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106-539: In geometry , a trapezoid ( / ˈ t r æ p ə z ɔɪ d / ) in North American English , or trapezium ( / t r ə ˈ p iː z i ə m / ) in British English , is a quadrilateral that has one pair of parallel sides. The parallel sides are called the bases of the trapezoid. The other two sides are called the legs (or the lateral sides ) if they are not parallel; otherwise,

212-404: A = 0 {\displaystyle d-c=b-a=0} , but it is an ex-tangential quadrilateral (which is not a trapezoid) when | d − c | = | b − a | ≠ 0 {\displaystyle |d-c|=|b-a|\neq 0} . Given a convex quadrilateral, the following properties are equivalent, and each implies that the quadrilateral

318-416: A across the point p is In the case where p is the origin, point reflection is simply the negation of the vector a . In Euclidean geometry , the inversion of a point X with respect to a point P is a point X * such that P is the midpoint of the line segment with endpoints X and X *. In other words, the vector from X to P is the same as the vector from P to X *. The formula for

424-520: 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

530-418: 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

636-471: A rotation of 180 degrees. In three dimensions, a point reflection can be described as a 180-degree rotation composed with reflection across the plane of rotation, perpendicular to the axis of rotation. In dimension n , point reflections are orientation -preserving if n is even, and orientation-reversing if n is odd. Given a vector a in the Euclidean space R , the formula for the reflection of

742-417: A special case the well-known formula for the area of a triangle , by considering a triangle as a degenerate trapezoid in which one of the parallel sides has shrunk to a point. The 7th-century Indian mathematician Bhāskara I derived the following formula for the area of a trapezoid with consecutive sides a , c , b , d : where a and b are parallel and b > a . This formula can be factored into

848-425: 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

954-537: A branch of mathematics concerned with properties of space such as 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

1060-405: 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 ,

1166-523: 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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1272-409: A hyperplane has a single −1 eigenvalue (and multiplicity n − 1 {\displaystyle n-1} on the 1 eigenvalue), while point reflection has only the −1 eigenvalue (with multiplicity n ). The term inversion should not be confused with inversive geometry , where inversion is defined with respect to a circle. In two dimensions, a point reflection is the same as

1378-440: 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,

1484-399: A more symmetric version When one of the parallel sides has shrunk to a point (say a = 0), this formula reduces to Heron's formula for the area of a triangle. Another equivalent formula for the area, which more closely resembles Heron's formula, is where s = 1 2 ( a + b + c + d ) {\displaystyle s={\tfrac {1}{2}}(a+b+c+d)}

1590-428: 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

1696-451: 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

1802-444: 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

1908-528: 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

2014-619: A point is reflection in respect to a plane , which can be thought of as a "inversion in a plane". Inversion symmetry plays a major role in the properties of materials, as also do other symmetry operations. Some molecules contain an inversion center when a point exists through which all atoms can reflect while retaining symmetry. In many cases they can be considered as polyhedra, categorized by their coordination number and bond angles. For example, four-coordinate polyhedra are classified as tetrahedra , while five-coordinate environments can be square pyramidal or trigonal bipyramidal depending on

2120-468: A point reflection among its symmetries is called centrosymmetric . Inversion symmetry is found in many crystal structures and molecules , and has a major effect upon their physical properties. The term reflection is loose, and considered by some an abuse of language, with inversion preferred; however, point reflection is widely used. Such maps are involutions , meaning that they have order 2 – they are their own inverse: applying them twice yields

2226-602: 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

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2332-483: A repetition of an atomic building block known as a unit cell, and these unit cells define which polyhedra form and in what order. In many materials such as oxides these polyhedra can link together via corner-, edge- or face sharing, depending on which atoms share common bonds and also the valence. In other cases such as for metals and alloys the structures are better considered as arrangements of close-packed atoms. Crystals which do not have inversion symmetry also display

2438-401: A single output based on a select signal. Typical designs will employ trapezoids without specifically stating they are multiplexors as they are universally equivalent. Geometry Geometry (from Ancient Greek γεωμετρία ( geōmetría )  'land measurement'; from γῆ ( gê )  'earth, land' and μέτρον ( métron )  'a measure') is

