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99-413: In geometry , a polygon ( / ˈ p ɒ l ɪ ɡ ɒ n / ) is a plane figure made up of line segments connected to form a closed polygonal chain . The segments of a closed polygonal chain are called its edges or sides . The points where two edges meet are the polygon's vertices or corners . An n -gon is a polygon with n sides; for example, a triangle is a 3-gon. A simple polygon

198-403: A cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra over the real numbers , the notion of orientation makes sense in arbitrary finite dimension, and is a kind of asymmetry that makes a reflection impossible to replicate by means of a simple displacement . Thus, in three dimensions, it is impossible to make

297-520: A geodesic is a generalization of the notion of a line to curved spaces . In Euclidean geometry a plane is a flat, two-dimensional surface that extends infinitely; the definitions for other types of geometries are generalizations of that. Planes are used in many areas of geometry. For instance, planes can be studied as a topological surface without reference to distances or angles; it can be studied as an affine space , where collinearity and ratios can be studied but not distances; it can be studied as

396-418: A parabola with the summation of an infinite series , and gave remarkably accurate approximations of pi . He also studied the spiral bearing his name and obtained formulas for the volumes of surfaces of revolution . Indian mathematicians also made many important contributions in geometry. The Shatapatha Brahmana (3rd century BC) contains rules for ritual geometric constructions that are similar to

495-425: A vector space and its dual space . Euclidean geometry is geometry in its classical sense. As it models the space of the physical world, it is used in many scientific areas, such as mechanics , astronomy , crystallography , and many technical fields, such as engineering , architecture , geodesy , aerodynamics , and navigation . The mandatory educational curriculum of the majority of nations includes

594-448: A , b ] is a one-dimensional manifold with boundary , and its boundary is the set { a , b } . In order to get the correct statement of the fundamental theorem of calculus, the point b should be oriented positively, while the point a should be oriented negatively. The one-dimensional case deals with an oriented line or directed line , which may be traversed in one of two directions. In real coordinate space , an oriented line

693-411: A basis is crucial. Two bases with a different ordering will differ by some permutation . They will have the same/opposite orientations according to whether the signature of this permutation is ±1. This is because the determinant of a permutation matrix is equal to the signature of the associated permutation. Similarly, let A be a nonsingular linear mapping of vector space R to R . This mapping

792-429: A choice of a privileged basis and therefore determines an orientation. More formally: π 0 ( GL ⁡ ( V ) ) = ( GL ⁡ ( V ) / GL + ⁡ ( V ) = { ± 1 } {\displaystyle \pi _{0}(\operatorname {GL} (V))=(\operatorname {GL} (V)/\operatorname {GL} ^{+}(V)=\{\pm 1\}} , and

891-430: A choice of one of them is an orientation. The various objects of geometric algebra are charged with three attributes or features : attitude, orientation, and magnitude. For example, a vector has an attitude given by a straight line parallel to it, an orientation given by its sense (often indicated by an arrowhead) and a magnitude given by its length. Similarly, a bivector in three dimensions has an attitude given by

990-405: A common endpoint, called the vertex of the angle. The size of an angle is formalized as an angular measure . In Euclidean geometry , angles are used to study polygons and triangles , as well as forming an object of study in their own right. The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry . In differential geometry and calculus ,

1089-523: A decimal place value system with a dot for zero." Aryabhata 's Aryabhatiya (499) includes the computation of areas and volumes. Brahmagupta wrote his astronomical work Brāhmasphuṭasiddhānta in 628. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain). In

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1188-440: A more rigorous foundation for geometry, treated congruence as an undefined term whose properties are defined by axioms . Congruence and similarity are generalized in transformation geometry , which studies the properties of geometric objects that are preserved by different kinds of transformations. Classical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically,

1287-428: A multitude of forms, including the graphics of Leonardo da Vinci , M. C. Escher , and others. In the second half of the 19th century, the relationship between symmetry and geometry came under intense scrutiny. Felix Klein 's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group , determines what geometry is . Symmetry in classical Euclidean geometry

1386-451: A number of apparently different definitions, which are all equivalent in the most common cases. 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

1485-444: A physical system, which has a dimension equal to the system's degrees of freedom . For instance, the configuration of a screw can be described by five coordinates. In general topology , the concept of dimension has been extended from natural numbers , to infinite dimension ( Hilbert spaces , for example) and positive real numbers (in fractal geometry ). In algebraic geometry , the dimension of an algebraic variety has received

