1.
Finite geometry
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A finite geometry is any geometric system that has only a finite number of points. The familiar Euclidean geometry is not finite, because a Euclidean line contains many points. A geometry based on the graphics displayed on a screen, where the pixels are considered to be the points, would be a finite geometry. Finite geometries can also be defined purely axiomatically. However, dimension two has projective planes that are not isomorphic to Galois geometries, namely the non-Desarguesian planes. Similar results hold for other kinds of finite geometries. The following remarks apply only to finite planes. There are two main kinds of finite geometry: affine and projective. In an affine plane, the normal sense of parallel lines applies. In a projective plane, by contrast, any two lines intersect at a unique point, so parallel lines do not exist. Finite projective plane geometry may be described by fairly simple axioms. There exists a set of four points, no three of which belong to the same line. The last axiom ensures that the geometry is not trivial, while the first two specify the nature of the geometry. The simplest plane contains only four points; it is called the affine plane of order 2. Since no three are collinear, any pair of points so this plane contains six lines.
Finite geometry
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Finite affine plane of order 2, containing 4 points and 6 lines. Lines of the same color are "parallel".
2.
Gino Fano
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Gino Fano was an Italian mathematician, best known as the founder of the finite geometry. He was died in Verona, also in Italy. Fano made various contributions on algebraic geometry. His work in the foundations of geometry predates the more popular, work of David Hilbert by about a decade. He was mathematician Robert Fano and uncle to physicist and mathematician Giulio Racah. Fano was an early writer in the area of finite projective spaces. In 1907 Gino Fano contributed two articles to Part III of Klein's encyclopedia. The first was a comparison of synthetic geometry through their historic development in the 19th century. The second was as a unifying principle in geometry. Collino, Alberto; Conte, Alberto; Verra, Alessandro. "scientific work of Gino Fano". ArXiv:1311.7177. Grattan-Guinness, Ivor. The Search for Mathematical Roots 1870–1940. Princeton University Press.
Gino Fano
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Gino Fano
Gino Fano
3.
Projective plane
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In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, there are some pairs of lines that do not intersect. A projective plane can be thought as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus any two lines in a projective plane intersect in only one point. Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic. The archetypical example is the real plane, also known as the extended Euclidean plane. There are both infinite, such as the complex projective plane, finite, such as the Fano plane. Not all projective planes can be embedded in 3-dimensional projective spaces. Such embeddability is a consequence of a property known as Desargues' theorem, not shared by all projective planes. The last condition excludes the so-called degenerate cases. The term "incidence" is used to emphasize the symmetric nature of the relationship between lines. To turn the ordinary Euclidean plane into a projective plane proceed as follows: To each class of parallel lines add a new point. That point is considered incident with each line of the class. Parallel classes get different points. These points are called points at infinity.
Projective plane
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These parallel lines appear to intersect in the vanishing point "at infinity". In a projective plane this is actually true.
Projective plane
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The Moulton plane. Lines sloping down and to the right are bent where they cross the y -axis.
4.
Projective space
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In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space V. All points that lie on a projection line, intersecting with the entrance pupil of the camera, are projected onto a common image point. In this case, the projective space corresponds to the image points. Projective spaces are also used in various applied fields, geometry in particular. Geometric objects, such as points, planes, can be given a representation as elements in projective spaces based on homogeneous coordinates. As a result, various relations between these objects can be described in a simpler way than is possible without homogeneous coordinates. Furthermore, various statements in geometry can be made more consistent and without exceptions. In the standard Euclidean geometry for the plane, two lines always intersect at a point except when the lines are parallel. Mathematical fields where projective spaces play a significant role are topology, the theory of Lie groups and algebraic groups, their representation theories. As outlined above, projective space is a geometric object that formalizes statements like "Parallel lines intersect at infinity." For concreteness, we give the construction of the projective plane P2 in some detail. There are three equivalent definitions: The set of all lines in R3 passing through the origin. Every such line meets the sphere of radius one centered in the origin twice, say in P = and its antipodal point. P2 can also be described on the sphere S2, where every point P and its antipodal point are not distinguished. For example, the point is identified with, etc.
Projective space
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In graphical perspective, parallel lines in the plane intersect in a vanishing point on the horizon.
5.
Projective Geometry
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Projective geometry is a topic of mathematics. Projective is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, a selective set of basic geometric concepts. The first issue for geometers is what kind of geometry is adequate for a situation. One source for projective geometry was indeed the theory of perspective. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a drawing. See projective plane for the basics of projective geometry in two dimensions. While the ideas were available earlier, projective geometry was mainly a development of the 19th century. This included the theory of complex space, the coordinates used being complex numbers. Major types of more abstract mathematics were based on projective geometry. Projective was also a subject with a large number of practitioners for its own sake, as synthetic geometry. Another topic that developed from axiomatic studies of projective geometry is finite geometry. The topic of projective geometry is itself now divided into two examples of which are projective algebraic geometry and projective differential geometry. Projective geometry is an non-metrical form of geometry, meaning that it is not based on a concept of distance. In two dimensions Projective begins with the study of configurations of lines.
