Geometry (from Ancient Greekγεωμετρία (geōmetría)'land measurement'; from γῆ (gê)'earth, land' and μέτρον (métron)'a measure')[1] is a branch ofmathematics concerned with properties of space such as the distance, shape, size, and relative position of figures.[2] Geometry is, along witharithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called ageometer. Until the 19th century, geometry was almost exclusively devoted toEuclidean geometry,[a] which includes the notions ofpoint,line,plane,distance,angle,surface, andcurve, as fundamental concepts.[3]
Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art,architecture, and other activities that are related to graphics.[4] Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental inWiles's proof ofFermat's Last Theorem, a problem that was stated in terms ofelementary arithmetic, and remained unsolved for several centuries.
During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries isCarl Friedrich Gauss'sTheorema Egregium ("remarkable theorem") that asserts roughly that theGaussian curvature of a surface is independent from any specificembedding in aEuclidean space. This implies that surfaces can be studiedintrinsically, that is, as stand-alone spaces, and has been expanded into the theory ofmanifolds andRiemannian geometry. Later in the 19th century, it appeared that geometries without theparallel postulate (non-Euclidean geometries) can be developed without introducing any contradiction. The geometry that underliesgeneral relativity is a famous application of non-Euclidean geometry.
Since the late 19th century, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods—differential geometry,algebraic geometry,computational geometry,algebraic topology,discrete geometry (also known ascombinatorial geometry), etc.—or on the properties of Euclidean spaces that are disregarded—projective geometry that consider only alignment of points but not distance and parallelism,affine geometry that omits the concept of angle and distance,finite geometry that omitscontinuity, and others. This enlargement of the scope of geometry led to a change of meaning of the word "space", which originally referred to the three-dimensionalspace of the physical world and itsmodel provided by Euclidean geometry; presently ageometric space, or simply aspace is amathematical structure on which some geometry is defined.
AEuropean and anArab practicing geometry in the 15th century
The earliest recorded beginnings of geometry can be traced to ancientMesopotamia andEgypt in the 2nd millennium BC.[5][6] 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 theEgyptianRhind Papyrus (2000–1800 BC) andMoscow Papyrus (c. 1890 BC), and theBabylonian clay tablets, such asPlimpton 322 (1900 BC). For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, orfrustum.[7] Later clay tablets (350–50 BC) demonstrate that Babylonian astronomers implementedtrapezoid procedures for computing Jupiter's position andmotion within time-velocity space. These geometric procedures anticipated theOxford Calculators, including themean speed theorem, by 14 centuries.[8] South of Egypt theancient Nubians established a system of geometry including early versions of sun clocks.[9][10]
In the 7th century BC, theGreek mathematicianThales 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 toThales's theorem.[11]Pythagoras established thePythagorean School, which is credited with the first proof of thePythagorean theorem,[12] though the statement of the theorem has a long history.[13][14]Eudoxus (408–c. 355 BC) developed themethod of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures,[15] as well as a theory of ratios that avoided the problem ofincommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whoseElements, widely considered the most successful and influential textbook of all time,[16] introducedmathematical rigor through theaxiomatic 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 theElements were already known, Euclid arranged them into a single, coherent logical framework.[17] TheElements 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.[18]Archimedes (c. 287–212 BC) ofSyracuse, Italy used the method of exhaustion to calculate thearea under the arc of aparabola with thesummation of an infinite series, and gave remarkably accurate approximations ofpi.[19] He also studied thespiral bearing his name and obtained formulas for thevolumes ofsurfaces of revolution.
Woman teaching geometry. Illustration at the beginning of a medieval translation ofEuclid's Elements, (c. 1310).
