangent toconic section s, the Pappus and Menelaus configurations of points and lines. In the Middle Ages new and more complicated questions of this type were considered: What is the maximum number of spheres simultaneously touching a given sphere of the same radius (kissing number problem )? What is the densest packing of spheres of equal size in space (Kepler conjecture )? Most of these questions involved 'rigid' geometrical shapes, such as lines or spheres. Projective, convex and discrete geometry are three subdisciplines within present day geometry that deal with these and related questions.A new chapter in "Geometria situs" was opened by

Leonhard Euler , who boldly cast out metric properties of geometric figures and considered their most fundamental geometrical structure based solely on shape.Topology , which grew out of geometry, but turned into a large independent discipline, does not differentiate between objects that can be continuously deformed into each other. The objects may nevertheless retain some geometry, as in the case ofhyperbolic knot s.**Geometry beyond Euclid**For nearly two thousand years since Euclid, while the range of geometrical questions asked and answered inevitably expanded, basic understanding of

space remained essentially the same.Immanuel Kant argued that there is only one, "absolute", geometry, which is known to be true "a priori" by an inner faculty of mind: Euclidean geometry wassynthetic 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 dominant view was overturned by the revolutionary discovery of non-Euclidean geometry in the works of Gauss (who never published his theory),Bolyai , andLobachevsky , who demonstrated that ordinaryEuclidean space is only one possibility for development of geometry. A broad vision of the subject of geometry was then expressed byRiemann in his inaugurational lecture "Über die Hypothesen, welche der Geometrie zu Grunde liegen" ("On the hypotheses on which geometry is based"), published only after his death. Riemann's new idea of space proved crucial inEinstein 'sgeneral relativity theory andRiemannian geometry , which considers very general spaces in which the notion of length is defined, is a mainstay of modern geometry.**Symmetry**The theme of

symmetry in geometry is nearly as old as the science of geometry itself. Thecircle ,regular polygon s andplatonic solid s held deep significance for many ancient philosophers and were investigated in detail by the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in a multitude of forms, including the bewildering graphics ofM. C. Escher . Nonetheless, it was not until the second half of nineteenth century that the unifying role of symmetry in foundations of geometry had been recognized.Felix Klein 'sErlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group, determines what geometry "is". Symmetry in classicalEuclidean geometry is represented by congruences and rigid motions, whereas inprojective geometry an analogous role is played bycollineation s, geometric transformations that take straight lines into straight lines. However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, andSophus Lie that Klein's idea to 'define a geometry via itssymmetry group ' proved most influential. Both discrete and continuous symmetries play prominent role in geometry, the former intopology andgeometric group theory , the latter inLie theory andRiemannian geometry .**Modern geometry**"Modern geometry" is the title of a popular textbook by Dubrovin, Novikov, and Fomenko first published in 1979 (in Russian). At close to 1000 pages, the book has one major thread: geometric structures of various types on

manifold s and their applications in contemporarytheoretical physics . A quarter century after its publication,differential geometry ,algebraic geometry ,symplectic geometry , andLie theory presented in the book remain among the most visible areas of modern geometry, with multiple connections with other parts of mathematics and physics.**Contemporary geometers**Some of the representative leading figures in modern geometry are

Michael Atiyah ,Mikhail Gromov , andWilliam Thurston . The common feature in their work is the use ofsmooth manifold s as the basic idea of "space"; they otherwise have rather different directions and interests. Geometry now is, in large part, the study of "structures" on manifolds that have a geometric meaning, in the sense of theprinciple of covariance that lies at the root ofgeneral relativity theory in theoretical physics. (See for a survey.)Much of this theory relates to the theory of "continuous symmetry", or in other words