Projective Geometry
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Growth measure and the polar vortices. Based on the work of Lawrence Edwards
Projective Geometry
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Projecting a sphere to a plane.
Projective Geometry
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Forms
6.
Linear algebra
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Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, subspaces, but is also concerned with properties common to all vector spaces. The set of points with coordinates that satisfy a linear equation forms a hyperplane in an n-dimensional space. The conditions under which a set of n hyperplanes intersect in a single point is an important focus of study in linear algebra. Such an investigation is initially motivated by a system of linear equations containing several unknowns. Such equations are naturally represented using the formalism of matrices and vectors. Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations. Because linear algebra is such a well-developed theory, nonlinear mathematical models are sometimes approximated by linear models. The study of linear algebra first emerged from the study of determinants, which were used to solve systems of linear equations. Determinants were used by Leibniz in 1693, subsequently, Gabriel Cramer devised Cramer's Rule for solving linear systems in 1750. Later, Gauss further developed the theory of solving linear systems by using Gaussian elimination, initially listed as an advancement in geodesy. The study of algebra first emerged in the mid-1800s.
Linear algebra
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The three-dimensional Euclidean space R 3 is a vector space, and lines and planes passing through the origin are vector subspaces in R 3.
7.
Homogeneous coordinates
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They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates. Formulas involving homogeneous coordinates are more symmetric than their Cartesian counterparts. If the homogeneous coordinates of a point are multiplied by a non-zero scalar then the resulting coordinates represent the same point. Parallel lines in the Euclidean plane are said to intersect to their common direction. Given a point for any non-zero real number Z, the triple is called a set of homogeneous coordinates for the point. By this definition, multiplying the three homogeneous coordinates by a non-zero factor gives a new set of homogeneous coordinates for the same point. In particular, is such a system of homogeneous coordinates for the point. For example, the Cartesian point can be represented in homogeneous coordinates as or. The Cartesian coordinates are recovered by dividing the first two positions by the third. Thus unlike Cartesian coordinates, a single point can be represented by many homogeneous coordinates. The equation of a line through the origin may be written nx + my = 0 where m are not both 0. In parametric form this can be written x = mt, y = −nt. Let Z = 1/t, so the coordinates of a point on the line may be written. In homogeneous coordinates this becomes. Thus we define as the homogeneous coordinates of the point at infinity corresponding to the direction of the line nx + my = 0.
Homogeneous coordinates
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Rational Bézier curve – polynomial curve defined in homogeneous coordinates (blue) and its projection on plane – rational curve (red)
8.
Vector space
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A vector space is a collection of objects called vectors, which may be added together and multiplied by numbers, called scalars in this context. There are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector scalar multiplication must satisfy certain requirements, called axioms, listed below. Euclidean vectors are an example of a space. In the same vein, but in a more geometric sense, vectors representing displacements in three-dimensional space also form vector spaces. Infinite-dimensional vector spaces arise naturally as function spaces, whose vectors are functions. These vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of continuity. Among these topologies, those that are defined by inner product are more commonly used, as having a notion of distance between two vectors. This is particularly the case of Banach spaces and Hilbert spaces, which are fundamental in mathematical analysis. Vector spaces are applied throughout mathematics, science and engineering. Furthermore, vector spaces furnish an coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques. Vector spaces may be generalized in several ways, leading in geometry and abstract algebra. This is used in physics to describe velocities. Given any two such arrows, w, the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too.
Vector space
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Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2 w.
9.
Desarguesian plane
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In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, there are some pairs of lines that do not intersect. A projective plane can be thought as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus any two lines in a projective plane intersect in only one point. Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic. The archetypical example is the real plane, also known as the extended Euclidean plane. There are both infinite, such as the complex projective plane, finite, such as the Fano plane. Not all projective planes can be embedded in 3-dimensional projective spaces. Such embeddability is a consequence of a property known as Desargues' theorem, not shared by all projective planes. The last condition excludes the so-called degenerate cases. The term "incidence" is used to emphasize the symmetric nature of the relationship between lines. To turn the ordinary Euclidean plane into a projective plane proceed as follows: To each class of parallel lines add a new point. That point is considered incident with each line of the class. Parallel classes get different points. These points are called points at infinity.
Desarguesian plane
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These parallel lines appear to intersect in the vanishing point "at infinity". In a projective plane this is actually true.
Desarguesian plane
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The Moulton plane. Lines sloping down and to the right are bent where they cross the y -axis.
10.
