Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. In mathematics Calabi&ndashYau manifolds are compact Kähler manifolds whose Canonical bundle is trivial Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the third century B. Length is the long Dimension of any object The length of a thing is the distance between its ends its linear extent as measured from end to end Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically C. , geometry was put into an axiomatic form by Euclid, whose treatment - Euclidean geometry - set a standard for many centuries to follow. In Mathematics, an axiomatic system is any set of Axioms from which some or all axioms can be used in conjunction to logically derive Theorems Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. The field of astronomy, especially mapping the positions of the stars and planets on the celestial sphere, served as an important source of geometric problems during the next one and a half millennia. Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study

Introduction of coordinates by René Descartes and the concurrent development of algebra marked a new stage for geometry, since geometric figures, such as plane curves, could now be represented analytically, i. In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. In mathematics a plane curve is a Curve in a Euclidian plane (cf Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry e. , with functions and equations. This played a key role in the emergence of calculus in the seventeenth century. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Furthermore, the theory of perspective showed that there is more to geometry than just the metric properties of figures. Perspective (from Latin perspicere to see through in the graphic arts such as drawing is an approximate representation on a flat surface (such as paper of an image as it is perceived The subject of geometry was further enriched by the study of intrinsic structure of geometric objects that originated with Euler and Gauss and led to the creation of topology and differential geometry. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry

Since the nineteenth century discovery of non-Euclidean geometry, the concept of space has undergone a radical transformation. In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another Contemporary geometry considers manifolds, spaces that are considerably more abstract than the familiar Euclidean space, which they only approximately resemble at small scales. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be These spaces may be endowed with additional structure, allowing one to speak about length. Modern geometry has multiple strong bonds with physics, exemplified by the ties between Riemannian geometry and general relativity. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Elliptic geometry is also sometimes called Riemannian geometry. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 One of the youngest physical theories, string theory, is also very geometric in flavour. String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings

The visual nature of geometry makes it initially more accessible than other parts of mathematics, such as algebra or number theory. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes However, the geometric language is also used in contexts that are far removed from its traditional, Euclidean provenance, for example, in fractal geometry, and especially in algebraic geometry. A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole" Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with [1]

## History

Main article: History of geometry
Woman teaching geometry. Geometry ( Greek γεωμετρία; geo = earth metria = measure arose as the field of knowledge dealing with spatial relationships Illustration at the beginning of a medieval translation of Euclid's Elements, (c. Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek 1310)

The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, Egypt, and the Indus Valley from around 3000 BC. Mesopotamia (from the Greek meaning "land between the rivers" is an area geographically located between the Tigris and Euphrates rivers largely corresponding Ancient Egypt was an Ancient Civilization in eastern North Africa, concentrated along the lower reaches of the Nile River in what is now The Indus Valley Civilization (Mature period 2600&ndash1900 BCE abbreviated IVC, was an ancient Civilization that flourished in the Indus River basin The 30th century BC is a Century which lasted from the year 3000 BC to 2901 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, construction, astronomy, and various crafts. Surveying is the technique and science of accurately determining the terrestrial or three-dimensional space Position of points and the distances and angles between In the fields of Architecture and Civil engineering, construction is a process that consists of the Building or assembling of Infrastructure Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets, and the Indian Shulba Sutras, while the Chinese had the work of Mozi, Zhang Heng, and the Nine Chapters on the Mathematical Art, edited by Liu Hui. Egyptian mathematics refers to the style and methods of Mathematics performed in Ancient Egypt. The Rhind Mathematical Papyrus (RMP (also designated as papyrus British Museum 10057 and pBM 10058 is named after Alexander Henry Rhind, a Scottish The Moscow Mathematical Papyrus is also called the Golenischev Mathematical Papyrus, after its first owner Egyptologist Vladimir Goleniščev. Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia (ancient Iraq) from the days of the early Sumerians to the fall of Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. The Shulba Sutras or Śulbasūtras ( Sanskrit śulba: "string cord rope" are Sutra texts belonging to the Mozi ( Lat as Micius, ca 470 BCE&ndashca 391 BCE was a Philosopher who lived in China during the Hundred Schools of Thought Zhang Heng ( (CE 78–139 was an astronomer, mathematician, inventor, geographer, cartographer, artist, poet The Nine Chapters on the Mathematical Art ( is a Chinese Mathematics book composed by several generations of scholars from the 10th&ndash2nd century BC and Liu Hui ( fl 3rd century) was a Chinese Mathematician who lived in the Wei Kingdom.

