Geometry (Ancient Greek Ancient Greek is the historical stage in the development of the Greek language spanning the Archaic , Classical (c. 5th–4th centuries BC), and Hellenistic (c. 3rd century BC – 6th century AD) periods of ancient Greece and the ancient world. It is predated in the 2nd millennium BC by Mycenaean Greek. Its Hellenistic phase is known as Koine (&: γεωμετρία; geo- "earth", -metri "measurement") "Earth Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets. It is sometimes referred to as the World, the Blue Planet,[note 6] or by its Latin name, Terra.[note 7]-measuring In science, measurement is the process of estimating or determining the magnitude of a quantity, such as length or mass, relative to a unit of measurement, such as a metre or a kilogram. The term measurement can also be used to refer to a specific result obtained from the measurement process" is a part of mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions concerned with questions of size, shape, relative position of figures, and the properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning lengths In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire thickness. Another example is FET transistors, in which the channel width may be larger than channel length, areas 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 surface area of a 3-dimensional solid is the total area of the exposed surface, such as the sum of the areas of the exposed sides of a polyhedron. Area is an important invariant in the differential geometry of, and volumes Volume is how much three-dimensional space a substance or shape occupies or contains, often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container, i. e. the amount of fluid (gas or liquid) that the container could hold, rather than the amount of, in the 3rd century BC geometry was put into an axiomatic form 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. A mathematical theory consists of an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system; usually though the effort towards by Euclid Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry." He was active in Alexandria during the reign of Ptolemy I (323–283 BC). His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics, whose treatment—Euclidean geometry Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, whose Elements is the earliest known systematic discussion of geometry. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these. Although many of Euclid's results had been—set a standard for many centuries to follow. The field of astronomy Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the Earth's atmosphere (such as the cosmic background radiation). It is concerned with the evolution, physics, chemistry, meteorology, and motion of celestial objects, as well as the formation and development of the universe, especially mapping the positions of the stars A star is a massive, luminous ball of plasma held together by gravity. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth. Other stars are visible in the night sky, when they are not outshone by the Sun. Historically, the most prominent stars on the celestial sphere were grouped together into constellations and planets A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.[a] on the celestial sphere In astronomy and navigation, the celestial sphere is an imaginary sphere of arbitrarily large radius, concentric with the Earth and rotating upon the same axis. All objects in the sky can be thought of as projected upon the celestial sphere. Projected upward from Earth's equator and poles are the celestial equator and the celestial poles. The, served as an important source of geometric problems during the next one and a half millennia. A mathematician who works in the field of geometry is called a geometer.

The introduction of coordinates In geometry, a coordinate system is a system which uses a set of numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in 'the x-coordinate'. In elementary by René Descartes René Descartes , (31 March 1596 – 11 February 1650), also known as Renatus Cartesius (Latinized form; adjectival form: "Cartesian"), was a French philosopher, mathematician, physicist, and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the "Father of Modern Philosophy", and much of and the concurrent development of algebra Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, topology, combinatorics, and number theory, algebra is one of the main branches of pure marked a new stage for geometry, since geometric figures, such as plane curves In mathematics, a plane curve is a curve in a Euclidian plane . The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves, could now be represented analytically Analytic geometry, also known as coordinate geometry, analytical geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis. This contrasts with the synthetic approach of Euclidean geometry, which treats certain geometric notions as primitive, and uses deductive reasoning based on, i.e., with functions and equations. This played a key role in the emergence of calculus Calculus is a branch in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change, in in the 17th century. Furthermore, the theory of perspective Perspective in the graphic arts, such as drawing, is an approximate representation, on a flat surface (such as paper), of an image as it is seen by the eye. The two most characteristic features of perspective are that objects are drawn: showed that there is more to geometry than just the metric properties of figures: perspective is the origin of projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more. The subject of geometry was further enriched by the study of intrinsic structure of geometric objects that originated with Euler Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also renowned for and Gauss Johann Carl Friedrich Gauss (pronounced /ˈɡaʊs/; German: Gauß listen , Latin: Carolus Fridericus Gauss) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and led to the creation of topology Topology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example, deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation and differential geometry Differential geometry is a mathematical discipline that uses the methods of differential and integral calculus, as well as linear and multilinear algebra, to study problems in geometry. The theory of plane and space curves and of surfaces in the three-dimensional Euclidean space formed the basis for its initial development in the eighteenth and.

In Euclid's time there was no clear distinction between physical space and geometrical space. Since the 19th-century discovery of non-Euclidean geometry A non-Euclidean geometry is the study of shapes and constructions that do not map directly to any n-dimensional Euclidean system, characterized by a non-vanishing Riemann curvature tensor. Examples of non-Euclidean geometries include the hyperbolic and elliptic geometry, which are contrasted with a Euclidean geometry. The essential difference, the concept of space Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of the boundless four-dimensional continuum known as spacetime. In mathematics one examines ' has undergone a radical transformation, and the question arose which geometrical space best fits physical space. With the rise of formal mathematics in the 20th century, also 'space' (and 'point', 'line', 'plane') lost its intuitive contents, so today we have to distinguish between physical space, geometrical spaces (in which 'space', 'point' etc. still have their intuitive meaning) and abstract spaces. Contemporary geometry considers manifolds In mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold. Thus a line and a circle are one-dimensional manifolds, a plane and sphere are two-dimensional manifolds, and so forth, spaces that are considerably more abstract than the familiar Euclidean space In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions. The term “Euclidean” is used to distinguish these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity, which they only approximately resemble at small scales. These spaces may be endowed with additional structure, allowing one to speak about length. Modern geometry has multiple strong bonds with physics Physics is a natural science that involves the study of matter and its motion through space-time, as well as all applicable concepts, including energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves, exemplified by the ties between pseudo-Riemannian In differential geometry, a pseudo-Riemannian manifold is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the metric tensor need not be positive-definite. Instead geometry and general relativity General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1915. It is the current description of gravitation in modern physics. It generalises special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space. One of the youngest physical theories, string theory String theory is a developing theory in particle physics which attempts to reconcile quantum mechanics and general relativity, is also very geometric in flavor.