2544-427: 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

2650-622: A strong influence on the optical properties; for instance molecules without inversion symmetry have a dipole moment and can directly interact with photons, while those with inversion have no dipole moment and only interact via Raman scattering . The later is named after C. V. Raman who was awarded the 1930 Nobel Prize in Physics for his discovery. In addition, in crystallography , the presence of inversion centers for periodic structures distinguishes between centrosymmetric and non-centrosymmetric compounds. All crystalline compounds come from

2756-600: 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

2862-518: 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

2968-494: 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

3074-537: A transposition of the terms. This was reversed in British English in about 1875, but it has been retained in American English to the present. The following table compares usages, with the most specific definitions at the top to the most general at the bottom. There is some disagreement whether parallelograms , which have two pairs of parallel sides, should be regarded as trapezoids. Some define

3180-433: A trapezoid as a quadrilateral having only one pair of parallel sides (the exclusive definition), thereby excluding parallelograms. Some sources use the term proper trapezoid to describe trapezoids under the exclusive definition, analogous to uses of the word proper in some other mathematical objects. Others define a trapezoid as a quadrilateral with at least one pair of parallel sides (the inclusive definition), making

3286-484: A trapezoid is given by where a and b are the lengths of the parallel sides, h is the height (the perpendicular distance between these sides), and m is the arithmetic mean of the lengths of the two parallel sides. In 499 AD Aryabhata , a great mathematician - astronomer from the classical age of Indian mathematics and Indian astronomy , used this method in the Aryabhatiya (section 2.8). This yields as

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3392-411: 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

3498-421: Is a longest element of the orthogonal group, with respect to the generating set of reflections: elements of the orthogonal group all have length at most n with respect to the generating set of reflections, and reflection through the origin has length n, though it is not unique in this: other maximal combinations of rotations (and possibly reflections) also have maximal length. In SO(2 r ), reflection through

3604-400: 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

3710-413: 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

3816-440: Is a trapezoid: Additionally, the following properties are equivalent, and each implies that opposite sides a and b are parallel: The midsegment of a trapezoid is the segment that joins the midpoints of the legs. It is parallel to the bases. Its length m is equal to the average of the lengths of the bases a and b of the trapezoid, The midsegment of a trapezoid is one of the two bimedians (the other bimedian divides

3922-443: Is an example of linear transformation . When P does not coincide with the origin, point reflection is equivalent to a special case of homothetic transformation : homothety with homothetic center coinciding with P, and scale factor −1. (This is an example of non-linear affine transformation .) The composition of two point reflections is a translation . Specifically, point reflection at p followed by point reflection at q

4028-409: 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

4134-438: Is equivalent to N rotations over π in each plane of an arbitrary set of N mutually orthogonal planes intersecting at P , combined with the reflection in the 2 N -dimensional subspace spanned by these rotation planes. Therefore, it reverses rather than preserves orientation , it is an indirect isometry . Geometrically in 3D it amounts to rotation about an axis through P by an angle of 180°, combined with reflection in

4240-437: 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

4346-415: 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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4452-431: Is possible for acute trapezoids or right trapezoids (as rectangles). A parallelogram is (under the inclusive definition) a trapezoid with two pairs of parallel sides. A parallelogram has central 2-fold rotational symmetry (or point reflection symmetry). It is possible for obtuse trapezoids or right trapezoids (rectangles). A tangential trapezoid is a trapezoid that has an incircle . A Saccheri quadrilateral

4558-482: 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,

4664-497: Is similar to a trapezoid in the hyperbolic plane, with two adjacent right angles, while it is a rectangle in the Euclidean plane . A Lambert quadrilateral in the hyperbolic plane has 3 right angles. Four lengths a , c , b , d can constitute the consecutive sides of a non-parallelogram trapezoid with a and b parallel only when The quadrilateral is a parallelogram when d − c = b −

4770-430: Is the identity matrix . In three dimensions, this sends ( x , y , z ) ↦ ( − x , − y , − z ) {\displaystyle (x,y,z)\mapsto (-x,-y,-z)} , and so forth. As a scalar matrix , it is represented in every basis by a matrix with − 1 {\displaystyle -1} on the diagonal, and, together with

4876-410: Is the semiperimeter of the trapezoid. (This formula is similar to Brahmagupta's formula , but it differs from it, in that a trapezoid might not be cyclic (inscribed in a circle). The formula is also a special case of Bretschneider's formula for a general quadrilateral ). From Bretschneider's formula, it follows that The bimedian connecting the parallel sides bisects the area. The lengths of