Indian mathematicians also made many important contributions in geometry. TheShatapatha Brahmana (3rd century BC) contains rules for ritual geometric constructions that are similar to theSulba Sutras.[20] According to (Hayashi 2005, p. 363), theŚulba Sūtras 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 ofPythagorean triples,[b] which are particular cases ofDiophantine equations.[21]In theBakhshali manuscript, there are 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'sAryabhatiya (499) includes the computation of areas and volumes.Brahmagupta wrote his astronomical workBrāhmasphuṭasiddhānta in 628. Chapter 12, containing 66Sanskrit 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).[23] In the latter section, he stated his famous theorem on the diagonals of acyclic quadrilateral. Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization ofHeron's formula), as well as a complete description ofrational triangles (i.e. triangles with rational sides and rational areas).[23]
In the early 17th century, there were two important developments in geometry. The first was the creation of analytic geometry, or geometry withcoordinates andequations, byRené Descartes (1596–1650) andPierre de Fermat (1601–1665).[30] This was a necessary precursor to the development ofcalculus and a precise quantitative science ofphysics.[31] The second geometric development of this period was the systematic study ofprojective geometry byGirard Desargues (1591–1661).[32] Projective geometry studies properties of shapes which are unchanged underprojections andsections, especially as they relate toartistic perspective.[33]
Two developments in geometry in the 19th century changed the way it had been studied previously.[34] These were the discovery ofnon-Euclidean geometries by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation ofsymmetry 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 (1826–1866), working primarily with tools frommathematical analysis, and introducing theRiemann surface, andHenri Poincaré, the founder ofalgebraic topology and the geometric theory ofdynamical 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 ascomplex analysis andclassical mechanics.[35]
Main concepts
The following are some of the most important concepts in geometry.[3][36]
Euclid took an abstract approach to geometry in hisElements,[37] one of the most influential books ever written.[38] Euclid introduced certainaxioms, orpostulates, expressing primary or self-evident properties of points, lines, and planes.[39] 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 asaxiomatic orsynthetic geometry.[40] At the start of the 19th century, the discovery ofnon-Euclidean geometries byNikolai Ivanovich Lobachevsky (1792–1856),János Bolyai (1802–1860),Carl Friedrich Gauss (1777–1855) and others[41] 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.[42]
Points 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",[43] or insynthetic geometry. In modern mathematics, they are generally defined aselements of aset calledspace, which is itselfaxiomatically 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.
Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself".[43] 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, inanalytic geometry, a line in the plane is often defined as the set of points whose coordinates satisfy a givenlinear equation,[46] but in a more abstract setting, such asincidence geometry, a line may be an independent object, distinct from the set of points which lie on it.[47] In differential geometry, ageodesic is a generalization of the notion of a line tocurved spaces.[48]
In Euclidean geometry a plane is a flat, two-dimensional surface that extends infinitely;[43] 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 atopological surface without reference to distances or angles;[49] it can be studied as anaffine space, where collinearity and ratios can be studied but not distances;[50] it can be studied as thecomplex plane using techniques ofcomplex analysis;[51] and so on.
Acurve is a 1-dimensional object that may be straight (like a line) or not; curves in 2-dimensional space are calledplane curves and those in 3-dimensional space are calledspace curves.[52]
In topology, a curve is defined by a function from an interval of the real numbers to another space.[49] In differential geometry, the same definition is used, but the defining function is required to be differentiable.[53] Algebraic geometry studiesalgebraic curves, which are defined asalgebraic varieties ofdimension one.[54]
Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique 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.[43] In modern terms, an angle is the figure formed by tworays, called thesides of the angle, sharing a common endpoint, called thevertex of the angle.[57]The size of an angle is formalized as anangular measure.
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.[61] Mathematicians have found many explicitformulas for area andformulas for volume of various geometric objects. Incalculus, area and volume can be defined in terms ofintegrals, such as theRiemann integral[63] or theLebesgue integral.[64]
In a different direction, the concepts of length, area and volume are extended bymeasure theory, which studies methods of assigning a size ormeasure tosets, where the measures follow rules similar to those of classical area and volume.[67]
Congruence andsimilarity are concepts that describe when two shapes have similar characteristics.[68] 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.[69]Hilbert, in his work on creating a more rigorous foundation for geometry, treated congruence as an undefined term whose properties are defined byaxioms.
Congruence and similarity are generalized intransformation geometry, which studies the properties of geometric objects that are preserved by different kinds of transformations.[70]
Classical 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 thecompass andstraightedge.[c] 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 usingneusis, parabolas and other curves, or mechanical devices, were found.