Euclid's The Elements of Geometry (c. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek 300 BCE) was one of the most important early texts on geometry, in which he presented geometry in an ideal axiomatic form, which came to be known as Euclidean geometry. Events By place Egypt Pyrrhus, the King of Epirus, is taken as a hostage to Egypt after the Battle of Ipsus In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. The treatise is not, as is sometimes thought, a compendium of all that Hellenistic mathematicians knew about geometry at that time; rather, it is an elementary introduction to it;[2] Euclid himself wrote eight more advanced books on geometry. This article focuses on the cultural aspects of the Hellenistic age for the historical aspects see Hellenistic period. We know from other references that Euclid’s was not the first elementary geometry textbook, but the others fell into disuse and were lost.

In the Middle Ages, Muslim mathematicians contributed to the development of geometry, especially algebraic geometry and geometric algebra. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In Mathematical physics, a geometric algebra is a Multilinear algebra described technically as a Clifford algebra over a real vector space equipped Al-Mahani (b. Abu-Abdullah Muhammad ibn Īsa Māhānī, was a Persian mathematician and astronomer from Mahan, Kerman, Persia. 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Thābit ibn Qurra (known as Thebit in Latin) (836-901) dealt with arithmetical operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry. (836 in Harran, Mesopotamia &ndash February 18, 901 in Baghdad) was an Arab astronomer, mathematician Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone A ratio is an expression which compares quantities relative to each other Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry Omar Khayyám (1048-1131) found geometric solutions to cubic equations, and his extensive studies of the parallel postulate contributed to the development of Non-Euclidian geometry. For the Thoroughbred racehorse see Omar Khayyam (horse Ghiyās od-Dīn Abol-Fath Omār ibn Ebrāhīm Khayyām Neyshābūri (غیاث الدین This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve. In Geometry, the parallel postulate, also called Euclid 's fifth postulate since it is the fifth postulate in Euclid's ''Elements'', is a distinctive In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry

In the early 17th century, there were two important developments in geometry. The first, and most important, was the creation of analytic geometry, or geometry with coordinates and equations, by René Descartes (1596–1650) and Pierre de Fermat (1601–1665). Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent Pierre de Fermat pjɛːʁ dəfɛʁ'ma ( 17 August 1601 or 1607/8 &ndash 12 January 1665) was a French Lawyer at the This was a necessary precursor to the development of calculus and a precise quantitative science of physics. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry is a non- metrical form of Geometry, notable for its principle of duality. Girard Desargues ( February 21 or March 2, 1591 -October 1661 was a French Mathematician and engineer who is considered one of Projective geometry is the study of geometry without measurement, just the study of how points align with each other.

Two developments in geometry in the nineteenth century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Lobachevsky, Bolyai and Gauss and of the formulation of symmetry as the central consideration in the Erlangen Programme of Felix Klein (which generalized the Euclidean and non Euclidean geometries). In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry Nikolai Ivanovich Lobachevsky (Никола́й Ива́нович Лобаче́вский ( December 1 1792 &ndash February 24 1856 ( N János Bolyai ( December 15, 1802 – January 27, 1860) was a Hungarian Mathematician, known for his work in Non-Euclidean Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen Felix Christian Klein ( 25 April 1849 &ndash 22 June 1925) was a German Mathematician, known for his work in Group Two of the master geometers of the time were Bernhard Riemann, 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. Analysis has its beginnings in the rigorous formulation of Calculus. In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic The dynamical system concept is a mathematical Formalization for any fixed "rule" which describes the Time dependence of a point's position

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. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects The traditional type of geometry was recognized as that of homogeneous spaces, those spaces which have a sufficient supply of symmetry, so that from point to point they look just the same. In Mathematics, particularly in the theories of Lie groups Algebraic groups and Topological groups a homogeneous space for a group

## What is geometry?

Visual proof of the Pythagorean theorem for the (3, 4, 5) triangle as in the Chou Pei Suan Ching 500–200 BC. In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line The Zhou Bi Suan Jing (周髀算经 The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven is one of the oldest and most famous Chinese