Desargues configuration
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In geometry, the Desargues configuration is a configuration of ten points and ten lines, with three points per line and three lines per point. It is named after Girard Desargues, closely related to Desargues' theorem, which proves the existence of the configuration. Abc are said to be in perspective centrally if the lines Aa, Bb, Cc meet in a common point. When this happens, ten lines of the two perspectivities together form an instance of the Desargues configuration. This construction is closely related to the property that every projective plane that can be embedded into a projective space obeys Desargues' theorem. This three-dimensional realization of the Desargues configuration is also called the complete pentahedron. A graph having one vertex for each point or line in the configuration, is known as the Desargues graph. Because of the self-duality of the Desargues configuration, the Desargues graph is a symmetric graph. Its ten points can be viewed in a unique way as a self-inscribed decagon. There also exist eight other configurations that are not incidence-isomorphic to the Desargues configuration, one of, shown at right. In all of these configurations, each point has three other points that are not collinear with it. As with the Desargues configuration, the other configuration can be viewed as a pair of mutually inscribed pentagons. MathWorld.
Desargues configuration
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Two perspective triangles, and their center and axis of perspectivity
11.
Inner product
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In linear algebra, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a quantity known as the inner product of the vectors. Inner products allow the rigorous introduction of the angle between two vectors. They also provide the means of defining orthogonality between vectors. Inner product spaces are studied in functional analysis. An inner product naturally induces an associated norm, thus an inner space is also a normed vector space. A complete space with an inner product is called a Hilbert space. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. In this article, the field of scalars denoted F is either the field of complex numbers C. Then the first argument becomes conjugate linear, rather than the second. In those disciplines we would write x, y ⟩ as ⟨ y | x ⟩, respectively y † x. This reverse order is now occasionally followed in the more abstract literature, taking ⟨ y ⟩ to be conjugate linear in x rather than y. There are technical reasons why it is necessary to restrict the basefield to R and C in the definition. Briefly, the basefield has to contain therefore has to have characteristic equal to 0. This immediately excludes finite fields.
Inner product
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Geometric interpretation of the angle between two vectors defined using an inner product
12.
Gray code
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The reflected binary code, also known as Gray code after Frank Gray, is a binary numeral system where two successive values differ in only one bit. The reflected code was originally designed to prevent spurious output from electromechanical switches. Gray codes are widely used to facilitate error correction in digital communications such as digital terrestrial television and some cable TV systems. Bell Labs researcher Frank Gray introduced the term reflected binary code in his 1947 application, remarking that the code had "as yet no recognized name". He derived the name from the fact that it "may be built up by a sort of reflection process". The code was later named by others who used it. A 1954 application refers to "the Bell Telephone Gray code". Many devices indicate position by opening switches. In the transition between the two states shown above, all three switches change state. In the brief period while all are changing, the switches will read some spurious position. Even without keybounce, the transition might look like 011 — 001 — 101 — 100. If the output feeds into a sequential system, possibly via combinational logic, then the sequential system may store a false value. This is called the "cyclic" property of a Gray code. These codes are also known as single-distance codes, reflecting the Hamming distance between adjacent codes. Reflected binary codes were applied to mathematical puzzles before they became known to engineers.
Gray code
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A Gray code absolute rotary encoder with 13 tracks. At the top can be seen the housing, interrupter disk, and light source; at the bottom can be seen the sensing element and support components.
Gray code
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Gray's patent introduces the term "reflected binary code"
13.
Symmetry
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Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, that an object is invariant to any of various transformations; including scaling. Although these two meanings of "symmetry" can sometimes be told apart, they are related, so they are here discussed together. The opposite of symmetry is asymmetry. A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object but doesn't change the overall shape. An object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has translational symmetry if it can be translated without changing its overall shape. An object rotated along a line known as a screw axis. An object has scale symmetry if it does not change shape when it is expanded or contracted. Fractals also exhibit a form of scale symmetry, where small portions of the fractal are similar in shape to large portions. Other symmetries include glide reflection symmetry. A dyadic R is only if, whenever it's true that Rab, it's true that Rba. Thus, "is the same age as" is symmetrical, for if Paul is the same age as Mary, then Mary is the same age as Paul. Or, biconditional, nand, nor.
Symmetry
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Symmetric arcades of a portico in the Great Mosque of Kairouan also called the Mosque of Uqba, in Tunisia.
Symmetry
Symmetry
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Many animals are approximately mirror-symmetric, though internal organs are often arranged asymmetrically.
Symmetry
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The ceiling of Lotfollah mosque, Isfahan, Iran has 8-fold symmetries.
14.
