Geometry (; ) is, with

differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra ...

, a differentiable manifold is a space where each neighborhood is diffeomorphic to Euclidean space.
Manifolds are used extensively in physics, including in

Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...

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. 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

Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...

is geometry in its classical sense. As it models the space of the physical world, it is used in many scientific areas, such as analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineerin ...

.

calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...

and general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current descr ...

postulation that the

algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. ...

developed from the

Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...

, 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,

general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current descr ...

. String theory makes use of several variants of geometry, as does quantum information theory.

''Unusual Geometry Problems''

''The Math Forum'' – Geometry

*

*

*

Nature Precedings – ''Pegs and Ropes Geometry at Stonehenge''

* ttps://web.archive.org/web/20071004174210/http://www.gresham.ac.uk/event.asp?PageId=45&EventId=618 "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

Interactive geometry reference with hundreds of applets

Geometry classes

at Khan Academy {{Authority control

arithmetic
Arithmetic () is an elementary part of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their chang ...

, one of the oldest branches of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. 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
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...

, which includes the notions of point, line
Line most often refers to:
* Line (geometry)
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional objects, though they may exist in Two-dimensional Euclide ...

, plane, distance, angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two ...

, surface, and curve, as fundamental concepts.
During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

. 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
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:
' ...

(non-Euclidean geometries
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...

) can be developed without introducing any contradiction. The geometry that underlies general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current descr ...

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
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra ...

, algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. ...

, computational geometry
Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems ar ...

, algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

, discrete geometry
Discrete geometry and combinatorial geometry are branches of geometry that study Combinatorics, combinatorial properties and constructive methods of discrete mathematics, discrete geometric objects. Most questions in discrete geometry involve fi ...

(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
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation) Finite number may refer to:
* A countable number less than infinity, being the cardinality of a finite set – i.e., some natural number
In mathematics, th ...

that omits continuity, and others.
Originally developed to model the physical world, geometry has applications in almost all science
Science is a systematic endeavor that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earli ...

s, and also in art
Art is a diverse range of human activity, and resulting product, that involves creative or imaginative talent expressive of technical proficiency, beauty, emotional power, or conceptual ideas.
There is no generally agreed definition of wha ...

, architecture
Architecture is the art and technique of designing and building, as distinguished from the skills associated with construction. It is both the process and the product of sketching, conceiving, planning, designing, and construction, constructin ...

, and other activities that are related to graphics
Graphics () are visual perception, visual images or designs on some surface, such as a wall, canvas, screen, paper, or stone, to inform, illustration, illustrate, or entertain. In contemporary usage, it includes a pictorial representation of dat ...

. 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

The earliest recorded beginnings of geometry can be traced to ancientMesopotamia
Mesopotamia ''Mesopotamíā''; ar, بِلَاد ٱلرَّافِدَيْن or ; syc, ܐܪܡ ܢܗܪ̈ܝܢ, or , ) is a historical region of Western Asia situated within the Tigris–Euphrates river system, in the northern part of the F ...

and Egypt
Egypt ( ar, مصر , ), officially the Arab Republic of Egypt, is a List of transcontinental countries, transcontinental country spanning the North Africa, northeast corner of Africa and Western Asia, southwest corner of Asia via a land bridg ...

in the 2nd millennium BC. 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
Surveying or land surveying is the technique, profession, art, and science of determining the land, terrestrial Two-dimensional space#In geometry, two-dimensional or Three-dimensional space#In Euclidean geometry, three-dimensional positions of ...

, construction
Construction is a general term meaning the art and science to form Physical object, objects, systems, or organizations,"Construction" def. 1.a. 1.b. and 1.c. ''Oxford English Dictionary'' Second Edition on CD-ROM (v. 4.0) Oxford University Pr ...

, astronomy
Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...

, and various crafts. The earliest known texts on geometry are the Egyptian Rhind Papyrus (2000–1800 BC) and Moscow Papyrus (), 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. 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. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks.
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. Pythagoras
Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samos, Samian, or simply ; in Ionian Greek; ) was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymou ...

established the Pythagorean School, which is credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history. Eudoxus (408–) developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, 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, 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. 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. Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, ...

