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Geometry

January 13, 2023

For other uses, see Geometry (disambiguation).

Geometry (from Ancient Greek γεωμετρία (geōmetría) 'land measurement'; from γῆ (gê) 'earth, land', and μέτρον (métron) 'a measure')[citation needed] is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures.[1] A mathematician who works in the field of geometry is called a geometer.

Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry,[a] which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.[2]

During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' Theorema Egregium ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied intrinsically, that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry.

Later in the 19th century, it appeared that geometries without the parallel postulate (non-Euclidean geometries) can be developed without introducing any contradiction. The geometry that underlies general relativity is a famous application of non-Euclidean geometry.

Since then, 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 as combinatorial 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 omits continuity, and others.

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.[3] Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, a problem that was stated in terms of elementary arithmetic, and remained unsolved for several centuries.

History

Main article: History of geometry
 
A European and an Arab practicing geometry in the 15th century

The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC.[4][5] Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts. The earliest known texts on geometry are the Egyptian Rhind Papyrus (2000–1800 BC) and Moscow Papyrus (c. 1890 BC), and the Babylonian clay tablets, such as Plimpton 322 (1900 BC). For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum.[6] Later clay tablets (350–50 BC) demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space. These geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries.[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 Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales's theorem.[10] Pythagoras established the Pythagorean School, which is credited with the first proof of the Pythagorean theorem,[11] though the statement of the theorem has a long history.[12][13] Eudoxus (408–c. 355 BC) developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures,[14] as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements, widely considered the most successful and influential textbook of all time,[15] introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework.[16] The Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today.[17] Archimedes (c. 287–212 BC) of Syracuse, Italy used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave remarkably accurate approximations of pi.[18] He also studied the spiral bearing his name and obtained formulas for the volumes of surfaces of revolution.

 
Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c. 1310).

Indian mathematicians also made many important contributions in geometry. The Shatapatha Brahmana (3rd century BC) contains rules for ritual geometric constructions that are similar to the Sulba Sutras.[19] According to (Hayashi 2005, p. 363), the Śulba 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 of Pythagorean triples,[20] which are particular cases of Diophantine equations.[21] In the Bakhshali 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's Aryabhatiya (499) includes the computation of areas and volumes. Brahmagupta wrote his astronomical work Brāhmasphuṭasiddhānta in 628. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain).[23] In the latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral. Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of Heron's formula), as well as a complete description of rational triangles (i.e. triangles with rational sides and rational areas).[23]

In the Middle Ages, mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry.[24][25] Al-Mahani (b. 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra.[26] Thābit ibn Qurra (known as Thebit in Latin) (836–901) dealt with arithmetic operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry.[27] Omar Khayyam (1048–1131) found geometric solutions to cubic equations.[28] The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were early results in hyperbolic geometry, and along with their alternative postulates, such as Playfair's axiom, these works had a considerable influence on the development of non-Euclidean geometry among later European geometers, including Vitello (c. 1230 – c. 1314), Gersonides (1288–1344), Alfonso, John Wallis, and Giovanni Girolamo Saccheri.[dubious – discuss][29]

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.[34] These were the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation of symmetry as the central consideration in the Erlangen programme of Felix Klein (which generalized the Euclidean and non-Euclidean geometries). Two of the master geometers of the time were Bernhard Riemann (1826–1866), working primarily with tools from mathematical analysis, and introducing the Riemann surface, and Henri Poincaré, the founder of algebraic topology and the geometric theory of dynamical systems. As a consequence of these major changes in the conception of geometry, the concept of "space" became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics.[35]

Main concepts

The following are some of the most important concepts in geometry.[2][36][37]

Axioms

 
An illustration of Euclid's parallel postulate
See also: Euclidean geometry and Axiom

Euclid took an abstract approach to geometry in his Elements,[38] one of the most influential books ever written.[39] Euclid introduced certain axioms, or postulates, expressing primary or self-evident properties of points, lines, and planes.[40] He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as axiomatic or synthetic geometry.[41] At the start of the 19th century, the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky (1792–1856), János Bolyai (1802–1860), Carl Friedrich Gauss (1777–1855) and 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]

Objects

Points

Main article: Point (geometry)

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",[44] or in synthetic geometry. In modern mathematics, they are generally defined as elements of a set called space, which is itself 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 point-free geometry, formulated by Alfred North Whitehead in 1919–1920.

Lines

Main article: Line (geometry)

Euclid 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]

Planes

Main article: Plane (geometry)

In Euclidean geometry a plane is a flat, two-dimensional 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.

Angles

Main article: Angle

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.[44] 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.[53]

 
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.[44] The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry.[54]

In differential geometry and calculus, the angles between plane curves or space curves or surfaces can be calculated using the derivative.[55][56]

Curves

Main article: Curve (geometry)

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.[57]

In topology, a curve is defined by a function from an interval of the real numbers to another space.[50] In differential geometry, the same definition is used, but the defining function is required to be differentiable [58] Algebraic geometry studies algebraic curves, which are defined as algebraic varieties of dimension one.[59]

Surfaces

Main article: Surface (mathematics)
 
A sphere is a surface that can be defined parametrically (by x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ) or implicitly (by x2 + y2 + z2 − r2 = 0.)