4982-418: Is the full isometry group of the line. Given the point P ( x , y ) {\displaystyle P(x,y)} and its reflection P ′ ( x ′ , y ′ ) {\displaystyle P'(x',y')} with respect to the point C ( x c , y c ) {\displaystyle C(x_{c},y_{c})} ,

5088-636: Is the problem of finding the distance between the parallel sides of a right trapezoid, given the diagonal lengths and the distance from the perpendicular leg to the diagonal intersection. In morphology , taxonomy and other descriptive disciplines in which a term for such shapes is necessary, terms such as trapezoidal or trapeziform commonly are useful in descriptions of particular organs or forms. In computer engineering, specifically digital logic and computer architecture, trapezoids are typically utilized to symbolize multiplexors . Multiplexors are logic elements that select between multiple elements and produce

5194-413: Is translation by the vector 2( q  − p ). The set consisting of all point reflections and translations is Lie subgroup of the Euclidean group . It is a semidirect product of R with a cyclic group of order 2, the latter acting on R by negation. It is precisely the subgroup of the Euclidean group that fixes the line at infinity pointwise. In the case n = 1, the point reflection group

5300-753: 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

5406-690: 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

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5512-523: 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

5618-518: 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

5724-509: 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

5830-399: 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,

5936-430: The identity map – which is also true of other maps called reflections . More narrowly, a reflection refers to a reflection in a hyperplane ( n − 1 {\displaystyle n-1} dimensional affine subspace – a point on the line , a line in the plane , a plane in 3-space), with the hyperplane being fixed, but more broadly reflection is applied to any involution of Euclidean space, and

6042-413: The piezoelectric effect . The presence or absence of inversion symmetry also has numerous consequences for the properties of solids, as does the mathematical relationships between the different crystal symmetries. Real polyhedra in crystals often lack the uniformity anticipated in their bonding geometry. Common irregularities found in crystallography include distortions and disorder. Distortion involves

6148-414: The trapezoidal rule for estimating areas under a curve. An acute trapezoid has two adjacent acute angles on its longer base edge. An obtuse trapezoid on the other hand has one acute and one obtuse angle on each base . An isosceles trapezoid is a trapezoid where the base angles have the same measure. As a consequence the two legs are also of equal length and it has reflection symmetry . This

6254-574: 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

6360-496: 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

6466-474: 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

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6572-533: The angle bisectors to angles C and D intersect at Q , then In architecture the word is used to refer to symmetrical doors, windows, and buildings built wider at the base, tapering toward the top, in Egyptian style. If these have straight sides and sharp angular corners, their shapes are usually isosceles trapezoids . This was the standard style for the doors and windows of the Inca . The crossed ladders problem

6678-590: 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

6784-576: The bonding angles. Six-coordinate octahedra are an example of centrosymmetric polyhedra, as the central atom acts as an inversion center through which the six bonded atoms retain symmetry. Tetrahedra, on the other hand, are non-centrosymmetric as an inversion through the central atom would result in a reversal of the polyhedron. Polyhedra with an odd (versus even) coordination number are not centrosymmtric. Polyhedra containing inversion centers are known as centrosymmetric, while those without are non-centrosymmetric. The presence or absence of an inversion center has

6890-531: The centrosymmetry of certain polyhedra as well, depending on whether or not the occupancy is split over an already-present inversion center. Centrosymmetry applies to the crystal structure as a whole, not just individual polyhedra. Crystals are classified into thirty-two crystallographic point groups which describe how the different polyhedra arrange themselves in space in the bulk structure. Of these thirty-two point groups, eleven are centrosymmetric. The presence of noncentrosymmetric polyhedra does not guarantee that

6996-412: 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

7102-513: 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

7208-419: The diagonals are where a is the short base, b is the long base, and c and d are the trapezoid legs. If the trapezoid is divided into four triangles by its diagonals AC and BD (as shown on the right), intersecting at O , then the area of △ {\displaystyle \triangle } AOD is equal to that of △ {\displaystyle \triangle } BOC , and