Lie group s. From the foundational point of view, on manifolds and their geometrical structures, important is the concept ofpseudogroup , defined formally byShiing-shen Chern in pursuing ideas introduced byÉlie Cartan . A pseudogroup can play the role of a Lie group of "infinite" dimension.**Dimension**Where the traditional geometry allowed dimensions 1 (a

line ), 2 (a plane) and 3 (our ambient world conceived of asthree-dimensional space ), mathematicians have usedhigher dimensions for nearly two centuries. Dimension has gone through stages of being anynatural number "n", possibly infinite with the introduction ofHilbert space , and any positive real number infractal geometry .Dimension theory is a technical area, initially withingeneral topology , that discusses "definitions"; in common with most mathematical ideas, dimension is now defined rather than an intuition. Connectedtopological manifold s have a well-defined dimension; this is a theorem (invariance of domain ) rather than anything "a priori".The issue of dimension still matters to geometry, in the absence of complete answers to classic questions. Dimensions 3 of space and 4 of

space-time are special cases ingeometric topology . Dimension 10 or 11 is a key number instring theory . Exactly why is something to which research may bring a satisfactory "geometric" answer.**Contemporary Euclidean geometry**The study of traditional

Euclidean geometry is by no means dead. It is now typically presented as the geometry ofEuclidean space s of any dimension, and of theEuclidean group ofrigid motion s. The fundamental formulae of geometry, such as thePythagorean theorem , can be presented in this way for a generalinner product space .Euclidean geometry has become closely connected with

computational geometry ,computer graphics ,convex geometry ,discrete geometry , and some areas ofcombinatorics . Momentum was given to further work on Euclidean geometry and the Euclidean groups bycrystallography and the work ofH. S. M. Coxeter , and can be seen in theories ofCoxeter group s andpolytope s.Geometric group theory is an expanding area of the theory of more generaldiscrete group s, drawing on geometric models and algebraic techniques.**Algebraic geometry**The field of

algebraic geometry is the modern incarnation of theCartesian geometry ofco-ordinates . After a turbulent period ofaxiomatization , its foundations are in the twenty-first century on a stable basis. Either one studies the 'classical' case where the spaces arecomplex manifold s that can be described byalgebraic equation s; or thescheme theory provides a technically sophisticated theory based on generalcommutative ring s.The geometric style which was traditionally called the Italian school is now known as

birational geometry . It has made progress in the fields of threefolds,singularity theory andmoduli space s, as well as recovering and correcting the bulk of the older results. Objects from algebraic geometry are now commonly applied instring theory , as well asdiophantine geometry .Methods of algebraic geometry rely heavily on

sheaf theory and other parts ofhomological algebra . TheHodge conjecture is an open problem that has gradually taken its place as one of the major questions for mathematicians. For practical applications,Gröbner basis theory andreal algebraic geometry are major subfields.**Differential geometry**Differential geometry , which in simple terms is the geometry ofcurvature , has been of increasing importance tomathematical physics since the suggestion that space is notflat space . Contemporary differential geometry is "intrinsic", meaning that space is a manifold and structure is given by aRiemannian metric , or analogue, locally determining a geometry that is variable from point to point.This approach contrasts with the "extrinsic" point of view, where curvature means the way a space "bends" within a larger space. The idea of 'larger' spaces is discarded, and instead manifolds carry

vector bundle s. Fundamental to this approach is the connection between curvature andcharacteristic class es, as exemplified by thegeneralized Gauss-Bonnet theorem .**Topology and geometry**The field of

topology , which saw massive development in the 20th century, is in a technical sense a type oftransformation geometry , in which transformations arehomeomorphism s. This has often been expressed in the form of the dictum 'topology is rubber-sheet geometry'. Contemporarygeometric topology anddifferential topology , and particular subfields such asMorse theory , would be counted by most mathematicians as part of geometry.Algebraic topology andgeneral topology have gone their own ways.**Axiomatic and open development**The model of Euclid's "Elements", a connected development of geometry as an