A surface is a two-dimensional object, such as a sphere or paraboloid.[60] In differential geometry[58] and topology,[50] 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.[59]

Manifolds

Main article: Manifold

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.[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 volume

Main articles: Length, Area, and Volume
See also: Area § List of formulas, and Volume § Volume formulas

Length, 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 3-dimensional space.[62] 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 integral[64] or the Lebesgue integral.[65]

Metrics and measures

Main articles: Metric (mathematics) and Measure (mathematics)
 
Visual checking of the Pythagorean theorem for the (3, 4, 5) triangle as in the Zhoubi Suanjing 500–200 BC. The Pythagorean theorem is a consequence of the Euclidean metric.

The concept of length or distance can be generalized, leading to the idea of metrics.[66] 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.[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 similarity

Main articles: Congruence (geometry) and Similarity (geometry)

Congruence and similarity are concepts that describe when two shapes have similar characteristics.[69] 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.[70] 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.[71]

Compass and straightedge constructions

Main article: 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 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.

Dimension

Main article: Dimension

Where the traditional geometry allowed dimensions 1 (a line), 2 (a plane) and 3 (our ambient world conceived of as three-dimensional space), mathematicians and physicists have used higher dimensions for nearly two centuries.[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 general topology, the concept of dimension has been extended from natural numbers, to infinite dimension (Hilbert spaces, for example) and positive real numbers (in fractal geometry).[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]

Symmetry

Main article: Symmetry

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 meta-phenomenon can roughly be described as follows: in any theorem, exchange point with plane, join with meet, lies in with contains, and the result is an equally true theorem.[86] A similar and closely related form of duality exists between a vector space and its dual space.[87]

Contemporary geometry

Euclidean geometry

Main article: Euclidean geometry

Euclidean geometry is geometry in its classical sense.[88] As it models the space of the physical world, it is used in many scientific areas, such as mechanics, astronomy, crystallography,[89] and many technical fields, such as engineering,[90] architecture,[91] geodesy,[92] aerodynamics,[93] and navigation.[94] The mandatory educational curriculum of the majority of nations includes the study of Euclidean concepts such as points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, and analytic geometry.[36]

Differential geometry

 
Differential geometry uses tools from calculus to study problems involving curvature.
Main article: Differential geometry

Differential 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 mathematical physics due to Albert Einstein's general relativity postulation that the universe is curved.[99] 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).[100]

Non-Euclidean geometry

Main article: Non-Euclidean geometry

Euclidean geometry was not the only historical form of geometry studied. Spherical geometry has long been used by astronomers, astrologers, and navigators.[101]

Immanuel Kant argued that there is only one, absolute, geometry, which is known to be true a priori by an inner faculty of mind: Euclidean geometry was synthetic a priori.[102] This view was at first somewhat challenged by thinkers such as Saccheri, then finally overturned by the revolutionary discovery of non-Euclidean geometry in the works of Bolyai, Lobachevsky, and Gauss (who never published his theory).[103] They demonstrated that ordinary Euclidean space is only one possibility for development of geometry. A broad vision of the subject of geometry was then expressed by Riemann in his 1867 inauguration lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen (On the hypotheses on which geometry is based),[104] published only after his death. Riemann's new idea of space proved crucial in Albert Einstein's general relativity theory. Riemannian geometry, which considers very general spaces in which the notion of length is defined, is a mainstay of modern geometry.[81]

Topology

Main article: Topology
 
A thickening of the trefoil knot

Topology is the field concerned with the properties of continuous mappings,[105] and can be considered a generalization of Euclidean geometry.[106] In practice, topology often means dealing with large-scale properties of spaces, such as connectedness and compactness.[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 rubber-sheet geometry'. Subfields of topology include geometric topology, differential topology, algebraic topology and general topology.[108]

Algebraic geometry

Main article: Algebraic geometry

The field of algebraic geometry developed from the Cartesian geometry of co-ordinates.[109] 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.[110] From the late 1950s through the mid-1970s it had undergone major foundational development, largely due to work of Jean-Pierre Serre and Alexander Grothendieck.[110] 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.[111] 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.[112] It has applications in many areas, including cryptography[113] and string theory.[114]

Complex geometry

Main article: Complex geometry

Complex geometry studies the nature of geometric structures modelled on, or arising out of, the complex plane.[115][116][117] Complex geometry lies at the intersection of differential geometry, algebraic geometry, and analysis of several 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 Jean-Pierre 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 complex manifolds, complex algebraic varieties, and complex analytic varieties, and holomorphic vector bundles and 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 geometry

Main article: Discrete geometry
 
Discrete geometry includes the study of various sphere packings.

Discrete 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 Kneser-Poulsen conjecture, etc.[127][128] It shares many methods and principles with combinatorics.

Computational geometry

Main article: 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.[129]

Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc.[130]

Geometric group theory

Main article: Geometric group theory
 
The Cayley graph of the free group on two generators a and b

Geometric group theory uses large-scale geometric techniques to study finitely generated groups.[131] It is closely connected to low-dimensional topology, such as in Grigori Perelman's proof of the Geometrization conjecture, which included the proof of the Poincaré conjecture, a Millennium Prize Problem.[132]

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.[131][133]

Convex geometry

Main article: 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.[134] It has close connections to convex analysis, optimization and functional analysis and important applications in number theory.