While the visual nature of geometry makes it initially more accessible than other parts of mathematics, such as algebra Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, topology, combinatorics, and number theory, algebra is one of the main branches of pure or number theory 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 of problems that arise from their study, geometric language is also used in contexts far removed from its traditional, Euclidean provenance (for example, in fractal geometry A fractal is "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity. Roots of mathematical interest in fractals can be traced back to the late 19th Century; however, the term "fractal" was coined by Benoît Mandelbrot in 1975 and and algebraic geometry Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. Initially a).^[1]

Overview

The recorded development of geometry spans more than two millennia A millennium is a period of time equal to one thousand years (from the Latin phrase mille, thousand, and annus, year), often but not necessarily related numerically to a particular dating system. It is hardly surprising that perceptions of what constituted geometry evolved throughout the ages.

Practical geometry

Geometry originated as a practical science concerned with surveying, measurements, areas, and volumes. Among the notable accomplishments one finds formulas for lengths In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire thickness. Another example is FET transistors, in which the channel width may be larger than channel length, areas 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 surface area of a 3-dimensional solid is the total area of the exposed surface, such as the sum of the areas of the exposed sides of a polyhedron. Area is an important invariant in the differential geometry of and volumes Volume is how much three-dimensional space a substance or shape occupies or contains, often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container, i. e. the amount of fluid (gas or liquid) that the container could hold, rather than the amount of, such as Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle . In terms of areas, it states:, circumference where r is the radius and d is the diameter of the circle, and π is defined as the ratio of the circumference of the circle to its diameter (the numerical value of pi is 3.141 592 653 589 793...) and area The area of a disk is πr2 when the circle has radius r. Here the symbol π (Greek letter pi) denotes, as usual, the constant ratio of the circumference of a circle to its diameter of a circle, area of a triangle 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 segments. A triangle with vertices A, B, and C is denoted ABC, volume of a cylinder, sphere, and a pyramid. A method of computing certain inaccessible distances or heights based on similarity of geometric figures is attributed to Thales. Development of astronomy led to emergence of trigonometry and spherical trigonometry, together with the attendant computational techniques.

Axiomatic geometry

Euclid took a more abstract approach 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 by Gauss and others led to a revival of interest, and in the 20th century David Hilbert employed axiomatic reasoning in an attempt to provide a modern foundation of geometry.

Geometric constructions

Ancient scientists paid special attention to constructing geometric objects that had been described in some other way. Classical instruments allowed in geometric constructions are those with compass and straightedge. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using parabolas and other curves, as well as mechanical devices, were found.

Numbers in geometry

In ancient Greece the Pythagoreans considered the role of numbers in geometry. However, the discovery of incommensurable lengths, which contradicted their philosophical views, made them abandon (abstract) numbers in favor of (concrete) geometric quantities, such as length and area of figures. 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. Analytic geometry applies methods of algebra to geometric questions, typically by relating geometric curves and algebraic equations. These ideas played a key role in the development of calculus in the 17th century and led to discovery of many new properties of plane curves. Modern algebraic geometry considers similar questions on a vastly more abstract level.

Geometry of position

Even in ancient times, geometers considered questions of relative position or spatial relationship of geometric figures and shapes. Some examples are given by inscribed and circumscribed circles of polygons, lines intersecting and tangent to conic sections, the Pappus and Menelaus configurations of points and lines. In the Middle Ages new and more complicated questions of this type were considered: What is the maximum number of spheres simultaneously touching a given sphere of the same radius (kissing number problem)? What is the densest packing of spheres of equal size in space (Kepler conjecture)? Most of these questions involved 'rigid' geometrical shapes, such as lines or spheres. Projective, convex and discrete geometry are three sub-disciplines within present day geometry that deal with these and related questions.

Leonhard Euler, in studying problems like the Seven Bridges of Königsberg, considered the most fundamental properties of geometric figures based solely on shape, independent of their metric properties. Euler called this new branch of geometry geometria situs (geometry of place), but it is now known as topology. Topology grew out of geometry, but turned into a large independent discipline. It does not differentiate between objects that can be continuously deformed into each other. The objects may nevertheless retain some geometry, as in the case of hyperbolic knots.

Geometry beyond Euclid

For nearly two thousand years since Euclid, while the range of geometrical questions asked and answered inevitably expanded, basic understanding of space remained essentially the same. Immanuel Kant argued that there is only one, absolute, geometry, which is known to be true a priori by an inner faculty of mind: Euclidean geometry was synthetic a priori.^[2] 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. A broad vision of the subject of geometry was then expressed by Riemann in his 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 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.

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