Recorded development of geometry spans more than two millennia. It is hardly surprising that perceptions of what constituted geometry evolved throughout the ages. The geometric paradigms presented below should be viewed as 'Pictures at an exhibition' of a sort: they do not exhaust the subject of geometry but rather reflect some of its defining themes. Pictures at an Exhibition (Картинки с выставки &ndash Воспоминание о Викторе Гартмане Kartinki s vystavki &ndash Vospominaniye

### Practical geometry

There is little doubt that geometry originated as a practical science, concerned with surveying, measurements, areas, and volumes. Among the notable accomplishments one finds formulas for lengths, areas and volumes, such as Pythagorean theorem, circumference and area of a circle, area of a triangle, volume of a cylinder, sphere, and a pyramid. Length is the long Dimension of any object The length of a thing is the distance between its ends its linear extent as measured from end to end Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry The circumference is the distance around a closed Curve. Circumference is a kind of Perimeter. The area of a disk (the region inside a Circle) is &pi r 2 when the circle has Radius r. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line A cylinder is one of the most basic curvilinear geometric shapes the Surface formed by the points at a fixed distance from a given Straight line, the axis "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe Volume The Volume of a pyramid is V = \frac{1}{3} Bh where B is the area of the base and h the height from the base to the apex Development of astronomy led to emergence of trigonometry and spherical trigonometry, together with the attendant computational techniques. Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Spherical trigonometry is a part of Spherical geometry that deals with Polygons (especially Triangles on the Sphere and explains how to find relations

### Axiomatic geometry

A method of computing certain inaccessible distances or heights based on similarity of geometric figures and attributed to Thales presaged more abstract approach to geometry taken by Euclid in his Elements, one of the most influential books ever written. Geometry Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking of the other Thales of Miletus According to Bertrand Russell, "Philosophy begins with Thales Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek Euclid introduced certain axioms, or postulates, expressing primary or self-evident properties of points, lines, and planes. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigour. In the twentieth century, David Hilbert employed axiomatic reasoning in his attempt to update Euclid and provide modern foundations of geometry. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most

### Geometric constructions

Ancient scientists paid special attention to constructing geometric objects that had been described in some other way. Classical instruments allowed in geometric constructions are the compass and straightedge. Pentagon constructgif|thumb|right|Construction of a regular pentagon]] Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or Angles However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using parabolas and other curves, as well as mechanical devices, were found. The approach to geometric problems with geometric or mechanical means is known as synthetic geometry. Synthetic geometry is the branch of Geometry which makes use of Theorems and synthetic observations to draw conclusions as opposed to Analytic geometry

### Numbers in geometry

Already Pythagoreans considered the role of numbers in geometry. Pythagoreanism is a term used for the Esoteric and metaphysical beliefs held by Pythagoras and his followers the Pythagoreans who were much influenced However, the discovery of incommensurable lengths, which contradicted their philosophical views, made them abandon (abstract) numbers in favour of (concrete) geometric quantities, such as length and area of figures. In Mathematics, two non- Zero Real numbers a and b are said to be commensurable Iff a / b Numbers were reintroduced into geometry in the form of coordinates by Descartes, who realized that the study of geometric shapes can be facilitated by their algebraic representation. In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point Analytic geometry applies methods of algebra to geometric questions, typically by relating geometric curves and algebraic equations. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent These ideas played a key role in the development of calculus in the seventeenth century and led to discovery of many new properties of plane curves. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Modern algebraic geometry considers similar questions on a vastly more abstract level. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with