Convex geometry dates back to antiquity.[134] 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

Main article: Mathematics and art
 
Bou Inania Madrasa, Fes, Morocco, zellige mosaic tiles forming elaborate geometric tessellations

Mathematics and art are related in a variety of ways. For instance, the theory of perspective showed that there is more to geometry than just the metric properties of figures: perspective is the origin of projective geometry.[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]

Architecture

Main articles: Mathematics and architecture and Architectural geometry

Geometry has many applications in architecture. In fact, it has been said that geometry lies at the core of architectural design.[142][143] Applications of geometry to architecture include the use of projective geometry to create forced perspective,[144] the use of conic sections in constructing domes and similar objects,[91] the use of tessellations,[91] and the use of symmetry.[91]

Physics

Main article: Mathematical 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.[145]

Riemannian geometry and pseudo-Riemannian geometry are used in general relativity.[146] String theory makes use of several variants of geometry,[147] as does quantum information theory.[148]

Other fields of mathematics

 
The Pythagoreans discovered that the sides of a triangle could have incommensurable lengths.

Calculus was strongly influenced by geometry.[30] For instance, the introduction of coordinates by René Descartes and the concurrent developments of algebra marked a new stage for geometry, since geometric figures such as plane curves could now be represented analytically in the form of functions and equations. This played a key role in the emergence of infinitesimal calculus in the 17th century. Analytic geometry continues to be a mainstay of pre-calculus and calculus curriculum.[149][150]

Another important area of application is number theory.[151] In ancient Greece the Pythagoreans considered the role of numbers in geometry. However, the discovery of incommensurable lengths contradicted their philosophical views.[152] 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.[153]

See also

Lists

Related topics

Other fields

Notes

  1. ^ 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 of hyperbolic geometry by Lobachevsky and other non-Euclidean geometries by Gauss and others. It was then realised that implicitly non-Euclidean geometry had appeared throughout history, including the work of Desargues in the 17th century, all the way back to the implicit use of spherical geometry to understand the Earth geodesy and to navigate the oceans since antiquity.
  2. ^ The ancient Greeks had some constructions using other instruments.
  1. ^ Vincenzo De Risi (2015). Mathematizing Space: The Objects of Geometry from Antiquity to the Early Modern Age. Birkhäuser. pp. 1–. ISBN 978-3-319-12102-4. from the original on 20 February 2021. Retrieved 14 September 2019.
  2. ^ a b Tabak, John (2014). Geometry: the language of space and form. Infobase Publishing. p. xiv. ISBN 978-0-8160-4953-0.
  3. ^ Walter A. Meyer (2006). Geometry and Its Applications. Elsevier. ISBN 978-0-08-047803-6. from the original on 1 September 2021. Retrieved 14 September 2019.
  4. ^ Friberg, Jöran (1981). "Methods and traditions of Babylonian mathematics". Historia Mathematica. 8 (3): 277–318. doi:10.1016/0315-0860(81)90069-0.
  5. ^ Neugebauer, Otto (1969) [1957]. "Chap. IV Egyptian Mathematics and Astronomy". The Exact Sciences in Antiquity (2 ed.). Dover Publications. pp. 71–96. ISBN 978-0-486-22332-2. from the original on 14 August 2020. Retrieved 27 February 2021..
  6. ^ (Boyer 1991, "Egypt" p. 19)
  7. ^ Ossendrijver, Mathieu (29 January 2016). "Ancient Babylonian astronomers calculated Jupiter's position from the area under a time-velocity graph". Science. 351 (6272): 482–484. Bibcode:2016Sci...351..482O. doi:10.1126/science.aad8085. PMID 26823423. S2CID 206644971.
  8. ^ Depuydt, Leo (1 January 1998). "Gnomons at Meroë and Early Trigonometry". The Journal of Egyptian Archaeology. 84: 171–180. doi:10.2307/3822211. JSTOR 3822211.
  9. ^ Slayman, Andrew (27 May 1998). "Neolithic Skywatchers". Archaeology Magazine Archive. from the original on 5 June 2011. Retrieved 17 April 2011.
  10. ^ (Boyer 1991, "Ionia and the Pythagoreans" p. 43)
  11. ^ Eves, Howard, An Introduction to the History of Mathematics, Saunders, 1990, ISBN 0-03-029558-0.
  12. ^ Kurt Von Fritz (1945). "The Discovery of Incommensurability by Hippasus of Metapontum". The Annals of Mathematics. Boston Studies in the Philosophy of Science. 240 (2): 211–231. doi:10.1007/978-1-4020-2640-9_11. ISBN 978-90-481-5850-8. JSTOR 1969021.
  13. ^ James R. Choike (1980). "The Pentagram and the Discovery of an Irrational Number". The Two-Year College Mathematics Journal. 11 (5): 312–316. doi:10.2307/3026893. JSTOR 3026893.
  14. ^ (Boyer 1991, "The Age of Plato and Aristotle" p. 92)
  15. ^ (Boyer 1991, "Euclid of Alexandria" p. 119)
  16. ^ (Boyer 1991, "Euclid of Alexandria" p. 104)
  17. ^ Howard Eves, An Introduction to the History of Mathematics, Saunders, 1990, ISBN 0-03-029558-0 p. 141: "No work, except The Bible, has been more widely used...."
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  147. ^ Shing-Tung Yau; Steve Nadis (2010). The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions. Basic Books. ISBN 978-0-465-02266-3. from the original on 24 December 2019. Retrieved 25 September 2019.
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Sources