Symmetry group
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For a space with a metric, it is a subgroup of the isometry group of the space concerned. The concept may also be studied in more general contexts as expanded below. The "objects" may be images, patterns, such as a wallpaper pattern. For symmetry of physical objects, one may also want to take their physical composition into account. The group of isometries of space induces a action on objects in it. The group is sometimes also called full symmetry group in order to emphasize that it includes the orientation-reversing isometries under which the figure is invariant. The subgroup of orientation-preserving isometries that leave the invariant is called its proper symmetry group. The proper group of an object is equal to its full symmetry group if and only if the object is chiral. The proper group is then a subgroup of the special orthogonal group SO, is therefore also called rotation group of the figure. There are also continuous symmetry groups, which contain rotations of small angles or translations of arbitrarily small distances. In general such continuous symmetry groups are studied as Lie groups. With a categorization of subgroups of the Euclidean group corresponds a categorization of symmetry groups. For example: 3D figures have mirror symmetry, but with respect to different mirror planes. 3D figures have 3-fold rotational symmetry, but with respect to different axes. 2D patterns have translational symmetry, each in one direction; the two translation vectors have the same length but a different direction.
Symmetry group
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A tetrahedron is invariant under 12 distinct rotations, reflections excluded. These are illustrated here in the cycle graph format, along with the 180° edge (blue arrows) and 120° vertex (reddish arrows) rotations that permute the tetrahedron through the positions. The 12 rotations form the rotation (symmetry) group of the figure.
15.
Projective linear group
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PSL, is defined analogously, as the induced action of the special linear group on the associated projective space. Explicitly: PSL = SL/SZ where SL is the linear group over V and SZ is the subgroup of scalar transformations with unit determinant. Here SZ is naturally identified with the group of nth roots of unity in K. If V is the n-dimensional space over a field F, namely V = Fn, the alternate notations PGL and PSL are also used. Note that PGL and PSL are equal if and only if every element of F has an nth root in F. PGL and PSL can also be defined over a ring, with an important example being the modular group, PSL. The name comes from projective geometry, where the projective group acting on homogeneous coordinates is the underlying group of the geometry. Stated differently, the natural action of GL on V descends on the projective space P. The linear groups therefore generalise the case PGL of Möbius transformations, which acts on the projective line. A related group is the group, defined axiomatically. A collineation is an invertible map which sends collinear points to collinear points. Correspondingly, the quotient group PΓL / PGL = Gal corresponds to "choices of linear structure", with the identity being the linear structure. One may also define collineation groups for axiomatically defined projective spaces, where there is no natural notion of a linear transform. Visually, this corresponds to standing at the origin, then projecting onto a flat plane. Every algebraic automorphism of a projective space is projective linear.
Projective linear group
16.
Projective special linear group
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PSL, is defined analogously, as the induced action of the special linear group on the associated projective space. Explicitly: PSL = SL/SZ where SL is the linear group over V and SZ is the subgroup of scalar transformations with unit determinant. Here SZ is naturally identified with the group of nth roots of unity in K. If V is the n-dimensional space over a field F, namely V = Fn, the alternate notations PGL and PSL are also used. Note that PGL and PSL are equal if and only if every element of F has an nth root in F. PGL and PSL can also be defined over a ring, with an important example being the modular group, PSL. The name comes from projective geometry, where the projective group acting on homogeneous coordinates is the underlying group of the geometry. Stated differently, the natural action of GL on V descends on the projective space P. The linear groups therefore generalise the case PGL of Möbius transformations, which acts on the projective line. A related group is the group, defined axiomatically. A collineation is an invertible map which sends collinear points to collinear points. Correspondingly, the quotient group PΓL / PGL = Gal corresponds to "choices of linear structure", with the identity being the linear structure. One may also define collineation groups for axiomatically defined projective spaces, where there is no natural notion of a linear transform. Visually, this corresponds to standing at the origin, then projecting onto a flat plane. Every algebraic automorphism of a projective space is projective linear.
Projective special linear group
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Relation between the projective special linear group PSL and the projective general linear group PGL.
17.
PSL(2,7)
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In mathematics, the projective special linear group PSL is a finite simple group that has important applications in algebra, geometry, number theory. It is the group of the Klein quartic as well as the symmetry group of the Fano plane. With 168 elements PSL is the second-smallest simple group after the alternating group A5 on five letters with 60 elements, or the isomorphic PSL. The general linear group GL consists of all invertible 2×2 matrices over F7, the finite field with 7 elements. These have determinant. The SL consists of all such matrices with unit determinant. Then PSL is defined to be the quotient group SL / obtained by − I, where I is the identity matrix. In this article, we let G denote any group isomorphic to PSL. G = PSL has 168 elements. The result is / = 168. It is a general result that PSL is simple for n, q ≥ 2, unless = or. PSL is isomorphic to alternating group A4. In fact, PSL is the second smallest nonabelian simple group, after the alternating group A5 = PSL = PSL. The number of conjugacy classes and irreducible representations is 6. The sizes of conjugacy classes are 1, 21, 42, 56, 24.