() of Syracuse, Italy used the method of exhaustion to calculate the area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an o ...

under the arc of a parabola with the summation of an infinite series, and gave remarkably accurate approximations of pi. He also studied the spiral bearing his name and obtained formulas for the volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...

s of surfaces of revolution.
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''.
According to , 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, which are particular cases of Diophantine equations
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknown (mathematics), unknowns with integer coefficients, such that the only equation solving, solutions of interest are the integer ones. A l ...

.: "The arithmetic content of the ''Śulva 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."
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." Aryabhata
Aryabhata (ISO 15919, ISO: ) or Aryabhata I (476–550 Common Era, CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta era, Gupta Era and produced works su ...

's '' Aryabhatiya'' (499) includes the computation of areas and volumes.
Brahmagupta wrote his astronomical work '' '' in 628. Chapter 12, containing 66 Sanskrit
Sanskrit (; attributively , ; nominalization, nominally , , ) is a classical language belonging to the Indo-Aryan languages, Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor languages had Trans-cul ...

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). 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).
In the Middle Ages
In the history of Europe, the Middle Ages or medieval period lasted approximately from the late 5th to the late 15th centuries, similar to the Post-classical, post-classical period of World history (field), global history. It began with t ...

, mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. ...

. Al-Mahani (b. 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Thābit ibn Qurra (known as Thebit in Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...

) (836–901) dealt with arithmetic
Arithmetic () is an elementary part of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their chang ...

operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineerin ...

. Omar Khayyam (1048–1131) found geometric solutions to cubic equations. 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
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For an ...

, 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 (), Gersonides (1288–1344), Alfonso, John Wallis, and Giovanni Girolamo Saccheri.
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 equation
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

s, by René Descartes
René Descartes ( or ; ; Latinisation of names, Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French people, French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of m ...

(1596–1650) and Pierre de Fermat (1601–1665). This was a necessary precursor to the development of calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...

and a precise quantitative science of physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...

. The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry studies properties of shapes which are unchanged under projections and sections, especially as they relate to artistic perspective.
Two developments in geometry in the 19th century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...

by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation of symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

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
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to mathematical analysis, analysis, number theory, and differential geometry. In the field of real analysis, he is mostly ...

(1826–1866), working primarily with tools from mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...

, and introducing the Riemann surface, and Henri Poincaré, the founder of algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

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
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

and classical mechanics
Classical mechanics is a Theoretical physics, physical theory describing the motion of macroscopic objects, from projectiles to parts of Machine (mechanical), machinery, and astronomical objects, such as spacecraft, planets, stars, and galax ...

.
Main concepts

The following are some of the most important concepts in geometry.Axioms

Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...

took an abstract approach to geometry in his Elements, one of the most influential books ever written. Euclid introduced certain axioms, or postulates, expressing primary or self-evident properties of points, lines, and planes. He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as ''axiomatic'' or '' synthetic'' geometry. At the start of the 19th century, the discovery of non-Euclidean geometries
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...

by Nikolai Ivanovich Lobachevsky (1792–1856), János Bolyai (1802–1860), Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

(1777–1855) and others 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.
Objects

Points

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",''Euclid's Elements – All thirteen books in one volume'', Based on Heath's translation, Green Lion Press . or in synthetic geometry. In modern mathematics, they are generally defined as elements of a set calledspace
Space is the boundless Three-dimensional space, three-dimensional extent in which Physical body, objects and events have relative position (geometry), position and direction (geometry), direction. In classical physics, physical space is often ...

, 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. One of the oldest such geometries is Whitehead's point-free geometry, formulated by Alfred North Whitehead in 1919–1920.
Lines

Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...

described a line as "breadthless length" which "lies equally with respect to the points on itself". In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineerin ...

, a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation, 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. In differential geometry, a geodesic is a generalization of the notion of a line to curved spaces.
Planes

In Euclidean geometry a plane is a flat, two-dimensional surface that extends infinitely; 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; it can be studied as an affine space, where collinearity and ratios can be studied but not distances; it can be studied as thecomplex plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, ...

using techniques of complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

; and so on.
Angles

Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...

defines a plane angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two ...

as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. In modern terms, an angle is the figure formed by two rays, called the ''sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle.
In Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...