  • Boyer, C.B. (1991) [1989]. A History of Mathematics (Second edition, revised by Uta C. Merzbach ed.). New York: Wiley. ISBN 978-0-471-54397-8.
  • Cooke, Roger (2005). The History of Mathematics. New York: Wiley-Interscience. ISBN 978-0-471-44459-6.
  • 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: The Johns Hopkins University Press. pp. 118–130. ISBN 978-0-8018-7396-6.
  • Hayashi, Takao (2005). "Indian Mathematics". In Flood, Gavin (ed.). The Blackwell Companion to Hinduism. Oxford: Basil Blackwell. pp. 360–375. ISBN 978-1-4051-3251-0.
  • Nikolai I. Lobachevsky (2010). Pangeometry. Heritage of European Mathematics Series. Vol. 4. translator and editor: A. Papadopoulos. European Mathematical Society.

Further reading

External links

 
Wikibooks has more on the topic of: Geometry
  • "Geometry" . Encyclopædia Britannica. Vol. 11 (11th ed.). 1911. pp. 675–736.
  • A geometry course from Wikiversity
  • Unusual Geometry Problems
  • The Math Forum – Geometry
    • The Math Forum – K–12 Geometry
    • The Math Forum – College Geometry
    • The Math Forum – Advanced Geometry
  • Nature Precedings – Pegs and Ropes Geometry at Stonehenge
  • , lecture by Robin Wilson given at Gresham College, 3 October 2007 (available for MP3 and MP4 download as well as a text file)
    • Finitism in Geometry at the Stanford Encyclopedia of Philosophy
  • The Geometry Junkyard
  • Interactive geometry reference with hundreds of applets
  • Geometry classes at Khan Academy