PSL(2,7)
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The Klein quartic can be realized as a quotient of the order-3 heptagonal tiling.
18.
General linear group
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In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. The inverse of an invertible matrix is invertible. To be more precise, it is necessary to specify what kind of objects may appear in the entries of the matrix. For example, the linear group over R is the group of n × n invertible matrices of real numbers, is denoted by GLn or GL. Typical notation is GLn or GL, or simply GL if the field is understood. More still, the general linear group of a vector space GL is the abstract automorphism group, not necessarily written as matrices. The linear group, written SL or SLn, is the subgroup of GL consisting of matrices with a determinant of 1. Its subgroups are often called linear groups or matrix groups. The modular group may be realised as a quotient of the linear group SL. If n ≥ 2, then the GL is not abelian. If V has finite n, then GL and GL are isomorphic. The isomorphism is not canonical; it depends on a choice of basis in V. In general this is not isomorphic to GL. Over a F, a matrix is invertible if and only if its determinant is nonzero. Therefore, an alternative definition of GL is as the group of matrices with determinant.
General linear group
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Algebraic structure → Group theory Group theory
19.
Cycles and fixed points
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The corresponding cycle of π is written as; this expression is not unique since c1 can be chosen to be any element of the orbit. For example, let π = = be a permutation that 6 to 8, etc.. Then one may write π = = = =... Here 5 and 7 are fixed points of π, since π=5 and π=7. It is typical, but not necessary, to not write the cycles of length one in such an expression. Thus, π =, would be an appropriate way to express this permutation. The unsigned Stirling number of the first kind, s counts the number of permutations of k elements with exactly j disjoint cycles. For every k > 0: s 1. For every k > 0: s =!. Every cycle of length k may be written as permutation of the number 1 to k; there are k! of these permutations. There are k different ways to write a given cycle of k, e.g. = = =. Finally: s = k!/k =!. The f counts the number of permutations of k elements with exactly j fixed points. For the main article on this topic, see rencontres numbers. For every j < 0 or j > k: f = 0.
Cycles and fixed points
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16-bit Gray code permutation G multiplied with the bit-reversal permutation B G has 2 fixed points, 1 2-cycle and 3 4-cycles B has 4 fixed points and 6 2-cycles GB has 2 fixed points and 2 7-cycles
20.
Transposition (mathematics)
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If S has k elements, the cycle is called a k-cycle. The set S is called the orbit of the cycle. Every permutation on many elements can be decomposed into a collection of cycles on disjoint orbits. A permutation is called a cyclic permutation if and only if it has a nontrivial cycle. For example, the permutation, written in two-line and also cycle notations, = =, is a six-cycle; its diagram is shown at right. Some authors restrict the definition to only those permutations which consist of one nontrivial cycle. For example, the permutation = is a cyclic permutation under this more restrictive definition, while the preceding example is not. This notion is most commonly used when X is a finite set; then of course S, is also finite. Let s 0 put s i = σ i for any i ∈ Z. If S is finite, there is a minimal k ≥ 1 for which s k = s 0. A cycle can be written using the compact cycle notation =. The length of a cycle is the number of elements of its largest orbit. A cycle of k is also called a k-cycle. As a permutation every 1-cycle is the identity permutation. When notation is used, the 1-cycles are often suppressed when no confusion will result.
Transposition (mathematics)
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This article is about group theory. For cycles in homological algebra, see Chain complex#Fundamental terminology. For cycles in graph theory, see Cycle (graph theory).
21.
Quadrilateral
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In Euclidean plane geometry, a quadrilateral is a polygon with four edges and four vertices or corners. Sometimes, the quadrangle is used, by analogy with triangle, sometimes tetragon for consistency with pentagon, hexagon and so on. The origin of the word "quadrilateral" is the two Latin words quadri, latus, meaning "side". Quadrilaterals are complex, also called crossed. Simple quadrilaterals are either concave. This is a special case of the n-gon interior angle formula × 180 °. All non-self-crossing quadrilaterals tile the plane by repeated rotation around the midpoints of their edges. Any quadrilateral, not self-intersecting is a simple quadrilateral. In a quadrilateral, all interior angles are less than 180 ° and the two diagonals both lie inside the quadrilateral. Irregular quadrilateral or trapezium: no sides are parallel. Trapezium or trapezoid: at least one pair of opposite sides are parallel. Trapezoids include parallelograms. Isosceles trapezium or isosceles trapezoid: the base angles are equal in measure. Alternative definitions are a quadrilateral with an axis of symmetry bisecting a trapezoid with diagonals of equal length. Parallelogram: a quadrilateral with two pairs of parallel sides.
Quadrilateral
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Six different types of quadrilaterals
22.