axiomatic system , is in a tension withRené Descartes 's reduction of geometry to algebra by means of acoordinate system . There were many champions ofsynthetic geometry , Euclid-style development of projective geometry, in the nineteenth century,Jakob Steiner being a particularly brilliant figure. In contrast to such approaches to geometry as a closed system, culminating inHilbert's axioms and regarded as of important pedagogic value, most contemporary geometry is a matter of style.Computational synthetic geometry is now a branch ofcomputer algebra .The Cartesian approach currently predominates, with geometric questions being tackled by tools from other parts of mathematics, and geometric theories being quite open and integrated. This is to be seen in the context of the axiomatization of the whole of

pure mathematics , which went on in the period c.1900–c.1950: in principle all methods are on a common axiomatic footing. This reductive approach has had several effects. There is a taxonomic trend, which following Klein and his Erlangen program (a taxonomy based on thesubgroup concept) arranges theories according to generalization and specialization. For exampleaffine geometry is more general than Euclidean geometry, and more special than projective geometry. The whole theory ofclassical group s thereby becomes an aspect of geometry. Theirinvariant theory , at one point in the nineteenth century taken to be the prospective master geometric theory, is just one aspect of the generalrepresentation theory ofalgebraic group s andLie group s. Usingfinite field s, the classical groups give rise tofinite group s, intensively studied in relation to thefinite simple group s; and associatedfinite geometry , which has both combinatorial (synthetic) and algebro-geometric (Cartesian) sides.An example from recent decades is the

twistor theory ofRoger Penrose , initially an intuitive and synthetic theory, then subsequently shown to be an aspect ofsheaf theory oncomplex manifold s. In contrast, thenon-commutative geometry ofAlain Connes is a conscious use of geometric language to express phenomena of the theory ofvon Neumann algebra s, and to extend geometry into the domain ofring theory where thecommutative law of multiplication is not assumed.Another consequence of the contemporary approach, attributable in large measure to the Procrustean bed represented by

Bourbaki ste axiomatization trying to complete the work ofDavid Hilbert , is to create winners and losers. The "Ausdehnungslehre " (calculus of extension) ofHermann Grassmann was for many years a mathematical backwater, competing in three dimensions against other popular theories in the area ofmathematical physics such as those derived fromquaternion s. In the shape of generalexterior algebra , it became a beneficiary of the Bourbaki presentation ofmultilinear algebra , and from 1950 onwards has been ubiquitous. In much the same way,Clifford algebra became popular, helped by a 1957 book "Geometric Algebra" byEmil Artin . The history of 'lost' geometric methods, for example "infinitely near point s", which were dropped since they did not well fit into the pure mathematical world post-"Principia Mathematica ", is yet unwritten. The situation is analogous to the expulsion ofinfinitesimal s fromdifferential calculus . As in that case, the concepts may be recovered by fresh approaches and definitions. Those may not be unique:synthetic differential geometry is an approach to infinitesimals from the side ofcategorical logic , asnon-standard analysis is by means ofmodel theory .**ee also****Lists***

List of basic geometry topics

*List of geometry topics

*List of geometers

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* Important publications in geometry

*List of mathematics articles **Related topics***

Interactive geometry software

* "Flatland ", a book written byEdwin Abbott Abbott about two andthree-dimensional space , to understand the concept of four dimensions

*Why 10 dimensions? **References****External links*** [

*http://www.mathforum.org/library/topics/geometry/ "The Math Forum" — Geometry*]

** [*http://www.mathforum.org/geometry/k12.geometry.html "The Math Forum" — K–12 Geometry*]

** [*http://www.mathforum.org/geometry/coll.geometry.html "The Math Forum" — College Geometry*]

** [*http://www.mathforum.org/advanced/geom.html "The Math Forum" — Advanced Geometry*]

* [*http://www.math.niu.edu/~rusin/known-math/index/tour_geo.html "The Mathematical Atlas" — Geometric Areas of Mathematics*]

* [*http://www.gresham.ac.uk/event.asp?PageId=45&EventId=618 "4000 Years of Geometry"*] , lecture by Robin Wilson given atGresham College , 3rd October 2007 (available for MP3 and MP4 download as well as a text file)

* [*http://www.cut-the-knot.org/WhatIs/WhatIsGeometry.shtml What Is Geometry?*] atcut-the-knot

* [*http://www.cut-the-knot.org/geometry.shtml Geometry*] atcut-the-knot

* [*http://agutie.homestead.com Geometry Step by Step from the Land of the Incas*] by Antonio Gutierrez.