January 13, 2023
geometry, other, uses, disambiguation, from, ancient, greek, γεωμετρία, geōmetría, land, measurement, from, γῆ, earth, land, μέτρον, métron, measure, citation, needed, with, arithmetic, oldest, branches, mathematics, concerned, with, properties, space, such, d. For other uses see Geometry disambiguation Geometry from Ancient Greek gewmetria geōmetria land measurement from gῆ ge earth land and metron metron a measure citation needed is with arithmetic one of the oldest branches of mathematics It is concerned with properties of space such as the distance shape size and relative position of figures 1 A mathematician who works in the field of geometry is called a geometer Until the 19th century geometry was almost exclusively devoted to Euclidean geometry a which includes the notions of point line plane distance angle surface and curve as fundamental concepts 2 During the 19th century several discoveries enlarged dramatically the scope of geometry One of the oldest such discoveries is Carl Friedrich Gauss Theorema Egregium code lat promoted to code la remarkable theorem that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space This implies that surfaces can be studied intrinsically that is as stand alone spaces and has been expanded into the theory of manifolds and Riemannian geometry Later in the 19th century it appeared that geometries without the parallel postulate non Euclidean geometries can be developed without introducing any contradiction The geometry that underlies general relativity is a famous application of non Euclidean geometry Since then 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 as combinatorial 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 omits continuity and others 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 3 Geometry also has applications in areas of mathematics that are apparently unrelated For example methods of algebraic geometry are fundamental in Wiles s proof of Fermat s Last Theorem a problem that was stated in terms of elementary arithmetic and remained unsolved for several centuries Contents 1 History 2 Main concepts 2 1 Axioms 2 2 Objects 2 2 1 Points 2 2 2 Lines 2 2 3 Planes 2 2 4 Angles 2 2 5 Curves 2 2 6 Surfaces 2 2 7 Manifolds 2 3 Length area and volume 2 3 1 Metrics and measures 2 4 Congruence and similarity 2 5 Compass and straightedge constructions 2 6 Dimension 2 7 Symmetry 3 Contemporary geometry 3 1 Euclidean geometry 3 2 Differential geometry 3 2 1 Non Euclidean geometry 3 3 Topology 3 4 Algebraic geometry 3 5 Complex geometry 3 6 Discrete geometry 3 7 Computational geometry 3 8 Geometric group theory 3 9 Convex geometry 4 Applications 4 1 Art 4 2 Architecture 4 3 Physics 4 4 Other fields of mathematics 5 See also 5 1 Lists 5 2 Related topics 5 3 Other fields 6 Notes 7 Sources 8 Further reading 9 External linksHistoryMain article History of geometry A European and an Arab practicing geometry in the 15th century The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC 4 5 Early geometry was a collection of empirically discovered principles concerning lengths angles areas and volumes which were developed to meet some practical need in surveying construction astronomy and various crafts The earliest known texts on geometry are the Egyptian Rhind Papyrus 2000 1800 BC and Moscow Papyrus c 1890 BC and the Babylonian clay tablets such as Plimpton 322 1900 BC For example the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid or frustum 6 Later clay tablets 350 50 BC demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter s position and motion within time velocity space These geometric procedures anticipated the Oxford Calculators including the mean speed theorem by 14 centuries 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 Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore He is credited with the first use of deductive reasoning applied to geometry by deriving four corollaries to Thales s theorem 10 Pythagoras established the Pythagorean School which is credited with the first proof of the Pythagorean theorem 11 though the statement of the theorem has a long history 12 13 Eudoxus 408 c 355 BC developed the method of exhaustion which allowed the calculation of areas and volumes of curvilinear figures 14 as well as a theory of ratios that avoided the problem of incommensurable magnitudes which enabled subsequent geometers to make significant advances Around 300 BC geometry was revolutionized by Euclid whose Elements widely considered the most successful and influential textbook of all time 15 introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today that of definition axiom theorem and proof Although most of the contents of the Elements were already known Euclid arranged them into a single coherent logical framework 16 The Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today 17 Archimedes c 287 212 BC of Syracuse Italy used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series and gave remarkably accurate approximations of pi 18 He also studied the spiral bearing his name and obtained formulas for the volumes of surfaces of revolution Woman teaching geometry Illustration at the beginning of a medieval translation of Euclid s Elements c 1310 Indian mathematicians also made many important contributions in geometry The Shatapatha Brahmana 3rd century BC contains rules for ritual geometric constructions that are similar to the Sulba Sutras 19 According to Hayashi 2005 p 363 the Sulba Sutras 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 20 which are particular cases of Diophantine equations 21 In the Bakhshali 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 s Aryabhatiya 499 includes the computation of areas and volumes Brahmagupta wrote his astronomical work Brahmasphuṭasiddhanta in 628 Chapter 12 containing 66 Sanskrit verses was divided into two sections basic operations including cube roots fractions ratio and proportion and barter and practical mathematics including mixture mathematical series plane figures stacking bricks sawing of timber and piling of grain 23 In the latter section he stated his famous theorem on the diagonals of a cyclic quadrilateral Chapter 12 also included a formula for the area of a cyclic quadrilateral a generalization of Heron s formula as well as a complete description of rational triangles i e triangles with rational sides and rational areas 23 In the Middle Ages mathematics in medieval Islam contributed to the development of geometry especially algebraic geometry 24 25 Al Mahani b 853 conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra 26 Thabit ibn Qurra known as Thebit in Latin 836 901 dealt with arithmetic operations applied to ratios of geometrical quantities and contributed to the development of analytic geometry 27 Omar Khayyam 1048 1131 found geometric solutions to cubic equations 28 The theorems of Ibn al Haytham Alhazen Omar Khayyam and Nasir al Din al Tusi on quadrilaterals including the Lambert quadrilateral and Saccheri quadrilateral were early results in hyperbolic geometry and along with their alternative postulates such as Playfair s axiom these works had a considerable influence on the development of non Euclidean geometry among later European geometers including Vitello c 1230 c 1314 Gersonides 1288 1344 Alfonso John Wallis and Giovanni Girolamo Saccheri dubious discuss 29 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 