7314-546: The element − 1 ∈ S p i n ( n ) {\displaystyle -1\in \mathrm {Spin} (n)} in the spin group . This is particularly confusing for even spin groups, as − I ∈ S O ( 2 n ) {\displaystyle -I\in SO(2n)} , and thus in Spin ⁡ ( n ) {\displaystyle \operatorname {Spin} (n)} there

7420-428: 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

7526-520: 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

7632-427: The fixed set (an affine space of dimension k , where 1 ≤ k ≤ n − 1 {\displaystyle 1\leq k\leq n-1} ) is called the mirror . In dimension 1 these coincide, as a point is a hyperplane in the line. In terms of linear algebra, assuming the origin is fixed, involutions are exactly the diagonalizable maps with all eigenvalues either 1 or −1. Reflection in

7738-526: 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

7844-542: 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

7950-537: 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

8056-624: The identity, is the center of the orthogonal group O ( n ) {\displaystyle O(n)} . It is a product of n orthogonal reflections (reflection through the axes of any orthogonal basis ); note that orthogonal reflections commute. In 2 dimensions, it is in fact rotation by 180 degrees, and in dimension 2 n {\displaystyle 2n} , it is rotation by 180 degrees in n orthogonal planes; note again that rotations in orthogonal planes commute. It has determinant ( − 1 ) n {\displaystyle (-1)^{n}} (from

8162-406: The inherent geometry of the polyhedra—a distorted octahedron is still classified as an octahedron, but strong enough distortions can have an effect on the centrosymmetry of a compound. Disorder involves a split occupancy over two or more sites, in which an atom will occupy one crystallographic position in a certain percentage of polyhedra and the other in the remaining positions. Disorder can influence

8268-428: The intersection point of the diagonals, bisects each base. The center of area (center of mass for a uniform lamina ) lies along the line segment joining the midpoints of the parallel sides, at a perpendicular distance x from the longer side b given by The center of area divides this segment in the ratio (when taken from the short to the long side) If the angle bisectors to angles A and B intersect at P , and

8374-413: The inversion in P is where p , x and x * are the position vectors of P , X and X * respectively. This mapping is an isometric involutive affine transformation which has exactly one fixed point , which is P . When the inversion point P coincides with the origin, point reflection is equivalent to a special case of uniform scaling : uniform scaling with scale factor equal to −1. This

8480-411: The inversion in a point P is equivalent to N rotations over angles π in each plane of an arbitrary set of N mutually orthogonal planes intersecting at P . These rotations are mutually commutative. Therefore, inversion in a point in even-dimensional space is an orientation-preserving isometry or direct isometry . In odd-dimensional Euclidean space , say (2 N  + 1)-dimensional space, it

8586-513: The last did not have two sets of parallel sides – a τραπέζια ( trapezia literally 'table', itself from τετράς ( tetrás ) 'four' + πέζα ( péza ) 'foot; end, border, edge'). Two types of trapezia were introduced by Proclus (AD 412 to 485) in his commentary on the first book of Euclid's Elements : All European languages follow Proclus's structure as did English until the late 18th century, until an influential mathematical dictionary published by Charles Hutton in 1795 supported without explanation

8692-455: The latter is the midpoint of the segment P P ′ ¯ {\displaystyle {\overline {PP'}}} ; Hence, the equations to find the coordinates of the reflected point are Particular is the case in which the point C has coordinates ( 0 , 0 ) {\displaystyle (0,0)} (see the paragraph below ) In even-dimensional Euclidean space , say 2 N -dimensional space,

8798-552: 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

8904-432: The lengths of the parallel sides. Let the trapezoid have vertices A , B , C , and D in sequence and have parallel sides AB and DC . Let E be the intersection of the diagonals, and let F be on side DA and G be on side BC such that FEG is parallel to AB and CD . Then FG is the harmonic mean of AB and DC : The line that goes through both the intersection point of the extended nonparallel sides and

9010-411: 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

9116-429: 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,

9222-494: 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 . Point reflection An object that is invariant under a point reflection is said to possess point symmetry (also called inversion symmetry or central symmetry ). A point group including

9328-441: 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

9434-405: The origin corresponds to additive inversion of the position vector, and also to scalar multiplication by −1. The operation commutes with every other linear transformation , but not with translation : it is in the center of the general linear group . "Inversion" without indicating "in a point", "in a line" or "in a plane", means this inversion; in physics 3-dimensional reflection through