1584-518: A plane or 3-dimensional space. Mathematicians have found many explicit formulas for area and formulas for volume of various geometric objects. In calculus , area and volume can be defined in terms of integrals , such as the Riemann integral or the Lebesgue integral . Other geometrical measures include the curvature and compactness . The concept of length or distance can be generalized, leading to

1683-598: A purely algebraic context. Scheme theory allowed to solve many difficult problems not only in geometry, but also in number theory . Wiles' proof of Fermat's Last Theorem is a famous example of a long-standing problem of number theory whose solution uses scheme theory and its extensions such as stack theory . One of seven Millennium Prize problems , the Hodge conjecture , is a question in algebraic geometry. Algebraic geometry has applications in many areas, including cryptography and string theory . Complex geometry studies

1782-454: A reflection by the XY Cartesian plane is not orientation-preserving: A 2 = ( 1 0 0 0 1 0 0 0 − 1 ) {\displaystyle \mathbf {A} _{2}={\begin{pmatrix}1&0&0\\0&1&0\\0&0&-1\end{pmatrix}}} The concept of orientation degenerates in

1881-401: A regular n -gon inscribed in a unit-radius circle, with side s and interior angle α , {\displaystyle \alpha ,} can also be expressed trigonometrically as: The area of a self-intersecting polygon can be defined in two different ways, giving different answers: Using the same convention for vertex coordinates as in the previous section, the coordinates of

1980-496: A simple formula for the polygon's area based on the numbers of interior and boundary grid points: the former number plus one-half the latter number, minus 1. In every polygon with perimeter p and area A , the isoperimetric inequality p 2 > 4 π A {\displaystyle p^{2}>4\pi A} holds. For any two simple polygons of equal area, the Bolyai–Gerwien theorem asserts that

2079-401: A simple polygon given by a sequence of line segments. This is called the point in polygon test. Geometry Geometry (from Ancient Greek γεωμετρία ( geōmetría )  'land measurement'; from γῆ ( gê )  'earth, land' and μέτρον ( métron )  'a measure') is a branch of mathematics concerned with properties of space such as

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2178-427: A size or measure to sets , where the measures follow rules similar to those of classical area and volume. Congruence and similarity are concepts that describe when two shapes have similar characteristics. In Euclidean geometry, similarity is used to describe objects that have the same shape, while congruence is used to describe objects that are the same in both size and shape. Hilbert , in his work on creating

2277-600: A technical sense a type of transformation geometry , in which transformations are homeomorphisms . This has often been expressed in the form of the saying 'topology is rubber-sheet geometry'. Subfields of topology include geometric topology , differential topology , algebraic topology and general topology . Algebraic geometry is fundamentally the study by means of algebraic methods of some geometrical shapes, called algebraic sets , and defined as common zeros of multivariate polynomials . Algebraic geometry became an autonomous subfield of geometry c.  1900 , with

2376-518: A theorem called Hilbert's Nullstellensatz that establishes a strong correspondence between algebraic sets and ideals of polynomial rings . This led to a parallel development of algebraic geometry, and its algebraic counterpart, called commutative algebra . From the late 1950s through the mid-1970s algebraic geometry had undergone major foundational development, with the introduction by Alexander Grothendieck of scheme theory , which allows using topological methods , including cohomology theories in

2475-494: A theory of ratios that avoided the problem of incommensurable magnitudes , which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements , widely considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of

2574-572: A zero-dimensional vector space is the same as a zero-dimensional vector space with ordered basis. Choosing { ∅ } ↦ + 1 {\displaystyle \{\emptyset \}\mapsto +1} or { ∅ } ↦ − 1 {\displaystyle \{\emptyset \}\mapsto -1} therefore chooses an orientation of every basis of every zero-dimensional vector space. If all zero-dimensional vector spaces are assigned this orientation, then, because all isomorphisms among zero-dimensional vector spaces preserve

2673-411: Is diffeomorphic to Euclidean space. Manifolds are used extensively in physics, including in general relativity and string theory . 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. In modern terms, an angle is the figure formed by two rays , called the sides of the angle, sharing