, angles are used to study polygons and triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...

s, as well as forming an object of study in their own right. The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry.
In differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra ...

and calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...

, the angles between plane curves or space curves or surfaces can be calculated using the derivative. Stewart, James (2012). ''Calculus: Early Transcendentals'', 7th ed., Brooks Cole Cengage Learning.
Curves

A curve is a 1-dimensional object that may be straight (like a line) or not; curves in 2-dimensional space are called plane curves and those in 3-dimensional space are called space curves. In topology, a curve is defined by a function from an interval of the real numbers to another space. In differential geometry, the same definition is used, but the defining function is required to be differentiable Algebraic geometry studies algebraic curves, which are defined as algebraic varieties of dimension one.Surfaces

A surface is a two-dimensional object, such as a sphere or paraboloid. Indifferential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra ...

and topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

, 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 equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For many authors, the term '' ...

s.
Manifolds

A manifold is a generalization of the concepts of curve and surface. Intopology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

, a manifold is a topological space where every point has a neighborhood
A neighbourhood (British English
British English (BrE, en-GB, or BE) is, according to Oxford Dictionaries, " English as used in Great Britain, as distinct from that used elsewhere". More narrowly, it can refer specifically to the En ...

that is homeomorphic to Euclidean space. In general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current descr ...

and string theory.
Length, area, and volume

Length,area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an o ...

, and volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...

describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively.
In Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...

and analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineerin ...

, the length of a line segment can often be calculated by the Pythagorean theorem.
Area and volume can be defined as fundamental quantities separate from length, or they can be described and calculated in terms of lengths in a plane or 3-dimensional space. Mathematicians have found many explicit formulas for area and formulas for volume of various geometric objects. In calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...

, area and volume can be defined in terms of integral
In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...

s, such as the Riemann integral or the Lebesgue integral
In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the -axis. The Lebesgue integral, ...

.
Metrics and measures

The concept of length or distance can be generalized, leading to the idea of metrics. 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 ofspecial relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between Spacetime, space and time. In Albert Einstein's original treatment, the theory is based on two Postulates of ...

and the semi- Riemannian metrics of general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current descr ...

.
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.
Congruence and similarity

Congruence and similarity are concepts that describe when two shapes have similar characteristics. 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. 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.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. 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

Where the traditional geometry allowed dimensions 1 (aline
Line most often refers to:
* Line (geometry)
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional objects, though they may exist in Two-dimensional Euclide ...

), 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. 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.
In general topology, the concept of dimension has been extended from natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...

s, to infinite dimension ( Hilbert spaces, for example) and positive real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...

s (in fractal geometry). In algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. ...

, the dimension of an algebraic variety has received a number of apparently different definitions, which are all equivalent in the most common cases.
Symmetry

The theme ofsymmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

in geometry is nearly as old as the science of geometry itself. Symmetric shapes such as the circle, regular polygons and platonic solids held deep significance for many ancient philosophers and were investigated in detail before the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in a multitude of forms, including the graphics of Leonardo da Vinci
Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, Drawing, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially res ...

, M. C. Escher, and others. 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''. Symmetry in classical symmetry group
In group theory, the symmetry group of a geometric object is the group (mathematics), group of all Transformation (geometry), transformations under which the object is invariant (mathematics), invariant, endowed with the group operation of Fu ...

' found its inspiration. Both discrete and continuous symmetries play prominent roles in geometry, the former in topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

and geometric group theory, the latter in Lie theory and Riemannian geometry.
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
In mathematics, a theorem is a statement (logic), statement that has been Mathematical proof, proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the th ...

, exchange ''point'' with ''plane'', ''join'' with ''meet'', ''lies in'' with ''contains'', and the result is an equally true theorem. A similar and closely related form of duality exists between a vector space
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

and its dual space.
Contemporary geometry

Euclidean geometry

mechanics
Mechanics (from Ancient Greek: wikt:μηχανική#Ancient_Greek, μηχανική, ''mēkhanikḗ'', "of machine, machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among Ph ...

, astronomy
Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...

, crystallography, and many technical fields, such as engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more specializ ...