### Geometry of position

Even in ancient times, geometers considered questions of relative position or spatial relationship of geometric figures and shapes. Some examples are given by inscribed and circumscribed circles of polygons, lines intersecting and tangent to conic sections, the Pappus and Menelaus configurations of points and lines. In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface In Projective geometry, the Pappus configuration consists of a pair (( A, B, C) ( D, E, F) of triplets of points Menelaus' theorem, attributed to Menelaus of Alexandria, is a theorem about Triangles in Plane geometry. In the Middle Ages new and more complicated questions of this type were considered: What is the maximum number of spheres simultaneously touching a given sphere of the same radius (kissing number problem)? What is the densest packing of spheres of equal size in space (Kepler conjecture)? Most of these questions involved 'rigid' geometrical shapes, such as lines or spheres. In Geometry, the kissing number is the maximum number of Spheres of radius 1 that can simultaneously touch the unit sphere in n -dimensional Euclidean In Mathematics, sphere packing problems are problems concerning arrangements of non-overlapping identical Spheres which fill a space In Mathematics, the Kepler conjecture is a Conjecture about Sphere packing in three-dimensional Euclidean space. Projective, convex and discrete geometry are three subdisciplines within present day geometry that deal with these and related questions. Projective geometry is a non- metrical form of Geometry, notable for its principle of duality. Convex geometry is the branch of Geometry studying Convex sets mainly in Euclidean space. Discrete geometry or combinatorial geometry may be loosely defined as study of geometrical objects and properties that are discrete or Combinatorial, either

A new chapter in Geometria situs was opened by Leonhard Euler, who boldly cast out metric properties of geometric figures and considered their most fundamental geometrical structure based solely on shape. Topology, which grew out of geometry, but turned into a large independent discipline, does not differentiate between objects that can be continuously deformed into each other. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of The objects may nevertheless retain some geometry, as in the case of hyperbolic knots. In Mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative

### Geometry beyond Euclid

For nearly two thousand years since Euclid, while the range of geometrical questions asked and answered inevitably expanded, basic understanding of space remained essentially the same. Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another 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. Immanuel Kant (ɪmanuəl kant 22 April 1724 12 February 1804 was an 18th-century German Philosopher from the Prussian city of Königsberg The analytic-synthetic distinction is a conceptual distinction used primarily in Philosophy to distinguish propositions into two types analytic propositions and [3] This dominant view was overturned by the revolutionary discovery of non-Euclidean geometry in the works of Gauss (who never published his theory), Bolyai, and Lobachevsky, who demonstrated that ordinary Euclidean space is only one possibility for development of geometry. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German János Bolyai ( December 15, 1802 – January 27, 1860) was a Hungarian Mathematician, known for his work in Non-Euclidean Nikolai Ivanovich Lobachevsky (Никола́й Ива́нович Лобаче́вский ( December 1 1792 &ndash February 24 1856 ( N A broad vision of the subject of geometry was then expressed by Riemann in his inaugurational lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen (On the hypotheses on which geometry is based), published only after his death. Riemann's new idea of space proved crucial in Einstein's general relativity theory and Riemannian geometry, which considers very general spaces in which the notion of length is defined, is a mainstay of modern geometry. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 Elliptic geometry is also sometimes called Riemannian geometry.

### Symmetry

A uniform tiling of the hyperbolic plane

The theme of symmetry in geometry is nearly as old as the science of geometry itself. A tessellation or tiling of the plane is a collection of Plane figures that fills the plane with no overlaps and no gaps In Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or The circle, regular polygons and platonic solids held deep significance for many ancient philosophers and were investigated in detail by the time of Euclid. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the General properties These properties apply to both convex and star regular polygons In Geometry, a Platonic solid is a convex Regular polyhedron. Symmetric patterns occur in nature and were artistically rendered in a multitude of forms, including the bewildering graphics of M. C. Escher. Maurits Cornelis Escher (17 June 1898 – 27 March 1972 usually referred to as M Nonetheless, it was not until the second half of nineteenth century that the unifying role of symmetry in foundations of geometry had been recognized. Felix Klein's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group, determines what geometry is. Felix Christian Klein ( 25 April 1849 &ndash 22 June 1925) was a German Mathematician, known for his work in Group An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element 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. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. In Geometry, two sets of points are called congruent if one can be transformed into the other by an Isometry, i Projective geometry is a non- metrical form of Geometry, notable for its principle of duality. A collineation is a one-to-one map from one Projective space to another or from a Projective plane onto itself such that the images of collinear points are themselves 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' proved most influential. William Clifford (christened 14 December 1811 at Bearsted, Kent; died 5 September 1841 at Gravesend, Kent was an English Cricketer Marius Sophus Lie (liː as "Lee" ( 17 December 1842 - 18 February 1899) was a Norwegian -born Mathematician. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is Both discrete and continuous symmetries play prominent role in geometry, the former in topology and geometric group theory, the latter in Lie theory and Riemannian geometry. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Geometric group theory is an area in Mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and Lie theory is an area of Mathematics, developed initially by Sophus Lie. Elliptic geometry is also sometimes called Riemannian geometry.