2772-613: Is orientation-preserving if its determinant is positive. For instance, in R a rotation around the Z Cartesian axis by an angle α is orientation-preserving: A 1 = ( cos ⁡ α − sin ⁡ α 0 sin ⁡ α cos ⁡ α 0 0 0 1 ) {\displaystyle \mathbf {A} _{1}={\begin{pmatrix}\cos \alpha &-\sin \alpha &0\\\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}} while

2871-458: Is a 2-dimensional example of the more general polytope in any number of dimensions. There are many more generalizations of polygons defined for different purposes. The word polygon derives from the Greek adjective πολύς ( polús ) 'much', 'many' and γωνία ( gōnía ) 'corner' or 'angle'. It has been suggested that γόνυ ( gónu ) 'knee' may be the origin of gon . Polygons are primarily classified by

2970-400: Is a part of some ambient flat Euclidean space). Topology is the field concerned with the properties of continuous mappings , and can be considered a generalization of Euclidean geometry. In practice, topology often means dealing with large-scale properties of spaces, such as connectedness and compactness . The field of topology, which saw massive development in the 20th century, is in

3069-408: Is a standard result in linear algebra that there exists a unique linear transformation A  : V → V that takes b 1 to b 2 . The bases b 1 and b 2 are said to have the same orientation (or be consistently oriented) if A has positive determinant ; otherwise they have opposite orientations . The property of having the same orientation defines an equivalence relation on

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3168-413: Is a three-dimensional object bounded by a closed surface; for example, a ball is the volume bounded by a sphere. 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. In differential geometry , a differentiable manifold is a space where each neighborhood

3267-422: Is also known as an axis . There are two orientations to a line just as there are two orientations to an oriented circle (clockwise and anti-clockwise). A semi-infinite oriented line is called a ray . In the case of a line segment (a connected subset of a line), the two possible orientations result in directed line segments . An orientable surface sometimes has the selected orientation indicated by

3366-533: Is an n -form we can evaluate it on an ordered set of n vectors, giving an element of R ). The form ω is called an orientation form . If { e i } is a privileged basis for V and { e i } is the dual basis , then the orientation form giving the standard orientation is e 1 ∧ e 2 ∧ … ∧ e n . The connection of this with the determinant point of view is: the determinant of an endomorphism T : V → V {\displaystyle T:V\to V} can be interpreted as

3465-437: Is commonly called the shoelace formula or surveyor's formula . The area A of a simple polygon can also be computed if the lengths of the sides, a 1 , a 2 , ..., a n and the exterior angles , θ 1 , θ 2 , ..., θ n are known, from: The formula was described by Lopshits in 1963. If the polygon can be drawn on an equally spaced grid such that all its vertices are grid points, Pick's theorem gives

3564-459: Is concerned only with simple and solid polygons, a polygon may refer only to a simple polygon or to a solid polygon. A polygonal chain may cross over itself, creating star polygons and other self-intersecting polygons . Some sources also consider closed polygonal chains in Euclidean space to be a type of polygon (a skew polygon ), even when the chain does not lie in a single plane. A polygon

3663-409: Is defined. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying , construction , astronomy , and various crafts. The earliest known texts on geometry are

3762-424: Is just a real line. There is no a priori choice of which direction on this line is positive. An orientation is just such a choice. Any nonzero linear form ω on Λ V determines an orientation of V by declaring that x is in the positive direction when ω ( x ) > 0. To connect with the basis point of view we say that the positively-oriented bases are those on which ω evaluates to a positive number (since ω

3861-451: Is large, this approaches one half. Or, each vertex inside the square mesh connects four edges (lines). The imaging system calls up the structure of polygons needed for the scene to be created from the database. This is transferred to active memory and finally, to the display system (screen, TV monitors etc.) so that the scene can be viewed. During this process, the imaging system renders polygons in correct perspective ready for transmission of

3960-437: Is not viewed as the set of the points through which it passes. However, there are modern geometries in which points are not primitive objects, or even without points. One of the oldest such geometries is Whitehead's point-free geometry , formulated by Alfred North Whitehead in 1919–1920. Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself". In modern mathematics, given

4059-415: Is of importance to mathematical physics due to Albert Einstein 's general relativity postulation that the universe is curved . Differential geometry can either be intrinsic (meaning that the spaces it considers are smooth manifolds whose geometric structure is governed by a Riemannian metric , which determines how distances are measured near each point) or extrinsic (where the object under study