Flag (geometry)
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In geometry, a flag is a sequence of faces of a polytope, each contained in the next, with just one face from each dimension. These latter two are called improper faces. A flag of a polyhedron is sometimes called a "dart". A polytope may be regarded as regular if, if its symmetry group is transitive on its flags. This definition excludes chiral polytopes. This level of abstraction generalizes both the polyhedral concept given above well as the related flag concept from linear algebra. A flag is maximal if it is not contained in a larger flag. When all maximal flags of an geometry have the same size, this common value is the rank of the geometry. ISBN 0-521-81496-0
Flag (geometry)
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Face diagram of a square pyramid showing one of its flags
23.
Triangle
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A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, C is denoted △ A B C. In Euclidean geometry any three points, when non-collinear, determine a unique plane. This article is about triangles in Euclidean geometry except where otherwise noted. Triangles can be classified according to the lengths of their sides: An equilateral triangle has the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length. Some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral isosceles triangles. The 45 -- 45 -- 90 right triangle, which appears in the square tiling, is isosceles. A scalene triangle has all its sides of different lengths. Equivalently, it has all angles of different measure. Hatch marks, also called tick marks, are used in diagrams of triangles and geometric figures to identify sides of equal lengths. In a triangle, the pattern is usually no more than 3 ticks.
Triangle
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The Flatiron Building in New York is shaped like a triangular prism
Triangle
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A triangle
24.
Bitangents of a quartic
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In real algebraic geometry, a general quartic plane curve has 28 bitangent lines, lines that are tangent to the curve in two places. Another quartic with 28 real bitangents can be formed with fixed axis lengths tangent to two non-parallel lines. These points form a quartic curve that has genus three and that has twenty-eight real bitangents. Additionally, each oval has one bitangent spanning the nonconvex portion of its boundary. The dual curve to a quartic curve has dual to the 28 bitangents of the primal curve. There are 64 choices for a, b, c, d, f, but only 28 of these choices produce an odd sum. The Levi graph of the Fano plane is the Heawood graph, in which the triangles of the Fano plane are represented by 6-cycles. The 28 6-cycles of the Heawood graph in turn correspond to the 28 vertices of the Coxeter graph. Blum, R.; Guinand, A. P. "A quartic with 28 real bitangents", Canadian Mathematical Bulletin, 7: 399–404, doi:10.4153/cmb-1964-038-6. Cayley, "On the bitangents of a quartic", Salmon's Higher Plane Curves, pp. 387 -- 389. In The collected mathematical papers of Arthur Cayley, Andrew Russell Forsyth, ed. The University Press, 1896, vol. 11, pp. 221–223. Gray, Jeremy, "From the history of a simple group", 4: 59 -- 67, doi:10.1007 / BF03023483, MR 0672918. Reprinted in Levy, Silvio, ed. The Eightfold Way, MSRI Publications, 35, Cambridge University Press, pp. 115–131, ISBN 0-521-66066-1, MR 1722415. Manivel, L. "Configurations of lines and models of Lie algebras", Journal of Algebra, 304: 457–486, doi:10.1016/j.jalgebra.2006.04.029.
Bitangents of a quartic
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The Trott curve and seven of its bitangents. The others are symmetric with respect to 90° rotations through the origin.
25.
Hexagon
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In geometry, a hexagon is a six sided polygon or 6-gon. The total of the internal angles of any hexagon is 720°. A regular hexagon can also be constructed as a truncated equilateral triangle, t, which alternates two types of edges. A regular hexagon is defined as a hexagon, both equiangular. It meaning that it is both cyclic and tangential. The common length of the sides equals the radius of the circumscribed circle, which equals 3 3 times the apothem. All internal angles are 120 degrees. A regular hexagon has 6 reflection symmetries, making up the dihedral group D6. The longest diagonals of a regular hexagon, connecting opposite vertices, are twice the length of one side. Like equilateral triangles, regular hexagons fit together without any gaps to tile the plane, so are useful for constructing tessellations. The cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of building materials. The Voronoi diagram of a regular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral. The maximal diameter, D is twice R, which equals the side length, t. The minimal diameter or the diameter of the inscribed circle, d, is twice the minimal radius or inradius, r.
Hexagon
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Giants causeway closeup
Hexagon
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The ideal crystalline structure of graphene is a hexagonal grid.
Hexagon
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Assembled E-ELT mirror segments
Hexagon
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A beehive honeycomb
26.