9540-575: The origin is also called a parity transformation . In mathematics, reflection through the origin refers to the point reflection of Euclidean space R across the origin of the Cartesian coordinate system . Reflection through the origin is an orthogonal transformation corresponding to scalar multiplication by − 1 {\displaystyle -1} , and can also be written as − I {\displaystyle -I} , where I {\displaystyle I}

9646-458: The origin is the farthest point from the identity element with respect to the usual metric. In O(2 r + 1), reflection through the origin is not in SO(2 r +1) (it is in the non-identity component), and there is no natural sense in which it is a "farther point" than any other point in the non-identity component, but it does provide a base point in the other component. It should not be confused with

9752-715: The parallelogram a special type of trapezoid. The latter definition is consistent with its uses in higher mathematics such as calculus . This article uses the inclusive definition and considers parallelograms as special cases of a trapezoid. This is also advocated in the taxonomy of quadrilaterals . Under the inclusive definition, all parallelograms (including rhombuses , squares and non-square rectangles ) are trapezoids. Rectangles have mirror symmetry on mid-edges; rhombuses have mirror symmetry on vertices, while squares have mirror symmetry on both mid-edges and vertices. A right trapezoid (also called right-angled trapezoid ) has two adjacent right angles . Right trapezoids are used in

9858-514: 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

9964-407: 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

10070-586: The plane through P which is perpendicular to the axis; the result does not depend on the orientation (in the other sense) of the axis. Notations for the type of operation, or the type of group it generates, are 1 ¯ {\displaystyle {\overline {1}}} , C i , S 2 , and 1×. The group type is one of the three symmetry group types in 3D without any pure rotational symmetry , see cyclic symmetries with n  = 1. The following point groups in three dimensions contain inversion: Closely related to inverse in

10176-450: The point group will be the same—two non-centrosymmetric shapes can be oriented in space in a manner which contains an inversion center between the two. Two tetrahedra facing each other can have an inversion center in the middle, because the orientation allows for each atom to have a reflected pair. The inverse is also true, as multiple centrosymmetric polyhedra can be arranged to form a noncentrosymmetric point group. Inversion with respect to

10282-422: The product of the areas of △ {\displaystyle \triangle } AOD and △ {\displaystyle \triangle } BOC is equal to that of △ {\displaystyle \triangle } AOB and △ {\displaystyle \triangle } COD . The ratio of the areas of each pair of adjacent triangles is the same as that between

10388-482: 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

10494-699: The representation by a matrix or as a product of reflections). Thus it is orientation-preserving in even dimension, thus an element of the special orthogonal group SO(2 n ), and it is orientation-reversing in odd dimension, thus not an element of SO(2 n  + 1) and instead providing a splitting of the map O ( 2 n + 1 ) → ± 1 {\displaystyle O(2n+1)\to \pm 1} , showing that O ( 2 n + 1 ) = S O ( 2 n + 1 ) × { ± I } {\displaystyle O(2n+1)=SO(2n+1)\times \{\pm I\}} as an internal direct product . Analogously, it

10600-554: 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

10706-589: 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

10812-409: 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

10918-442: The trapezoid into equal areas). The height (or altitude) is the perpendicular distance between the bases. In the case that the two bases have different lengths ( a ≠ b ), the height of a trapezoid h can be determined by the length of its four sides using the formula where c and d are the lengths of the legs and p = a + b + c + d {\displaystyle p=a+b+c+d} . The area K of

11024-689: The trapezoid is a parallelogram, and there are two pairs of bases. A scalene trapezoid is a trapezoid with no sides of equal measure, in contrast with the special cases below. A trapezoid is usually considered to be a convex quadrilateral in Euclidean geometry , but there are also crossed cases. If ABCD is a convex trapezoid, then ABDC is a crossed trapezoid. The metric formulas in this article apply in convex trapezoids. The ancient Greek mathematician Euclid defined five types of quadrilateral, of which four had two sets of parallel sides (known in English as square, rectangle, rhombus and rhomboid) and

11130-405: The warping of polyhedra due to nonuniform bonding lengths, often due to differing electrostatic interactions between heteroatoms or electronic effects such as Jahn–Teller distortions . For instance, a titanium center will likely bond evenly to six oxygens in an octahedra, but distortion would occur if one of the oxygens were replaced with a more electronegative fluorine. Distortions will not change

11236-596: 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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