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4158-429: Is one which does not intersect itself. More precisely, the only allowed intersections among the line segments that make up the polygon are the shared endpoints of consecutive segments in the polygonal chain. A simple polygon is the boundary of a region of the plane that is called a solid polygon . The interior of a solid polygon is its body , also known as a polygonal region or polygonal area . In contexts where one

4257-442: Is regular (and therefore cyclic). Many specialized formulas apply to the areas of regular polygons . The area of a regular polygon is given in terms of the radius r of its inscribed circle and its perimeter p by This radius is also termed its apothem and is often represented as a . The area of a regular n -gon in terms of the radius R of its circumscribed circle can be expressed trigonometrically as: The area of

4356-482: Is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations , geometric transformations that take straight lines into straight lines. However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group ' found its inspiration. Both discrete and continuous symmetries play prominent roles in geometry,

4455-440: Is the empty set. This means that an orientation of a zero-dimensional space is a function { { ∅ } } → { ± 1 } . {\displaystyle \{\{\emptyset \}\}\to \{\pm 1\}.} It is therefore possible to orient a point in two different ways, positive and negative. Because there is only a single ordered basis ∅ {\displaystyle \emptyset } ,

4554-726: The Sulba Sutras . According to ( Hayashi 2005 , p. 363), the Śulba Sūtras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. They contain lists of Pythagorean triples , which are particular cases of Diophantine equations . In the Bakhshali manuscript , there are a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs

4653-667: The Egyptian Rhind Papyrus (2000–1800 BC) and Moscow Papyrus ( c.  1890 BC ), and the Babylonian clay tablets , such as Plimpton 322 (1900 BC). For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum . Later clay tablets (350–50 BC) demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space. These geometric procedures anticipated

4752-613: The Giant's Causeway in Northern Ireland , or at the Devil's Postpile in California . In biology , the surface of the wax honeycomb made by bees is an array of hexagons , and the sides and base of each cell are also polygons. In computer graphics , a polygon is a primitive used in modelling and rendering. They are defined in a database, containing arrays of vertices (the coordinates of

4851-518: The Lambert quadrilateral and Saccheri quadrilateral , were part of a line of research on the parallel postulate continued by later European geometers, including Vitello ( c.  1230  – c.  1314 ), Gersonides (1288–1344), Alfonso, John Wallis , and Giovanni Girolamo Saccheri , that by the 19th century led to the discovery of hyperbolic geometry . In the early 17th century, there were two important developments in geometry. The first

4950-459: The Oxford Calculators , including the mean speed theorem , by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with

5049-509: 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 . The following are some of the most important concepts in geometry. Euclid took an abstract approach to geometry in his Elements , one of

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5148-533: The Stiefel manifold of n -frames in V {\displaystyle V} is a GL ⁡ ( V ) {\displaystyle \operatorname {GL} (V)} - torsor , so V n ( V ) / GL + ⁡ ( V ) {\displaystyle V_{n}(V)/\operatorname {GL} ^{+}(V)} is a torsor over { ± 1 } {\displaystyle \{\pm 1\}} , i.e., its 2 points, and

5247-399: The complex plane using techniques of complex analysis ; and so on. A curve is a 1-dimensional object that may be straight (like a line) or not; curves in 2-dimensional space are called plane curves and those in 3-dimensional space are called space curves . In topology, a curve is defined by a function from an interval of the real numbers to another space. In differential geometry,

5346-452: The geometrical vertices , as well as other attributes of the polygon, such as color, shading and texture), connectivity information, and materials . Any surface is modelled as a tessellation called polygon mesh . If a square mesh has n + 1 points (vertices) per side, there are n squared squares in the mesh, or 2 n squared triangles since there are two triangles in a square. There are ( n + 1) / 2( n ) vertices per triangle. Where n

5445-486: The regular star pentagon is also known as the pentagram . To construct the name of a polygon with more than 20 and fewer than 100 edges, combine the prefixes as follows. The "kai" term applies to 13-gons and higher and was used by Kepler , and advocated by John H. Conway for clarity of concatenated prefix numbers in the naming of quasiregular polyhedra , though not all sources use it. Polygons have been known since ancient times. The regular polygons were known to

5544-621: 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

5643-491: The 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss 's Theorema Egregium ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space . This implies that surfaces can be studied intrinsically , that is, as stand-alone spaces, and has been expanded into