### Modern geometry

Modern geometry is the title of a popular textbook by Dubrovin, Novikov, and Fomenko first published in 1979 (in Russian). Sergei Petrovich Novikov (also Serguei) ( Russian Сергей Петрович Новиков (born 20 March 1938) is a Russian At close to 1000 pages, the book has one major thread: geometric structures of various types on manifolds and their applications in contemporary theoretical physics. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be Theoretical physics employs Mathematical models and Abstractions of Physics in an attempt to explain experimental data taken of the natural world A quarter century after its publication, differential geometry, algebraic geometry, symplectic geometry, and Lie theory presented in the book remain among the most visible areas of modern geometry, with multiple connections with other parts of mathematics and physics. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with Symplectic geometry is a branch of differential topology/geometry which studies Symplectic manifolds that is Differentiable manifolds equipped with a Lie theory is an area of Mathematics, developed initially by Sophus Lie.

## Contemporary geometers

Some of the representative leading figures in modern geometry are Michael Atiyah, Mikhail Gromov, and William Thurston. Sir Michael Francis Atiyah, OM, FRS, FRSE (b April 22, 1929) is a British Mathematician, and one of the Mikhail Leonidovich Gromov Russian Михаил Леонидович Громов (born December 23, 1943, also known William Paul Thurston (born October 30, 1946) is an American Mathematician. The common feature in their work is the use of smooth manifolds as the basic idea of space; they otherwise have rather different directions and interests. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. Geometry now is, in large part, the study of structures on manifolds that have a geometric meaning, in the sense of the principle of covariance that lies at the root of general relativity theory in theoretical physics. In Theoretical physics, general covariance (also known as Diffeomorphism covariance or general invariance) is the Invariance of the General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 (See Category:Structures on manifolds for a survey. )

Much of this theory relates to the theory of continuous symmetry, or in other words Lie groups. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group From the foundational point of view, on manifolds and their geometrical structures, important is the concept of pseudogroup, defined formally by Shiing-shen Chern in pursuing ideas introduced by Élie Cartan. In Mathematics, a pseudogroup is an extension of the group concept but one that grew out of the geometric approach of Sophus Lie, rather than out of Shiing-Shen Chern (陳省身 pinyin: Chén Xǐngshēn October 26 1911 &ndash December 3 2004) was a Chinese American Mathematician Élie Joseph Cartan ( 9 April 1869 &ndash 6 May 1951) was an influential French Mathematician, who did fundamental A pseudogroup can play the role of a Lie group of infinite dimension.

## Dimension

Where the traditional geometry allowed dimensions 1 (a line), 2 (a plane) and 3 (our ambient world conceived of as three-dimensional space), mathematicians have used higher dimensions for nearly two centuries. Three-dimensional space is a geometric model of the physical Universe in which we live Higher dimension as a term in Mathematics most commonly refers to any number of spatial Dimensions greater than three Dimension has gone through stages of being any natural number n, possibly infinite with the introduction of Hilbert space, and any positive real number in fractal geometry. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an This article assumes some familiarity with Analytic geometry and the concept of a limit. A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole" Dimension theory is a technical area, initially within general topology, that discusses definitions; in common with most mathematical ideas, dimension is now defined rather than an intuition. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it In Mathematics, general topology or point-set topology is the branch of Topology which studies properties of Topological spaces and structures Connected topological manifolds have a well-defined dimension; this is a theorem (invariance of domain) rather than anything a priori. In Mathematics, a topological manifold is a Hausdorff Topological space which looks locally like Euclidean space in a sense defined below Invariance of domain is a theorem in Topology about Homeomorphic Subsets of Euclidean space R n.

The issue of dimension still matters to geometry, in the absence of complete answers to classic questions. Dimensions 3 of space and 4 of space-time are special cases in geometric topology. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS In Mathematics, geometric topology is the study of Manifolds and their Embeddings Low-dimensional topology, concerning questions of dimensions Dimension 10 or 11 is a key number in string theory. String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings Exactly why is something to which research may bring a satisfactory geometric answer.