, architecture
Architecture is the art and technique of designing and building, as distinguished from the skills associated with construction. It is both the process and the product of sketching, conceiving, planning, designing, and construction, constructin ...

, geodesy
Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure ( geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equiv ...

, aerodynamics
Aerodynamics, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dyn ...

, and navigation
Navigation is a field of study that focuses on the process of monitoring and controlling the motion, movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navi ...

. The mandatory educational curriculum of the majority of nations includes the study of Euclidean concepts such as points, lines, planes, angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two ...

s, triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...

s, congruence, similarity, solid figure
In mathematics, solid geometry or stereometry is the traditional name for the geometry of Three-dimensional space, three-dimensional, Euclidean spaces (i.e., 3D geometry).
Stereometry deals with the measurements of volumes of various solid fig ...

s, circles, and Differential geometry

Differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra ...

uses techniques of linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (mat ...

to study problems in geometry. It has applications in physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...

, econometrics
Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. ...

, and bioinformatics
Bioinformatics () is an interdisciplinary field that develops methods and software tools for understanding biological data, in particular when the data sets are large and complex. As an interdisciplinary field of science, bioinformatics combi ...

, among others.
In particular, differential geometry is of importance to mathematical physics due to Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born Theoretical physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for d ...

's universe
The universe is all of space and time and their contents, including planets, stars, galaxy, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmology, cosmological description of the development of ...

is curved. 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).
Non-Euclidean geometry

Euclidean geometry was not the only historical form of geometry studied.Spherical geometry
file:Spherical_triangle_3d.png, 300px, A sphere with a spherical triangle on it.
Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other ...

has long been used by astronomers, astrologers, and navigators.
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. 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). They demonstrated that ordinary Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

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''), published only after his death. Riemann's new idea of space proved crucial in Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born Theoretical physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for d ...

's general relativity theory
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current descr ...

. Riemannian geometry, which considers very general spaces in which the notion of length is defined, is a mainstay of modern geometry.
Topology

Topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

is the field concerned with the properties of continuous mappings, and can be considered a generalization of Euclidean geometry. In practice, topology often means dealing with large-scale properties of spaces, such as connectedness and compactness.
The field of topology, which saw massive development in the 20th century, is in a technical sense a type of transformation geometry, in which transformations are homeomorphisms. 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
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

and general topology.
Algebraic geometry

The field ofCartesian geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineerin ...

of co-ordinates. 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. 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. 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. Wiles' proof of Fermat's Last Theorem uses advanced methods of algebraic geometry for solving a long-standing problem of number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative intege ...

.
In general, algebraic geometry studies geometry through the use of concepts in commutative algebra such as multivariate polynomials. It has applications in many areas, including cryptography
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logy, -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of ...

and string theory.
Complex geometry

Complex geometry studies the nature of geometric structures modelled on, or arising out of, thecomplex plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, ...

. 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.
Complex geometry first appeared as a distinct area of study in the work of Bernhard Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to mathematical analysis, analysis, number theory, and differential geometry. In the field of real analysis, he is mostly ...

in his study of Riemann surfaces. 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.
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
In physics, spacetime is a mathematical model that combines the three-dimensional space, three dimensions of space and one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visualize S ...

may be modelled by Calabi–Yau manifolds.
Discrete geometry

Discrete geometry
Discrete geometry and combinatorial geometry are branches of geometry that study Combinatorics, combinatorial properties and constructive methods of discrete mathematics, discrete geometric objects. Most questions in discrete geometry involve fi ...

is a subject that has close connections with convex geometry. 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. It shares many methods and principles with combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite set, finite Mathematical structure, structures. It is closely related to many other ar ...

.
Computational geometry

Computational geometry
Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems ar ...

deals with algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...

s 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.
Although being a young area of geometry, it has many applications in computer vision
Computer vision is an Interdisciplinarity, interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate t ...

, image processing
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...

, computer-aided design
Computer-aided design (CAD) is the use of computers (or ) to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve co ...

, medical imaging, etc.
Geometric group theory

Geometric group theory uses large-scale geometric techniques to study finitely generated groups. 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. 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.Convex geometry

Convex geometry investigatesconvex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope, ...

shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis and discrete mathematics. It has close connections to convex analysis, optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...

and functional analysis and important applications in number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative intege ...