5742-474: 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 are generally considered fundamental objects for building geometry. They may be defined by

5841-523: The ancient Greeks, with the pentagram , a non-convex regular polygon ( star polygon ), appearing as early as the 7th century B.C. on a krater by Aristophanes , found at Caere and now in the Capitoline Museum . The first known systematic study of non-convex polygons in general was made by Thomas Bradwardine in the 14th century. In 1952, Geoffrey Colin Shephard generalized the idea of polygons to

5940-584: The angles between plane curves or space curves or surfaces can be calculated using the derivative . Length , area , and volume describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively. In Euclidean geometry and analytic geometry , the length of a line segment can often be calculated by the Pythagorean theorem . Area and volume can be defined as fundamental quantities separate from length, or they can be described and calculated in terms of lengths in

6039-453: The centroid of a solid simple polygon are In these formulas, the signed value of area A {\displaystyle A} must be used. For triangles ( n = 3 ), the centroids of the vertices and of the solid shape are the same, but, in general, this is not true for n > 3 . The centroid of the vertex set of a polygon with n vertices has the coordinates The idea of a polygon has been generalized in various ways. Some of

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6138-425: The complex plane, where each real dimension is accompanied by an imaginary one, to create complex polygons . Polygons appear in rock formations, most commonly as the flat facets of crystals , where the angles between the sides depend on the type of mineral from which the crystal is made. Regular hexagons can occur when the cooling of lava forms areas of tightly packed columns of basalt , which may be seen at

6237-412: The concept of angle and distance, finite geometry that omits continuity , and others. This enlargement of the scope of geometry led to a change of meaning of the word "space", which originally referred to the three-dimensional space of the physical world and its model provided by Euclidean geometry; presently a geometric space , or simply a space is a mathematical structure on which some geometry

6336-409: The connected components of B . These orbits are precisely the equivalence classes referred to above. Since B does not have a distinguished element (i.e. a privileged basis) there is no natural choice of which component is positive. Contrast this with GL( V ) which does have a privileged component: the component of the identity. A specific choice of homeomorphism between B and GL( V ) is equivalent to

6435-504: The contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework. The Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes ( c.  287–212 BC ) of Syracuse, Italy used the method of exhaustion to calculate the area under the arc of

6534-448: The determinant of the transformation is positive or negative (except for GL 0 , which is the trivial group and thus has a single connected component; this corresponds to the canonical orientation on a zero-dimensional vector space). The identity component of GL( V ) is denoted GL ( V ) and consists of those transformations with positive determinant. The action of GL ( V ) on B is not transitive: there are two orbits which correspond to

6633-468: The distance, shape, size, and relative position of figures. Geometry is, along with arithmetic , one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer . Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry , which includes the notions of point , line , plane , distance , angle , surface , and curve , as fundamental concepts. Originally developed to model

6732-407: The family of planes associated with it (possibly specified by the normal line common to these planes ), an orientation (sometimes denoted by a curved arrow in the plane) indicating a choice of sense of traversal of its boundary (its circulation ), and a magnitude given by the area of the parallelogram defined by its two vectors. Each point p on an n -dimensional differentiable manifold has

6831-428: The field has been split in many subfields that depend on the underlying methods— differential geometry , algebraic geometry , computational geometry , algebraic topology , discrete geometry (also known as combinatorial geometry ), etc.—or on the properties of Euclidean spaces that are disregarded— projective geometry that consider only alignment of points but not distance and parallelism, affine geometry that omits

6930-410: The first can be cut into polygonal pieces which can be reassembled to form the second polygon. The lengths of the sides of a polygon do not in general determine its area. However, if the polygon is simple and cyclic then the sides do determine the area. Of all n -gons with given side lengths, the one with the largest area is cyclic. Of all n -gons with a given perimeter, the one with the largest area

7029-512: The first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales's theorem . Pythagoras established the Pythagorean School , which is credited with the first proof of the Pythagorean theorem , though the statement of the theorem has a long history. Eudoxus (408– c.  355 BC ) developed the method of exhaustion , which allowed the calculation of areas and volumes of curvilinear figures, as well as

7128-523: 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 , among other fields. This meta-phenomenon can roughly be described as follows: in any theorem , exchange point with plane , join with meet , lies in with contains , and the result is an equally true theorem. A similar and closely related form of duality exists between