## Contemporary Euclidean geometry

Main article: Euclidean geometry

The study of traditional Euclidean geometry is by no means dead. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. It is now typically presented as the geometry of Euclidean spaces of any dimension, and of the Euclidean group of rigid motions. In Mathematics, the Euclidean group E ( n) sometimes called ISO( n) or similar is the Symmetry group of n -dimensional In Physics, a rigid body is an idealization of a solid body of finite size in which Deformation is neglected The fundamental formulae of geometry, such as the Pythagorean theorem, can be presented in this way for a general inner product space. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry In Mathematics, an inner product space is a Vector space with the additional Structure of inner product.

Euclidean geometry has become closely connected with computational geometry, computer graphics, convex geometry, discrete geometry, and some areas of combinatorics. Computational geometry is a branch of Computer science devoted to the study of algorithms which can be stated in terms of Geometry. Computer graphics are Graphics created by Computers and more generally the Representation and Manipulation of Pictorial Data Convex geometry is the branch of Geometry studying Convex sets mainly in Euclidean space. Discrete geometry or combinatorial geometry may be loosely defined as study of geometrical objects and properties that are discrete or Combinatorial, either Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects Momentum was given to further work on Euclidean geometry and the Euclidean groups by crystallography and the work of H. S. M. Coxeter, and can be seen in theories of Coxeter groups and polytopes. Crystallography is the experimental science of determining the arrangement of Atoms in Solids In older usage it is the scientific study of Crystals The Harold Scott MacDonald "Donald" Coxeter CC ( February 9, 1907 – March 31, 2003) is regarded as one of the great In Mathematics, a Coxeter group, named after HSM Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries In Geometry, polytope is a generic term that can refer to a two-dimensional Polygon, a three-dimensional Polyhedron, or any of the various generalizations Geometric group theory is an expanding area of the theory of more general discrete groups, drawing on geometric models and algebraic techniques. Geometric group theory is an area in Mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and In Mathematics, a discrete group is a group G equipped with the Discrete topology.

## Algebraic geometry

The field of algebraic geometry is the modern incarnation of the Cartesian geometry of co-ordinates. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point After a turbulent period of axiomatization, its foundations are in the twenty-first century on a stable basis. In Mathematics, an axiomatic system is any set of Axioms from which some or all axioms can be used in conjunction to logically derive Theorems Either one studies the 'classical' case where the spaces are complex manifolds that can be described by algebraic equations; or the scheme theory provides a technically sophisticated theory based on general commutative rings. In Differential geometry, a complex manifold is a Manifold with an atlas of charts to the open unit disk in C n, In Mathematics, an algebraic equation over a given field is an Equation of the form P = Q where P and Q In Mathematics, a scheme is an important concept connecting the fields of Algebraic geometry, Commutative algebra and Number theory. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property

The geometric style which was traditionally called the Italian school is now known as birational geometry. In relation with the history of Mathematics, the Italian school of Algebraic geometry refers to the work over half a century or more (flourishing roughly 1885-1935 In Mathematics, birational geometry is a part of the subject of Algebraic geometry, that deals with the geometry of an Algebraic variety that is dependent It has made progress in the fields of threefolds, singularity theory and moduli spaces, as well as recovering and correcting the bulk of the older results. For other mathematical uses see Mathematical singularity. For non-mathematical uses see Gravitational singularity. In Algebraic geometry, a moduli space is a geometric space (usually a scheme or an Algebraic stack) whose points represent algebro-geometric objects of Objects from algebraic geometry are now commonly applied in string theory, as well as diophantine geometry. String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings In Mathematics, a Diophantine equation is an indeterminate Polynomial Equation that allows the variables to be Integers only

Methods of algebraic geometry rely heavily on sheaf theory and other parts of homological algebra. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. Homological algebra is the branch of Mathematics which studies homology in a general algebraic setting The Hodge conjecture is an open problem that has gradually taken its place as one of the major questions for mathematicians. The Hodge conjecture is a major unsolved problem in Algebraic geometry which relates the Algebraic topology of a Non-singular complex Algebraic For practical applications, Gröbner basis theory and real algebraic geometry are major subfields. In Computer algebra, computational Algebraic geometry, and computational Commutative algebra, a Gröbner basis is a particular kind of generating subset In Mathematics, real algebraic geometry is the study of Real number solutions to Algebraic equations with real number coefficients