Traditional geometry allowed dimensions 1 (aline or curve), 2 (aplane or surface), and 3 (our ambient world conceived of asthree-dimensional space). Furthermore, mathematicians and physicists have usedhigher dimensions for nearly two centuries.[71] One example of a mathematical use for higher dimensions is theconfiguration space of a physical system, which has a dimension equal to the system'sdegrees of freedom. For instance, the configuration of a screw can be described by five coordinates.[72]
The theme ofsymmetry in geometry is nearly as old as the science of geometry itself.[75] Symmetric shapes such as thecircle,regular polygons andplatonic solids held deep significance for many ancient philosophers[76] and were investigated in detail before the time of Euclid.[39] Symmetric patterns occur in nature and were artistically rendered in a multitude of forms, including the graphics ofLeonardo da Vinci,M. C. Escher, and others.[77] In the second half of the 19th century, the relationship between symmetry and geometry came under intense scrutiny.Felix Klein'sErlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformationgroup, determines what geometryis.[78] Symmetry in classicalEuclidean geometry is represented bycongruences and rigid motions, whereas inprojective geometry an analogous role is played bycollineations,geometric transformations that take straight lines into straight lines.[79] 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' found its inspiration.[80] Both discrete and continuous symmetries play prominent roles in geometry, the former intopology andgeometric group theory,[81][82] the latter inLie theory andRiemannian geometry.[83][84]
A different type of symmetry is the principle ofduality inprojective geometry, among other fields. This meta-phenomenon can roughly be described as follows: in anytheorem, exchangepoint withplane,join withmeet,lies in withcontains, and the result is an equally true theorem.[85] A similar and closely related form of duality exists between avector space and itsdual space.[86]
In particular, differential geometry is of importance tomathematical physics due toAlbert Einstein'sgeneral relativity postulation that theuniverse iscurved.[99] Differential geometry can either beintrinsic (meaning that the spaces it considers aresmooth manifolds whose geometric structure is governed by aRiemannian metric, which determines how distances are measured near each point) orextrinsic (where the object under study is a part of some ambient flat Euclidean space).[100]
Topology is the field concerned with the properties ofcontinuous mappings,[101] and can be considered a generalization of Euclidean geometry.[102] In practice, topology often means dealing with large-scale properties of spaces, such asconnectedness andcompactness.[49]
Convex geometry dates back to antiquity.[131]Archimedes gave the first known precise definition of convexity. Theisoperimetric problem, a recurring concept in convex geometry, was studied by the Greeks as well, includingZenodorus. Archimedes,Plato,Euclid, and laterKepler andCoxeter all studiedconvex 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 andlattices.
Applications
Geometry has found applications in many fields, some of which are described below.
Mathematics and art are related in a variety of ways. For instance, the theory ofperspective showed that there is more to geometry than just the metric properties of figures: perspective is the origin ofprojective geometry.[132]
Artists have long used concepts ofproportion in design.Vitruvius developed a complicated theory ofideal proportions for the human figure.[133] These concepts have been used and adapted by artists fromMichelangelo to modern comic book artists.[134]
Thegolden 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.[135]
Cézanne advanced the theory that all images can be built up from thesphere, thecone, and thecylinder. This is still used in art theory today, although the exact list of shapes varies from author to author.[137][138]
Geometry has many applications in architecture. In fact, it has been said that geometry lies at the core of architectural design.[139][140] Applications of geometry to architecture include the use ofprojective geometry to createforced perspective,[141] the use ofconic sections in constructing domes and similar objects,[90] the use oftessellations,[90] and the use of symmetry.[90]
The field ofastronomy, especially as it relates to mapping the positions ofstars andplanets on thecelestial sphere and describing the relationship between movements of celestial bodies, have served as an important source of geometric problems throughout history.[142]
The Pythagoreans discovered that the sides of a triangle could haveincommensurable lengths.
Calculus was strongly influenced by geometry.[30] For instance, the introduction ofcoordinates byRené Descartes and the concurrent developments ofalgebra marked a new stage for geometry, since geometric figures such asplane curves could now be representedanalytically in the form of functions and equations. This played a key role in the emergence ofinfinitesimal calculus in the 17th century. Analytic geometry continues to be a mainstay of pre-calculus and calculus curriculum.[146][147]
^Until the 19th century, geometry was dominated by the assumption that all geometric constructions were Euclidean. In the 19th century and later, this was challenged by the development ofhyperbolic geometry byLobachevsky and othernon-Euclidean geometries byGauss and others. It was then realised that implicitly non-Euclidean geometry had appeared throughout history, including the work ofDesargues in the 17th century, all the way back to the implicit use ofspherical geometry to understand theEarth geodesy and to navigate the oceans since antiquity.
^Pythagorean triples are triples of integers with the property:. Thus,,, etc.
