All rights reserved. Examrace is a subsidiary of Mindsprite Solutions. By visiting this website you agree to our terms & conditions. Read our copyright policy, customer and visitor privacy policy, shipping policy and return, refund and cancellation policy . For concerns or suggestions please email us at [email protected] or call +91-999-800-8851 or +91-79-40061016 (India) or +1-646-926-7223 (International). © Mindsprite Solutions 2008-2017. This website is best viewed on IE8 and above, Chrome, Opera, Firefox and Safari. Read more about Examrace or contact us. Also check out Examrace YouTube Channel for hundreds of free tution videos

Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC.[1] By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's *Elements*, set a standard for many centuries to follow.[2] Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC.[3] Islamic scientists preserved Greek ideas and expanded on them during the Middle Ages.[4] By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.

**Discrete geometry**is concerned mainly with questions of relative position of simple geometric objects, such as points, lines and circles. It shares many methods and principles with combinatorics.

Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics.

## Contents

## History

The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC.[8][9] Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts. The earliest known texts on geometry are the Egyptian *Rhind Papyrus* (2000–1800 BC) and *Moscow Papyrus* (c.

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' Theorem.[1] Pythagoras established the Pythagorean School, which is credited with the first proof of the Pythagorean theorem,[14] though the statement of the theorem has a long history[15][16]Eudoxus (408–c. 355 BC) developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures,[17] as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances.

Although most of the contents of the*Elements*were already known, Euclid arranged them into a single, coherent logical framework.[19] 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.[20]Archimedes (c. 287–212 BC) of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave remarkably accurate approximations of Pi.[21] He also studied the spiral bearing his name and obtained formulas for the volumes of surfaces of revolution.

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

Indian mathematicians also made many important contributions in geometry. The *Satapatha Brahmana* (3rd century BC) contains rules for ritual geometric constructions that are similar to the *Sulba Sutras*.

*Aryabhatiya*(499) includes the computation of areas and volumes. Brahmagupta wrote his astronomical work

*Brāhma Sphuṭa Siddhānta*in 628. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain).[25] 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.*[26][27]Al-Mahani (b. 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra.[28]Thābit ibn Qurra (known as Thebit in Latin) (836–901) dealt with arithmetic operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry.[4]Omar Khayyám (1048–1131) found geometric solutions to cubic equations.[29] The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were early results in hyperbolic geometry, and along with their alternative postulates, such as Playfair's axiom, these works had a considerable influence on the development of non-Euclidean geometry among later European geometers, including Witelo (c. 1230–c. 1314), Gersonides (1288–1344), Alfonso, John Wallis, and Giovanni Girolamo Saccheri.[30]

In the early 17th century, there were two important developments in geometry. The first was the creation of analytic geometry, or geometry with coordinates and equations, by René Descartes (1596–1650) and Pierre de Fermat (1601–1665).

Projective geometry is a geometry without measurement or parallel lines, just the study of how points are related to each other.Two developments in geometry in the 19th century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation of symmetry as the central consideration in the Erlangen Programme of Felix Klein (which generalized the Euclidean and non-Euclidean geometries). Two of the master geometers of the time were Bernhard Riemann (1826–1866), working primarily with tools from mathematical analysis, and introducing the Riemann surface, and Henri Poincaré, the founder of algebraic topology and the geometric theory of dynamical systems. As a consequence of these major changes in the conception of geometry, the concept of "space" became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics.

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 Nikolai Ivanovich Lobachevsky (1792–1856), János Bolyai (1802–1860), Carl Friedrich Gauss (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.

### Points

Points are considered fundamental objects in Euclidean geometry. They have been defined in a variety of ways, including Euclid's definition as 'that which has no part'[31] and through the use of algebra or nested sets.[32] In many areas of geometry, such as analytic geometry, differential geometry, and topology, all objects are considered to be built up from points.

[31] In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry, a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation,[34] 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.[35] In differential geometry, a geodesic is a generalization of the notion of a line to curved spaces.[36]### Planes

A plane is a flat, two-dimensional surface that extends infinitely far.[31] Planes are used in every area of geometry. For instance, planes can be studied as a topological surface without reference to distances or angles;[37] it can be studied as an affine space, where collinearity and ratios can be studied but not distances;[38] it can be studied as the complex plane using techniques of complex analysis;[39] and so on.

### Angles

Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other.

The acute and obtuse angles are also known as oblique angles.In Euclidean geometry, angles are used to study polygons and triangles, as well as forming an object of study in their own right.[31] The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry.[41]

In differential geometry and calculus, the angles between plane curves or space curves or surfaces can be calculated using the derivative.[42][43]

### 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.[44]

In topology, a curve is defined by a function from an interval of the real numbers to another space.[37] In differential geometry, the same definition is used, but the defining function is required to be differentiable [45] Algebraic geometry studies algebraic curves, which are defined as algebraic varieties of dimension one.

)A surface is a two-dimensional object, such as a sphere or paraboloid.[47] In differential geometry[45] and topology,[37] surfaces are described by two-dimensional 'patches' (or neighborhoods) that are assembled by diffeomorphisms or homeomorphisms, respectively. In algebraic geometry, surfaces are described by polynomial equations.[46]

### Manifolds

A manifold is a generalization of the concepts of curve and surface. In topology, a manifold is a topological space where every point has a neighborhood that is homeomorphic to Euclidean space.[37] In differential geometry, a differentiable manifold is a space where each neighborhood is diffeomorphic to Euclidean space.[45]

Manifolds are used extensively in physics, including in general relativity and string theory[48]

### Topologies and metrics

A topology is a mathematical structure on a set that tells how elements of the set relate spatially to each other.

Other important examples of metrics include the Lorentz metric of special relativity and the semi-Riemannian metrics of general relativity.[50]### 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 allowed in 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 parabolas and other curves, as well as mechanical devices, were found.

### 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. 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.

*a priori*.

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. Dimension 10 or 11 is a key number in string theory. Research may bring a satisfactory *geometric* reason for the significance of 10 and 11 dimensions.

### Symmetry

The theme of symmetry 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 M. C. Escher. Nonetheless, it was not until the second half of 19th century that the unifying role of symmetry in foundations of geometry was recognized.

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. Both discrete and continuous symmetries play prominent roles in geometry, the former in topology 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 (see Duality (projective geometry)) among other fields. This meta-phenomenon can roughly be described as follows: in any theorem, exchange *point* with *plane*, *join* with *meet*, *lies in* with *contains*, and you will get an equally true theorem. A similar and closely related form of duality exists between a vector space and its dual space.

### Non-Euclidean geometry

In the nearly two thousand years since Euclid, while the range of geometrical questions asked and answered inevitably expanded, the basic understanding of space remained essentially the same.

## Comments