## Differential geometry

Differential geometry, which in simple terms is the geometry of curvature, has been of increasing importance to mathematical physics since the suggestion that space is not flat space. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry In Mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry Mathematical physics is the scientific discipline concerned with the interface of Mathematics and Physics. The intuitive idea of flatness is important in several fields Contemporary differential geometry is intrinsic, meaning that space is a manifold and structure is given by a Riemannian metric, or analogue, locally determining a geometry that is variable from point to point. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M

This approach contrasts with the extrinsic point of view, where curvature means the way a space bends within a larger space. The idea of 'larger' spaces is discarded, and instead manifolds carry vector bundles. In Mathematics, a vector bundle is a topological construction which makes precise the idea of a family of Vector spaces parameterized by another space Fundamental to this approach is the connection between curvature and characteristic classes, as exemplified by the generalized Gauss-Bonnet theorem. In Mathematics, a characteristic class is a way of associating to each Principal bundle on a Topological space X a Cohomology class In Mathematics, the generalized-Gauss-Bonnet theorem presents the Euler characteristic of a closed even-dimensional Riemannian manifold as an integral of a certain

## Topology and geometry

A thickening of the trefoil knot

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. In Knot theory, the trefoil knot is the simplest nontrivial knot. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Mathematics, transformation geometry is a name for a Pedagogic theory for teaching Euclidean geometry, based on the Erlangen programme. Topological equivalence redirects here see also Topological equivalence (dynamical systems. This has often been expressed in the form of the dictum 'topology is rubber-sheet geometry'. Contemporary geometric topology and differential topology, and particular subfields such as Morse theory, would be counted by most mathematicians as part of geometry. In Mathematics, geometric topology is the study of Manifolds and their Embeddings Low-dimensional topology, concerning questions of dimensions In Mathematics, differential topology is the field dealing with differentiable functions on Differentiable manifolds It is closely related to Differential "Morse function" redirects here In another context a "Morse function" can also mean an Anharmonic oscillator. Algebraic topology and general topology have gone their own ways. Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic In Mathematics, general topology or point-set topology is the branch of Topology which studies properties of Topological spaces and structures

## Axiomatic and open development

The model of Euclid's Elements, a connected development of geometry as an axiomatic system, is in a tension with René Descartes's reduction of geometry to algebra by means of a coordinate system. In Mathematics, an axiomatic system is any set of Axioms from which some or all axioms can be used in conjunction to logically derive Theorems In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point There were many champions of synthetic geometry, Euclid-style development of projective geometry, in the nineteenth century, Jakob Steiner being a particularly brilliant figure. Synthetic geometry is the branch of Geometry which makes use of Theorems and synthetic observations to draw conclusions as opposed to Analytic geometry Jakob Steiner ( 18 March, 1796 &ndash April 1, 1863) was a Swiss Mathematician. In contrast to such approaches to geometry as a closed system, culminating in Hilbert's axioms and regarded as of important pedagogic value, most contemporary geometry is a matter of style. Hilbert's axioms are a set of 20 assumptions (originally 21 David Hilbert proposed in 1899 as the foundation for a modern treatment of Euclidean geometry. Computational synthetic geometry is now a branch of computer algebra. A computer algebra system ( CAS) is a software program that facilitates Symbolic mathematics.

The Cartesian approach currently predominates, with geometric questions being tackled by tools from other parts of mathematics, and geometric theories being quite open and integrated. This is to be seen in the context of the axiomatization of the whole of pure mathematics, which went on in the period c. Broadly speaking pure mathematics is Mathematics motivated entirely for reasons other than application 1900–c. 1950: in principle all methods are on a common axiomatic footing. This reductive approach has had several effects. There is a taxonomic trend, which following Klein and his Erlangen program (a taxonomy based on the subgroup concept) arranges theories according to generalization and specialization. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of For example affine geometry is more general than Euclidean geometry, and more special than projective geometry. Affine geometry is a form of Geometry featuring the unique parallel line property (see the parallel postulate) but where the notion of angle is undefined and lengths The whole theory of classical groups thereby becomes an aspect of geometry. The classical Lie groups are four infinite families of Lie groups closely related to the symmetries of Euclidean spaces There is a certain leeway in using the term Their invariant theory, at one point in the nineteenth century taken to be the prospective master geometric theory, is just one aspect of the general representation theory of Lie groups. Invariant theory is a branch of Abstract algebra that studies actions of groups on algebraic varieties from the point of view of their effect In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of Using finite fields, the classical groups give rise to finite groups, intensively studied in relation to the finite simple groups; and associated finite geometry, which has both combinatorial (synthetic) and algebro-geometric (Cartesian) sides. In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements In Mathematics, a finite group is a group which has finitely many elements In Mathematics, the Classification of finite simple groups states thatevery finite Simple group is cyclic, or alternating, or in one of 16 families A finite geometry is any geometric system that has only a finite number of points.