Rene 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 34 These were the discovery of non Euclidean geometries by Nikolai Ivanovich Lobachevsky Janos Bolyai and Carl Friedrich Gauss and of the formulation of symmetry as the central consideration in the Erlangen programme of Felix Klein which generalized the Euclidean and non Euclidean geometries Two of the master geometers of the time were Bernhard Riemann 1826 1866 working primarily with tools from mathematical analysis and introducing the Riemann surface and Henri Poincare 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 Axioms An illustration of Euclid s parallel postulate See also Euclidean geometry and Axiom Euclid took an abstract approach to geometry in his Elements 38 one of the most influential books ever written 39 Euclid introduced certain axioms or postulates expressing primary or self evident properties of points lines and planes 40 He proceeded to rigorously deduce other properties by mathematical reasoning The characteristic feature of Euclid s approach to geometry was its rigor and it has come to be known as axiomatic or synthetic geometry 41 At the start of the 19th century the discovery of non Euclidean geometries by Nikolai Ivanovich Lobachevsky 1792 1856 Janos 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 Objects Points Main article Point geometry 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 44 or in synthetic geometry In modern mathematics they are generally defined as elements of a set called space which is itself 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 point free geometry formulated by Alfred North Whitehead in 1919 1920 Lines Main article Line geometry Euclid 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 Planes Main article Plane geometry In Euclidean geometry a plane is a flat two dimensional 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 Angles Main article Angle 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 44 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 53 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 44 The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry 54 In differential geometry and calculus the angles between plane curves or space curves or surfaces can be calculated using the derivative 55 56 Curves Main article Curve geometry 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 57 In topology a curve is defined by a function from an interval of the real numbers to another space 50 In differential geometry the same definition is used but the defining function is required to be differentiable 58 Algebraic geometry studies algebraic curves which are defined as algebraic varieties of dimension one 59 Surfaces Main article Surface mathematics A sphere is a surface that can be defined parametrically by x r sin 8 cos f y r sin 8 sin f z r cos 8 or implicitly by x2 y2 z2 r2 0 A surface is a two dimensional object such as a sphere or paraboloid 60 In differential geometry 58 and topology 50 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 59 Manifolds Main article Manifold 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 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 volume Main articles Length Area and Volume See also Area List of formulas and Volume Volume formulas Length 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 3 dimensional space 62 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 integral 64 or the Lebesgue integral 65 Metrics and measures Main articles Metric mathematics and Measure mathematics Visual checking of the Pythagorean theorem for the 3 4 5 triangle as in the Zhoubi Suanjing 500 200 BC The Pythagorean theorem is a consequence of the Euclidean metric The concept of length or distance can be generalized leading to the idea of metrics 66 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 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 similarity Main articles Congruence geometry and Similarity geometry Congruence and similarity are concepts that describe when two shapes have similar characteristics 69 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 70 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 71 Compass and straightedge constructions Main article 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 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 Dimension Main article Dimension The Koch snowflake with fractal dimension log4 log3 and topological dimension 1 Where the traditional geometry allowed dimensions 1 a line 2 a plane and 3 our ambient world conceived of as three dimensional space mathematicians and physicists have used higher dimensions for nearly two centuries 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 general topology the concept of dimension has been extended from natural numbers to infinite dimension Hilbert spaces for example and positive real numbers in fractal geometry 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 Symmetry Main article Symmetry A tiling 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 meta phenomenon can roughly be described as follows in any theorem exchange point with plane join with meet lies in with contains and the result is an equally true theorem 86 A similar and closely related form of duality exists between a vector space and its dual space 87 Contemporary geometryEuclidean geometry Main article Euclidean geometry Euclidean geometry is geometry in its classical sense 88 As it models the space of the physical world it is used in many scientific areas such as mechanics astronomy crystallography 89 and many technical fields such as engineering 90 architecture 91 geodesy 92 aerodynamics 93 and navigation 94 The mandatory educational curriculum of the majority of nations includes the study of Euclidean concepts such as points lines planes angles triangles congruence similarity solid figures circles and analytic geometry 36 Differential geometry Differential geometry uses tools from calculus to study problems involving curvature Main article Differential geometry Differential 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 mathematical physics due to Albert Einstein s general relativity postulation that the universe is curved 99 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 100 Non Euclidean geometry Main article Non Euclidean geometry Euclidean geometry was not the only historical form of geometry studied Spherical geometry has long been used by astronomers astrologers and navigators 101 Immanuel Kant argued that there is only one absolute geometry which is known to be true a priori by an inner faculty of mind Euclidean geometry was synthetic a priori 102 This view was at first somewhat challenged by thinkers such as Saccheri then finally overturned by the revolutionary discovery of non Euclidean geometry in the works of Bolyai Lobachevsky and Gauss who never published his theory 103 They demonstrated that ordinary Euclidean space is only one possibility for development of geometry A broad vision of the subject of geometry was then expressed by Riemann in his 1867 inauguration lecture Uber die Hypothesen welche der Geometrie zu Grunde liegen On the hypotheses on which geometry is based 104 published only after his death Riemann s new idea of space proved crucial in Albert Einstein s general relativity theory Riemannian geometry which considers very general spaces in