* [*http://www.islamicarchitecture.org/art/islamic-geometry-and-floral-patterns.html Islamic Geometry*]

* Stanford Encyclopedia of Philosophy:

** [*http://plato.stanford.edu/entries/geometry-finitism/ Finitism in Geometry*]

** [*http://plato.stanford.edu/entries/geometry-19th/ Geometry in the 19th Century*]

* [*http://www.egwald.ca/geometry/index.php Online Interactive Geometric Objects*] by Elmer G. Wiens

* [*http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Arabic_mathematics.html Arabic mathematics : forgotten brilliance?*]

* [*http://www.ics.uci.edu/~eppstein/junkyard/topic.html The Geometry Junkyard*]

* [*http://mrperezonlinemathtutor.com/A_Geometry.html Geometry lessons in PowerPoint*]

*Wikimedia Foundation.
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**Geometry**— Ge*om e*try, n.; pl. {Geometries}[F. g[ e]om[ e]trie, L. geometria, fr. Gr. ?, fr. ? to measure land; ge a, gh^, the earth + ? to measure. So called because one of its earliest and most important applications was to the measurement of the earth s … The Collaborative International Dictionary of English**geometry**— [jē äm′ə trē] n. pl. geometries [ME geometrie < OFr < L geometria < Gr geōmetria < geōmetrein, to measure the earth < gē, earth + metria, measurement < metrein, to measure: for IE base, see METER1] 1. the branch of mathematics… … English World dictionary**geometry**— early 14c., from O.Fr. géométrie (12c.), from L. geometria, from Gk. geometria measurement of earth or land; geometry, from comb. form of ge earth, land + metria (see METRY (Cf. metry)) … Etymology dictionary**geometry**— ► NOUN (pl. geometries) 1) the branch of mathematics concerned with the properties and relations of points, lines, surfaces, and solids. 2) the shape and relative arrangement of the parts of something. DERIVATIVES geometrician noun. ORIGIN Greek … English terms dictionary**geometry**— /jee om i tree/, n. 1. the branch of mathematics that deals with the deduction of the properties, measurement, and relationships of points, lines, angles, and figures in space from their defining conditions by means of certain assumed properties… … Universalium**geometry**— Although various laws concerning lines and angles were known to the Egyptians and the Pythagoreans, the systematic treatment of geometry by the axiomatic method began with the Elements of Euclid . From a small number of explicit axioms,… … Philosophy dictionary**geometry**— [[t]ʤiɒ̱mɪtri[/t]] 1) N UNCOUNT Geometry is the branch of mathematics concerned with the properties and relationships of lines, angles, curves, and shapes. ...the very ordered way in which mathematics and geometry describe nature. 2) N UNCOUNT:… … English dictionary**Geometry**— Mathematics Math e*mat ics, n. [F. math[ e]matiques, pl., L. mathematica, sing., Gr. ? (sc. ?) science. See {Mathematic}, and { ics}.] That science, or class of sciences, which treats of the exact relations existing between quantities or… … The Collaborative International Dictionary of English**geometry**— noun a) The branch of mathematics dealing with spatial relationships. spherical geometry b) A type of geometry with particular properties. See Also: geometer, geometrical … Wiktionary**geometry**— n. descriptive; Euclidean; plane; projective; solid geometry * * * [dʒɪ ɒmɪtrɪ] Euclidean plane projective solid geometry descriptive … Combinatory dictionary