An example from recent decades is the twistor theory of Roger Penrose, initially an intuitive and synthetic theory, then subsequently shown to be an aspect of sheaf theory on complex manifolds. The twistor theory, originally developed by Roger Penrose in 1967, is the mathematical theory which maps the Geometric objects of the four dimensional space-time Sir Roger Penrose, PhD, OM, FRS (born 8 August 1931) is an English Mathematical physicist and Emeritus In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. In Differential geometry, a complex manifold is a Manifold with an atlas of charts to the open unit disk in C n, In contrast, the non-commutative geometry of Alain Connes is a conscious use of geometric language to express phenomena of the theory of von Neumann algebras, and to extend geometry into the domain of ring theory where the commutative law of multiplication is not assumed. Noncommutative geometry, or NCG, is a branch of Mathematics concerned with the possible spatial interpretations of Algebraic structures for which the Alain Connes (born 1 April 1947 is a French Mathematician, currently Professor at the College de France, IHÉS and Vanderbilt University In Mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those In Mathematics, commutativity is the ability to change the order of something without changing the end result

Another consequence of the contemporary approach, attributable in large measure to the Procrustean bed represented by Bourbakiste axiomatization trying to complete the work of David Hilbert, is to create winners and losers. Nicolas Bourbaki is the collective Pseudonym under which a group of (mainly French) 20th-century Mathematicians wrote a series of books presenting an exposition David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most The Ausdehnungslehre (calculus of extension) of Hermann Grassmann was for many years a mathematical backwater, competing in three dimensions against other popular theories in the area of mathematical physics such as those derived from quaternions. Hermann Günther Grassmann ( April 15, 1809, Stettin ( Szczecin) &ndash September 26, 1877, Stettin) was a Hermann Günther Grassmann ( April 15, 1809, Stettin ( Szczecin) &ndash September 26, 1877, Stettin) was a Mathematical physics is the scientific discipline concerned with the interface of Mathematics and Physics. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician In the shape of general exterior algebra, it became a beneficiary of the Bourbaki presentation of multilinear algebra, and from 1950 onwards has been ubiquitous. In Mathematics, multilinear algebra extends the methods of Linear algebra. In much the same way, Clifford algebra became popular, helped by a 1957 book Geometric Algebra by Emil Artin. In Mathematics, Clifford algebras are a type of Associative algebra. Emil Artin ( March 3, 1898, in Vienna – December 20, 1962, in Hamburg) was an Austrian Mathematician The history of 'lost' geometric methods, for example infinitely near points, which were dropped since they did not well fit into the pure mathematical world post-Principia Mathematica, is yet unwritten. In Mathematics, the notion of infinitely near points was initially part of the intuitive foundations of Differential calculus. The Principia Mathematica is a 3-volume work on the Foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell The situation is analogous to the expulsion of infinitesimals from differential calculus. Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have Differential Calculus, a field in Mathematics, is the study of how functions change when their inputs change As in that case, the concepts may be recovered by fresh approaches and definitions. Those may not be unique: synthetic differential geometry is an approach to infinitesimals from the side of categorical logic, as non-standard analysis is by means of model theory. In Mathematics, synthetic differential geometry is a reformulation of Differential geometry in the language of Topos theory. Categorical logic is a branch of Category theory within Mathematics, adjacent to Mathematical logic but in fact more notable for its connections to Non-standard analysis is a branch of Mathematics that formulates analysis using a rigorous notion of an Infinitesimal number In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models