which the notion of length is defined is a mainstay of modern geometry 81 Topology Main article Topology A thickening of the trefoil knot Topology is the field concerned with the properties of continuous mappings 105 and can be considered a generalization of Euclidean geometry 106 In practice topology often means dealing with large scale properties of spaces such as connectedness and compactness 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 rubber sheet geometry Subfields of topology include geometric topology differential topology algebraic topology and general topology 108 Algebraic geometry Main article Algebraic geometry Quintic Calabi Yau threefold The field of algebraic geometry developed from the Cartesian geometry of co ordinates 109 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 110 From the late 1950s through the mid 1970s it had undergone major foundational development largely due to work of Jean Pierre Serre and Alexander Grothendieck 110 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 111 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 112 It has applications in many areas including cryptography 113 and string theory 114 Complex geometry Main article Complex geometry Complex geometry studies the nature of geometric structures modelled on or arising out of the complex plane 115 116 117 Complex geometry lies at the intersection of differential geometry algebraic geometry and analysis of several 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 Jean Pierre 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 complex manifolds complex algebraic varieties and complex analytic varieties and holomorphic vector bundles and 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 geometry Main article Discrete geometry Discrete geometry includes the study of various sphere packings Discrete 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 Kneser Poulsen conjecture etc 127 128 It shares many methods and principles with combinatorics Computational geometry Main article 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 129 Although being a young area of geometry it has many applications in computer vision image processing computer aided design medical imaging etc 130 Geometric group theory Main article Geometric group theory The Cayley graph of the free group on two generators a and b Geometric group theory uses large scale geometric techniques to study finitely generated groups 131 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 Poincare conjecture a Millennium Prize Problem 132 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 131 133 Convex geometry Main article 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 134 It has close connections to convex analysis optimization and functional analysis and important applications in number theory Convex geometry dates back to antiquity 134 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 ApplicationsGeometry has found applications in many fields some of which are described below Art Main article Mathematics and art Bou Inania Madrasa Fes Morocco zellige mosaic tiles forming elaborate geometric tessellations Mathematics and art are related in a variety of ways For instance the theory of perspective showed that there is more to geometry than just the metric properties of figures perspective is the origin of projective geometry 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 Cezanne 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 Architecture Main articles Mathematics and architecture and Architectural geometry Geometry has many applications in architecture In fact it has been said that geometry lies at the core of architectural design 142 143 Applications of geometry to architecture include the use of projective geometry to create forced perspective 144 the use of conic sections in constructing domes and similar objects 91 the use of tessellations 91 and the use of symmetry 91 Physics Main article Mathematical 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 145 Riemannian geometry and pseudo Riemannian geometry are used in general relativity 146 String theory makes use of several variants of geometry 147 as does quantum information theory 148 Other fields of mathematics The Pythagoreans discovered that the sides of a triangle could have incommensurable lengths Calculus was strongly influenced by geometry 30 For instance the introduction of coordinates by Rene 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 149 150 Another important area of application is number theory 151 In ancient Greece the Pythagoreans considered the role of numbers in geometry However the discovery of incommensurable lengths contradicted their philosophical views 152 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 153 See also Mathematics portalLists List of geometers Category Algebraic geometers Category Differential geometers Category Geometers Category Topologists List of formulas in elementary geometry List of geometry topics List of important publications in geometry Lists of mathematics topicsRelated topics Descriptive geometry Finite geometry Flatland a book written by Edwin Abbott Abbott about two and three dimensional space to understand the concept of four dimensions List of interactive geometry softwareOther fields Molecular geometryNotes 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 of hyperbolic geometry by Lobachevsky and other non Euclidean geometries by Gauss and others It was then realised that implicitly non Euclidean geometry had appeared throughout history including the work of Desargues in the 17th century all the way back to the implicit use of spherical geometry to understand the Earth geodesy and to navigate the oceans since antiquity The ancient Greeks had some constructions using other instruments Vincenzo De Risi 2015 Mathematizing Space The Objects of Geometry from Antiquity to the Early Modern Age Birkhauser pp 1 ISBN 978 3 319 12102 4 Archived from the original on 20 February 2021 Retrieved 14 September 2019 a b Tabak John 2014 Geometry the language of space and form Infobase Publishing p xiv ISBN 978 0 8160 4953 0 Walter A Meyer 2006 Geometry and Its Applications Elsevier ISBN 978 0 08 047803 6 Archived from the original on 1 September 2021 Retrieved 14 September 2019 Friberg Joran 1981 Methods and traditions of Babylonian mathematics Historia Mathematica 8 3 277 318 doi 10 1016 0315 0860 81 90069 0 Neugebauer Otto 1969 1957 Chap IV Egyptian Mathematics and Astronomy The Exact Sciences in Antiquity 2 ed Dover Publications pp 71 96 ISBN 978 0 486 22332 2 Archived from the original on 14 August 2020 Retrieved 27 February 2021 Boyer 1991 Egypt p 19 Ossendrijver Mathieu 29 January 2016 Ancient Babylonian astronomers calculated Jupiter s position from the area under a time velocity graph Science 351 6272 482 484 Bibcode 2016Sci 351 482O doi 10 1126 science aad8085 PMID 26823423 S2CID 206644971 Depuydt Leo 1 January 1998 Gnomons at Meroe and Early Trigonometry The Journal of Egyptian Archaeology 84 171 180 doi 10 2307 3822211 JSTOR 3822211 Slayman Andrew 27 May 1998 Neolithic Skywatchers Archaeology Magazine Archive Archived from the original on 5 June 2011 Retrieved 17 April 2011 Boyer 1991 Ionia and the Pythagoreans p 43 Eves Howard An Introduction to the History of Mathematics Saunders 1990 ISBN 0 03 029558 0 Kurt Von Fritz 1945 The Discovery of Incommensurability