7227-537: The idea of metrics . For instance, the Euclidean metric measures the distance between points in the Euclidean plane , while the hyperbolic metric measures the distance in the hyperbolic plane . Other important examples of metrics include the Lorentz metric of special relativity and the semi- Riemannian metrics of general relativity . In a different direction, the concepts of length, area and volume are extended by measure theory , which studies methods of assigning

7326-533: The idea of reducing geometrical problems such as duplicating the cube to problems in algebra. 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 . Omar Khayyam (1048–1131) found geometric solutions to cubic equations . The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals , including

7425-418: The induced action on the top exterior power. Let B be the set of all ordered bases for V . Then the general linear group GL( V ) acts freely and transitively on B . (In fancy language, B is a GL( V )- torsor ). This means that as a manifold , B is (noncanonically) homeomorphic to GL( V ). Note that the group GL( V ) is not connected , but rather has two connected components according to whether

7524-546: The latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral . Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of Heron's formula ), as well as a complete description of rational triangles ( i.e. triangles with rational sides and rational areas). In the Middle Ages , mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry . Al-Mahani (b. 853) conceived

7623-453: The left hand of a human figure into the right hand of the figure by applying a displacement alone, but it is possible to do so by reflecting the figure in a mirror. As a result, in the three-dimensional Euclidean space , the two possible basis orientations are called right-handed and left-handed (or right-chiral and left-chiral). Let V be a finite-dimensional real vector space and let b 1 and b 2 be two ordered bases for V . It

7722-832: The more important include: The word polygon comes from Late Latin polygōnum (a noun), from Greek πολύγωνον ( polygōnon/polugōnon ), noun use of neuter of πολύγωνος ( polygōnos/polugōnos , the masculine adjective), meaning "many-angled". Individual polygons are named (and sometimes classified) according to the number of sides, combining a Greek -derived numerical prefix with the suffix -gon , e.g. pentagon , dodecagon . The triangle , quadrilateral and nonagon are exceptions. Beyond decagons (10-sided) and dodecagons (12-sided), mathematicians generally use numerical notation, for example 17-gon and 257-gon. Exceptions exist for side counts that are easily expressed in verbal form (e.g. 20 and 30), or are used by non-mathematicians. Some special polygons also have their own names; for example

7821-411: 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

7920-429: 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 , but in a more abstract setting, such as incidence geometry , a line may be an independent object, distinct from the set of points which lie on it. In differential geometry,

8019-435: The nature of geometric structures modelled on, or arising out of, the complex plane . Complex geometry lies at the intersection of differential geometry, algebraic geometry, and analysis of several complex variables , and has found applications to string theory and mirror symmetry . Orientation (vector space) In mathematics , orientability is a broader notion that, in two dimensions, allows one to say when

8118-550: The notation ( x n , y n ) = ( x 0 , y 0 ) will also be used. If the polygon is non-self-intersecting (that is, simple ), the signed area is or, using determinants where Q i , j {\displaystyle Q_{i,j}} is the squared distance between ( x i , y i ) {\displaystyle (x_{i},y_{i})} and ( x j , y j ) . {\displaystyle (x_{j},y_{j}).} The signed area depends on

8217-525: The number of sides. Polygons may be characterized by their convexity or type of non-convexity: The property of regularity may be defined in other ways: a polygon is regular if and only if it is both isogonal and isotoxal, or equivalently it is both cyclic and equilateral. A non-convex regular polygon is called a regular star polygon . Euclidean geometry is assumed throughout. Any polygon has as many corners as it has sides. Each corner has several angles. The two most important ones are: In this section,

8316-441: The only instruments used in most geometric constructions are the compass and straightedge . Also, every construction had to be complete in a finite number of steps. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using neusis , parabolas and other curves, or mechanical devices, were found. The geometrical concepts of rotation and orientation define part of

8415-434: The ordered basis, they also preserve the orientation. This is unlike the case of higher-dimensional vector spaces where there is no way to choose an orientation so that it is preserved under all isomorphisms. However, there are situations where it is desirable to give different orientations to different points. For example, consider the fundamental theorem of calculus as an instance of Stokes' theorem . A closed interval [