^The ancient Greeks had some constructions using other instruments.
^Depuydt, Leo (1 January 1998). "Gnomons at Meroë and Early Trigonometry".The Journal of Egyptian Archaeology.84:171–180.doi:10.2307/3822211.JSTOR3822211.
^Slayman, Andrew (27 May 1998)."Neolithic Skywatchers".Archaeology Magazine Archive.Archived from the original on 5 June 2011. Retrieved17 April 2011.
^Kurt Von Fritz (1945). "The Discovery of Incommensurability by Hippasus of Metapontum".Classics in the History of Greek Mathematics. Annals of Mathematics; Boston Studies in the Philosophy of Science. Vol. 240. Annals of Mathematics, Trustees of Princeton University on Behalf of the Annals of Mathematics, Mathematics Department, Princeton University. pp. 211–231.doi:10.1007/978-1-4020-2640-9_11.ISBN978-90-481-5850-8.JSTOR1969021.
^(Cooke 2005, p. 198): "The arithmetic content of theŚulva Sūtras 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."
^(Boyer 1991, "The Arabic Hegemony" pp. 241–242) "Omar Khayyam (c. 1050–1123), the "tent-maker," wrote anAlgebra 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."".
"Three scientists, Ibn al-Haytham, Khayyam, and al-Tusi, had made the most considerable contribution to this branch of geometry whose importance came to be completely recognized only in the 19th century. In essence, their propositions concerning the properties of quadrangles which they considered, assuming that some of the angles of these figures were acute of obtuse, embodied the first few theorems of the hyperbolic and the elliptic geometries. Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V. It is extremely important that these scholars established the mutual connection between this postulate and the sum of the angles of a triangle and a quadrangle. By their works on the theory of parallel lines Arab mathematicians directly influenced the relevant investigations of their European counterparts. The first European attempt to prove the postulate on parallel lines—made by Witelo, the Polish scientists of the 13th century, while revising Ibn al-Haytham'sBook of Optics (Kitab al-Manazir)—was undoubtedly prompted by Arabic sources. The proofs put forward in the 14th century by the Jewish scholar Levi ben Gerson, who lived in southern France, and by the above-mentioned Alfonso from Spain directly border on Ibn al-Haytham's demonstration. Above, we have demonstrated thatPseudo-Tusi's Exposition of Euclid had stimulated both J. Wallis's and G. Saccheri's studies of the theory of parallel lines."
^abcdeEuclid's Elements – All thirteen books in one volume, Based on Heath's translation, Green Lion PressISBN1-888009-18-7.
^Gerla, G. (1995)."Pointless Geometries"(PDF). In Buekenhout, F.; Kantor, W. (eds.).Handbook of incidence geometry: buildings and foundations. North-Holland. pp. 1015–1031. Archived fromthe original(PDF) on 17 July 2011.
^Briggs, William L., and Lyle Cochran Calculus. "Early Transcendentals."ISBN978-0-321-57056-7.
^Yau, Shing-Tung; Nadis, Steve (2010).The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions. Basic Books.ISBN978-0-465-02023-2.
^Nihat Ay; Jürgen Jost; Hông Vân Lê; Lorenz Schwachhöfer (2017).Information Geometry. Springer. p. 185.ISBN978-3-319-56478-4.Archived from the original on 24 December 2019. Retrieved23 September 2019.
^Jon Rogawski; Colin Adams (2015).Calculus. W. H. Freeman.ISBN978-1-4641-7499-5.Archived from the original on 1 January 2020. Retrieved25 September 2019.
Cooke, Roger (2005).The History of Mathematics. New York: Wiley-Interscience.ISBN978-0-471-44459-6.
Hayashi, Takao (2003). "Indian Mathematics". In Grattan-Guinness, Ivor (ed.).Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences. Vol. 1. Baltimore, MD: TheJohns Hopkins University Press. pp. 118–130.ISBN978-0-8018-7396-6.
Hayashi, Takao (2005). "Indian Mathematics". In Flood, Gavin (ed.).The Blackwell Companion to Hinduism. Oxford:Basil Blackwell. pp. 360–375.ISBN978-1-4051-3251-0.
Nikolai I. Lobachevsky (2010).Pangeometry. Heritage of European Mathematics Series. Vol. 4. translator and editor: A. Papadopoulos. European Mathematical Society.