by Hippasus of Metapontum The Annals of Mathematics Boston Studies in the Philosophy of Science 240 2 211 231 doi 10 1007 978 1 4020 2640 9 11 ISBN 978 90 481 5850 8 JSTOR 1969021 James R Choike 1980 The Pentagram and the Discovery of an Irrational Number The Two Year College Mathematics Journal 11 5 312 316 doi 10 2307 3026893 JSTOR 3026893 Boyer 1991 The Age of Plato and Aristotle p 92 Boyer 1991 Euclid of Alexandria p 119 Boyer 1991 Euclid of Alexandria p 104 Howard Eves An Introduction to the History of Mathematics Saunders 1990 ISBN 0 03 029558 0 p 141 No work except The Bible has been more widely used O Connor J J Robertson E F February 1996 A history of calculus University of St Andrews Archived from the original on 15 July 2007 Retrieved 7 August 2007 Staal Frits 1999 Greek and Vedic Geometry Journal of Indian Philosophy 27 1 2 105 127 doi 10 1023 A 1004364417713 S2CID 170894641 Pythagorean triples are triples of integers a b c displaystyle a b c with the property a 2 b 2 c 2 displaystyle a 2 b 2 c 2 Thus 3 2 4 2 5 2 displaystyle 3 2 4 2 5 2 8 2 15 2 17 2 displaystyle 8 2 15 2 17 2 12 2 35 2 37 2 displaystyle 12 2 35 2 37 2 etc Cooke 2005 p 198 The arithmetic content of the Sulva Sutras 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 Hayashi 2005 p 371 a b Hayashi 2003 pp 121 122 Rashid Rushdi 1994 The development of Arabic mathematics between arithmetic and algebra Boston Studies in the Philosophy of Science Vol 156 p 35 doi 10 1007 978 94 017 3274 1 ISBN 978 0 7923 2565 9 OCLC 29181926 Boyer 1991 The Arabic Hegemony pp 241 242 Omar Khayyam c 1050 1123 the tent maker wrote an Algebra that went beyond that of al Khwarizmi to include equations of third degree Like his Arab predecessors Omar Khayyam provided for quadratic equations both arithmetic and geometric solutions for general cubic equations he believed mistakenly as the 16th century later showed arithmetic solutions were impossible hence he gave only geometric solutions The scheme of using intersecting conics to solve cubics had been used earlier by Menaechmus Archimedes and Alhazan but Omar Khayyam took the praiseworthy step of generalizing the method to cover all third degree equations having positive roots For equations of higher degree than three Omar Khayyam evidently did not envision similar geometric methods for space does not contain more than three dimensions One of the most fruitful contributions of Arabic eclecticism was the tendency to close the gap between numerical and geometric algebra The decisive step in this direction came much later with Descartes but Omar Khayyam was moving in this direction when he wrote Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain No attention should be paid to the fact that algebra and geometry are different in appearance Algebras are geometric facts which are proved O Connor John J Robertson Edmund F Al Mahani MacTutor History of Mathematics archive University of St Andrews O Connor John J Robertson Edmund F Al Sabi Thabit ibn Qurra al Harrani MacTutor History of Mathematics archive University of St Andrews O Connor John J Robertson Edmund F Omar Khayyam MacTutor History of Mathematics archive University of St Andrews Boris A Rosenfeld and Adolf P Youschkevitch 1996 Geometry in Roshdi Rashed ed Encyclopedia of the History of Arabic Science Vol 2 pp 447 494 470 Routledge London and New York 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 s Book 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 that Pseudo Tusi s Exposition of Euclid had stimulated both J Wallis s and G Saccheri s studies of the theory of parallel lines a b Carl B Boyer 2012 History of Analytic Geometry Courier 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Archived from the original on 21 December 2019 Retrieved 25 September 2019 Dmitriĭ Vladimirovich Alekseevskiĭ 2008 Recent Developments in Pseudo Riemannian Geometry European Mathematical Society ISBN 978 3 03719 051 7 Archived from the original on 28 December 2019 Retrieved 25 September 2019 Shing Tung Yau Steve Nadis 2010 The Shape of Inner Space String Theory and the Geometry of the Universe s Hidden Dimensions Basic Books ISBN 978 0 465 02266 3 Archived from the original on 24 December 2019 Retrieved 25 September 2019 Bengtsson Ingemar Zyczkowski Karol 2017 Geometry of Quantum States An Introduction to Quantum Entanglement 2nd ed Cambridge University Press ISBN 978 1 107 02625 4 OCLC 1004572791 Harley Flanders Justin J Price 2014 Calculus with Analytic Geometry Elsevier Science ISBN 978 1 4832 6240 6 Archived from the original on 24 December 2019 Retrieved 25 September 2019 Jon Rogawski Colin Adams 2015 Calculus W H Freeman ISBN 978 1 4641 7499 5 Archived from the original on 1 January 2020 Retrieved 25 September 2019 Alvaro Lozano Robledo 2019 Number Theory and Geometry An Introduction to Arithmetic Geometry American Mathematical Soc ISBN 978 1 4704 5016 8 Archived from the original on 27 December 2019 Retrieved 25 September 2019 Arturo Sangalli 2009 Pythagoras Revenge A Mathematical Mystery Princeton University Press p 57 ISBN 978 0 691 04955 7 Gary Cornell Joseph H Silverman Glenn Stevens 2013 Modular Forms and Fermat s Last Theorem Springer Science amp Business Media ISBN 978 1 4612 1974 3 Archived from the original on 30 December 2019 Retrieved 25 September 2019 SourcesBoyer C B 1991 1989 A History of Mathematics Second edition revised by Uta C Merzbach ed New York Wiley ISBN 978 0 471 54397 8 Cooke Roger 2005 The History of Mathematics New York Wiley Interscience ISBN 978 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 The Johns Hopkins University Press pp 118 130 ISBN 978 0 8018 7396 6 Hayashi Takao 2005 Indian Mathematics In Flood Gavin ed The Blackwell Companion to Hinduism Oxford Basil Blackwell pp 360 375 ISBN 978 1 4051 3251 0 Nikolai I Lobachevsky 2010 Pangeometry Heritage of European Mathematics Series Vol 4 translator and editor A Papadopoulos European Mathematical Society Further readingJay Kappraff 2014 A Participatory Approach to Modern Geometry World Scientific Publishing doi 10 1142 8952 ISBN 978 981 4556 70 5 Zbl 1364 00004 Leonard Mlodinow 2002 Euclid s Window The Story of Geometry from Parallel Lines to Hyperspace UK ed Allen Lane ISBN 978 0 7139 9634 0 External linksGeometry at Wikipedia s sister projects Definitions from Wiktionary Media from Commons News from Wikinews Quotations from Wikiquote Texts from Wikisource Textbooks from Wikibooks Resources from Wikiversity Wikibooks has more on the topic of Geometry Geometry Encyclopaedia Britannica Vol 11 11th ed 1911 pp 675 736 A geometry course from Wikiversity Unusual Geometry Problems The Math Forum Geometry The Math Forum K 12 Geometry The Math Forum College Geometry The Math Forum Advanced Geometry Nature Precedings Pegs and Ropes Geometry at Stonehenge The Mathematical Atlas Geometric Areas of Mathematics 4000 Years of Geometry lecture by Robin Wilson given at Gresham College 3 October 2007 available for MP3 and MP4 download as well as a text file Finitism in Geometry at the Stanford Encyclopedia of Philosophy The Geometry Junkyard Interactive geometry reference with hundreds of applets Dynamic Geometry Sketches with some Student Explorations Geometry classes at Khan Academy Retrieved from https en wikipedia org w index php title Geometry amp oldid 1132231432, wikipedia, wiki, book, books, library,

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