8514-423: The ordering of the vertices and of the orientation of the plane. Commonly, the positive orientation is defined by the (counterclockwise) rotation that maps the positive x -axis to the positive y -axis. If the vertices are ordered counterclockwise (that is, according to positive orientation), the signed area is positive; otherwise, it is negative. In either case, the area formula is correct in absolute value . This

8613-446: The orientation of a surface normal . An oriented plane can be defined by a pseudovector . For any n -dimensional real vector space V we can form the k th- exterior power of V , denoted Λ V . This is a real vector space of dimension ( n k ) {\displaystyle {\tbinom {n}{k}}} . The vector space Λ V (called the top exterior power ) therefore has dimension 1. That is, Λ V

8712-510: The physical world, geometry has applications in almost all sciences, and also in art, architecture , and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem , a problem that was stated in terms of elementary arithmetic , and remained unsolved for several centuries. During

8811-407: The placement of objects embedded in the plane or in space. Traditional geometry allowed dimensions 1 (a line or curve), 2 (a plane or surface), and 3 (our ambient world conceived of as three-dimensional space ). Furthermore, mathematicians and physicists have used higher dimensions for nearly two centuries. One example of a mathematical use for higher dimensions is the configuration space of

8910-450: The privileged basis is declared to be positive. For example, the standard basis on R provides a standard orientation on R (in turn, the orientation of the standard basis depends on the orientation of the Cartesian coordinate system on which it is built). Any choice of a linear isomorphism between V and R will then provide an orientation on V . The ordering of elements in

9009-426: The processed data to the display system. Although polygons are two-dimensional, through the system computer they are placed in a visual scene in the correct three-dimensional orientation. In computer graphics and computational geometry , it is often necessary to determine whether a given point P = ( x 0 , y 0 ) {\displaystyle P=(x_{0},y_{0})} lies inside

9108-478: The properties that they must have, as in Euclid's definition as "that which has no part", or in synthetic geometry . In modern mathematics, they are generally defined as elements of a set called space , which is itself axiomatically defined. With these modern definitions, every geometric shape is defined as a set of points; this is not the case in synthetic geometry, where a line is another fundamental object that

9207-554: The same definition is used, but the defining function is required to be differentiable. Algebraic geometry studies algebraic curves , which are defined as algebraic varieties of dimension one. A surface is a two-dimensional object, such as a sphere or paraboloid. In differential geometry and topology , 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 . A solid

9306-403: The set of all ordered bases for V . If V is non-zero, there are precisely two equivalence classes determined by this relation. An orientation on V is an assignment of +1 to one equivalence class and −1 to the other. Every ordered basis lives in one equivalence class or another. Thus any choice of a privileged ordered basis for V determines an orientation: the orientation class of

9405-589: The study of Euclidean concepts such as points , lines , planes , angles , triangles , congruence , similarity , solid figures , circles , and analytic geometry . Euclidean vectors are used for a myriad of applications in physics and engineering, such as position , displacement , deformation , velocity , acceleration , force , etc. Differential geometry uses techniques of calculus and linear algebra to study problems in geometry. It has applications in physics , econometrics , and bioinformatics , among others. In particular, differential geometry

9504-409: The theory of manifolds and Riemannian geometry . Later in the 19th century, it appeared that geometries without the parallel postulate ( non-Euclidean geometries ) can be developed without introducing any contradiction. The geometry that underlies general relativity is a famous application of non-Euclidean geometry. Since the late 19th century, the scope of geometry has been greatly expanded, and

9603-417: The vertices of the polygon under consideration are taken to be ( x 0 , y 0 ) , ( x 1 , y 1 ) , … , ( x n − 1 , y n − 1 ) {\displaystyle (x_{0},y_{0}),(x_{1},y_{1}),\ldots ,(x_{n-1},y_{n-1})} in order. For convenience in some formulas,

9702-423: The zero-dimensional case. A zero-dimensional vector space has only a single point, the zero vector. Consequently, the only basis of a zero-dimensional vector space is the empty set ∅ {\displaystyle \emptyset } . Therefore, there is a single equivalence class of ordered bases, namely, the class { ∅ } {\displaystyle \{\emptyset \}} whose sole member

9801-596: Was the creation of analytic geometry, or geometry with coordinates and equations , by René Descartes (1596–1650) and Pierre de Fermat (1601–1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics . The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry studies properties of shapes which are unchanged under projections and sections , especially as they relate to artistic perspective . Two developments in geometry in

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