Cyclic order
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In mathematics, a cyclic order is a way to arrange a set of objects in a circle. Unlike most structures in theory, a cyclic order is not modeled as a binary relation, such as "a < b". One does not say that east is "more clockwise" than west. Instead, a cyclic order is defined as a ternary relation, meaning "one reaches b before c". For example. A ternary relation is called a cyclic order if it is cyclic, transitive, total. Dropping the "total" requirement results in a cyclic order. A set with a cyclic order is called a cyclically ordered set or simply a cycle. In a finite cycle, each element has a "previous element". There are also continuously variable cycles such as the oriented unit circle in the plane. Cyclic orders are closely related to the more familiar linear orders, which arrange objects in a line. Any cyclic order can be cut at a point, resulting in a line. A cyclic order on a set X with n elements is for an n-hour clock. There are a equivalent ways to state this definition. A cyclic order on X is the same as a permutation that makes all of X into a single cycle.
Cyclic order
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Contents
27.
Group (mathematics)
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The operation satisfies four conditions called the group axioms, namely closure, invertibility. It allows entities beyond to be handled while retaining their essential structural aspects. The ubiquity of groups in numerous areas outside mathematics makes a central organizing principle of contemporary mathematics. Groups share a fundamental kinship with the notion of symmetry. The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into better-understandable pieces, such as subgroups, simple groups. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric theory, which studies finitely generated groups as geometric objects, has become a particularly active area in theory. The following properties of integer addition serve as a model for the abstract group axioms given in the definition below. For a + b is also an integer. That is, addition of integers always yields an integer. This property is known as closure under addition. For c, + c = a +.
Group (mathematics)
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A periodic wallpaper pattern gives rise to a wallpaper group.
Group (mathematics)
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The manipulations of this Rubik's Cube form the Rubik's Cube group.
28.
GL(3,2)
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In mathematics, the projective special linear group PSL is a finite simple group that has important applications in algebra, geometry, number theory. It is the group of the Klein quartic as well as the symmetry group of the Fano plane. With 168 elements PSL is the second-smallest simple group after the alternating group A5 on five letters with 60 elements, or the isomorphic PSL. The general linear group GL consists of all invertible 2×2 matrices over F7, the finite field with 7 elements. These have determinant. The SL consists of all such matrices with unit determinant. Then PSL is defined to be the quotient group SL / obtained by − I, where I is the identity matrix. In this article, we let G denote any group isomorphic to PSL. G = PSL has 168 elements. The result is / = 168. It is a general result that PSL is simple for n, q ≥ 2, unless = or. PSL is isomorphic to alternating group A4. In fact, PSL is the second smallest nonabelian simple group, after the alternating group A5 = PSL = PSL. The number of conjugacy classes and irreducible representations is 6. The sizes of conjugacy classes are 1, 21, 42, 56, 24.
GL(3,2)
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The Klein quartic can be realized as a quotient of the order-3 heptagonal tiling.
29.
Matroid theory
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In combinatorics, a branch of mathematics, a matroid /ˈmeɪtrɔɪd/ is a structure that abstracts and generalizes the notion of linear independence in vector spaces. Matroids have found applications in geometry, coding theory. There are many equivalent ways to define a matroid. Alternatively, at least one subset of E is independent, i.e. I ≠ ∅. Every subset of an independent set is independent, i.e. for each A ′ ⊂ A ⊂ E, if A ∈ I then A ′ ∈ I. This is sometimes called the hereditary property. This is sometimes called the augmentation property or the independent set exchange property. The first two properties define a combinatorial structure known as an independence system. A subset of the ground set E, not independent is called dependent. A maximal independent set—, an independent set which becomes dependent on adding any element of E —is called a basis for the matroid. A circuit in a matroid M is a minimal dependent subset of E —, a dependent set whose proper subsets are all independent. The terminology arises because the circuits of graphic matroids are cycles in the corresponding graphs. The collection of dependent sets, or of bases, or of circuits each has simple properties that may be taken as axioms for a matroid. It follows from the basis exchange property that no member of B can be a proper subset of another. This number is called the rank of M.
Matroid theory
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The Fano matroid, derived from the Fano plane. It is GF(2) -linear but not real-linear.
30.
Matroid
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In combinatorics, a branch of mathematics, a matroid /ˈmeɪtrɔɪd/ is a structure that abstracts and generalizes the notion of linear independence in vector spaces. Matroids have found applications in geometry, topology, combinatorial optimization, coding theory. There are many equivalent ways to define a matroid. Alternatively, at least one subset of E is independent, i.e. I ≠ ∅. Every subset of an independent set is independent, i.e. for each A ′ ⊂ A ⊂ E, if A ∈ I then A ′ ∈ I. This is sometimes called the hereditary property. This is sometimes called the independent set exchange property. The first two properties define a combinatorial structure known as an system. A subset of the ground E, not independent is called dependent. A independent set --, an independent set which becomes dependent on adding any element of E -- is called a basis for the matroid. A circuit in a matroid M is a dependent subset of E --, a dependent set whose proper subsets are all independent. The terminology arises because the circuits of graphic matroids are cycles in the corresponding graphs. Of bases, or of circuits each has simple properties that may be taken as axioms for a matroid. It follows from the basis property that no member of B can be a proper subset of another. This number is called the rank of M.