.
Convex geometry dates back to antiquity. Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, ...

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
Plato ( ; grc-gre, wikt:Πλάτων, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greeks, Greek philosopher born in Athens during the Classical Greece, Classical period in Ancient Greece. He founded the Platonist school of thou ...

, algorithms
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represen ...

, tilings and lattices.
Applications

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

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. Artists have long used concepts of proportion in design.Vitruvius
Vitruvius (; c. 80–70 BC – after c. 15 BC) was a Roman architect and engineer during the 1st century BC, known for his multi-volume work entitled ''De architectura''. He originated the idea that all buildings should have three attribute ...

developed a complicated theory of ''ideal proportions'' for the human figure. These concepts have been used and adapted by artists from Michelangelo
Michelangelo di Lodovico Buonarroti Simoni (; 6 March 1475 – 18 February 1564), known as Michelangelo (), was an Italian sculptor, painter, architect, and poet of the High Renaissance. Born in the Republic of Florence, his work was insp ...

to modern comic book artists.
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.
Tilings, or tessellations, have been used in art throughout history. Islamic art
Islamic art is a part of Islamic culture and encompasses the visual arts produced since the 7th century CE by people who lived within territories inhabited or ruled by Muslims, Muslim populations. Referring to characteristic traditions across ...

makes frequent use of tessellations, as did the art of M. C. Escher. Escher's work also made use of hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For an ...

.
Cézanne advanced the theory that all images can be built up from the sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

, the cone, and the cylinder
A cylinder (from ) has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a Prism (geometry), prism with a circle as its base.
A cylinder ...

. This is still used in art theory today, although the exact list of shapes varies from author to author.
Architecture

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

The field ofastronomy
Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...

, especially as it relates to mapping the positions of star
A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...

s and planet
A planet is a large, rounded Astronomical object, astronomical body that is neither a star nor its Stellar remnant, remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud colla ...

s on the celestial sphere
In astronomy and navigation, the celestial sphere is an abstraction, abstract sphere that has an arbitrarily large radius and is concentric objects, concentric to Earth. All objects in the sky can be conceived as being projective geometry, proje ...

and describing the relationship between movements of celestial bodies, have served as an important source of geometric problems throughout history.
Riemannian geometry and pseudo-Riemannian geometry are used in Other fields of mathematics

Calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...

was strongly influenced by geometry. For instance, the introduction of coordinates by René Descartes
René Descartes ( or ; ; Latinisation of names, Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French people, French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of m ...

and the concurrent developments of algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...

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.
Another important area of application is number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative intege ...

. In ancient Greece
Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean Sea, Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of Classical Antiquity, classical antiquity ( AD 600), th ...

the Pythagoreans considered the role of numbers in geometry. However, the discovery of incommensurable lengths contradicted their philosophical views. 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.
See also

Lists

*List of geometers
A geometer is a mathematician whose area of study is geometry.
Some notable geometers and their main fields of work, chronologically listed, are:
1000 BCE to 1 BCE
* Baudhayana (fl. c. 800 BC) – Euclidean geometry, geometric algebra
* M ...

** :Algebraic geometers
** :Differential geometers
** :Geometers
** :Topologists
* List of formulas in elementary geometry
* List of geometry topics
* List of important publications in geometry
* Lists of mathematics topics
Related topics

* Descriptive geometry *Finite geometry
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation) Finite number may refer to:
* A countable number less than infinity, being the cardinality of a finite set – i.e., some natural number
In mathematics, th ...

* '' 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 software
Other fields

* Molecular geometryNotes

Sources

* * * * *Further reading

* *External links

* Ageometry
Geometry (; ) 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. A mathematician who works in the field of geometry is ca ...

course from Wikiversity
''Unusual Geometry Problems''

''The Math Forum'' – Geometry

*

*

*

Nature Precedings – ''Pegs and Ropes Geometry at Stonehenge''

* ttps://web.archive.org/web/20071004174210/http://www.gresham.ac.uk/event.asp?PageId=45&EventId=618 "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

Interactive geometry reference with hundreds of applets

Geometry classes

at Khan Academy {{Authority control