Matroid
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The Fano matroid, derived from the Fano plane. It is GF(2) -linear but not real-linear.
31.
Graphic matroid
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In matroid theory, a discipline within mathematics, a graphic matroid is a matroid whose independent sets are the forests in a given undirected graph. The dual matroids of graphic matroids are called co-graphic matroids or bond matroids. A matroid, both co-graphic is called a planar matroid; these are exactly the graphic matroids formed from planar graphs. Thus, F forms the independent sets of a matroid, called the graphic matroid of G or M. The cycles of M are the simple cycles of G. The corank of the graphic matroid is known as cyclomatic number. The closure cl of a set S of edges in M is a flat consisting of the edges that are not independent of S. Thus, the lattice of flats of the graphic matroid of K n is naturally isomorphic to the lattice of partitions of an n -element set. Since the lattices of flats of matroids are exactly the geometric lattices, this implies that the lattice of partitions is also geometric. The graphic matroid of a graph G can be defined as the matroid of any oriented incidence matrix of G. Such a matrix has one column for each edge. The matroid of this matrix has as its independent sets the linearly independent subsets of columns. If a set of edges contains a cycle, then the corresponding columns is not independent. Therefore, the matrix is isomorphic to M. This method of representing graphic matroids works regardless of the field over which the incidence is defined.
Graphic matroid
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Two different graphs (red) that are duals of the same planar graph (pale blue). Despite being non-isomorphic as graphs, they have isomorphic graphic matroids.
32.
Steiner system
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In combinatorial mathematics, a Steiner system is a type of block design, specifically a t-design with λ = 1 and t ≥ 2. In an alternate notation for block designs, an S would be a t- design. This definition is relatively modern, generalizing the classical definition of Steiner systems which in addition required that k = + 1. An S was called a Steiner triple system, while an S was called a Steiner system, so on. With the generalization of the definition, this system is no longer strictly adhered to. A long-standing problem in theory was if any nontrivial Steiner systems have t ≥ 6; also if infinitely many have t = 4 or 5. This was solved by Peter Keevash in 2014. A finite affine plane of order q, with the lines as blocks, is an S. Choosing different blocks to remove in this way can lead to non-isomorphic affine planes. Its blocks are called triples. It is common to see the STS for a Steiner triple system of order n. Hence the total number of triples is n/6. This shows that n must be of the form 6k 6k + 3 for some k. The fact that this condition on n is sufficient for the existence of an S was proved by Raj Chandra Bose and T. Skolem. The affine plane of order 3 is an STS.
Steiner system
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The Fano plane is an S(2,3,7) Steiner triple system. The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line.
33.
Quasigroup
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In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible. Quasigroups differ from groups mainly in that they need not be associative. A quasigroup with an element is called a loop. There are at least two structurally equivalent formal definitions of quasigroup. The other, from universal algebra, defines a quasigroup as having three primitive operations. The homomorphic image of a quasigroup defined with a single operation, however, need not be a quasigroup. We begin with the first definition. A quasigroup is Q, with a binary operation, ∗, obeying the Latin square property. The unique solutions to these equations are y = b / a. The operations'\' and'/' are called, respectively, left and right division. The empty set equipped with the empty operation satisfies this definition of a quasigroup. Others explicitly exclude it. Algebraic structures axiomatized solely by identities are called varieties. Standard results in universal algebra hold only for varieties. Quasigroups are varieties if left and right division are taken as primitive.
Quasigroup
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Algebraic structures between magmas and groups.
34.
Octonion
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There are three normed division algebras over the reals: the quaternions H. The octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity, namely they are alternative. Octonions are not well known as the complex numbers, which are much more widely used. Despite this, they are related among them the exceptional Lie groups. Additionally, octonions have applications in fields such as logic. The octonions were discovered in 1843 by John T. Graves, inspired by his friend W. R. Hamilton's discovery of quaternions. The octonions were discovered independently by Cayley and are sometimes referred to as Cayley numbers or the Cayley algebra. Hamilton described the early history of Graves' discovery. Hamilton invented the word associative so that he could say that octonions were not associative. The octonions can be thought of as octets of real numbers. Addition and subtraction of octonions is done by adding and subtracting corresponding terms and hence their coefficients, like quaternions. Multiplication is more complex. Multiplication is distributive over addition, so the product of two octonions can be calculated by summing the product of all the terms, again like quaternions. 7.
Octonion
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A mnemonic for the products of the unit octonions.
35.
Isomorphism
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In mathematics, an isomorphism is a homomorphism or morphism that admits an inverse. Two mathematical objects are isomorphic if an isomorphism exists between them. An automorphism is an isomorphism whose target coincide. Including groups and rings, a homomorphism is an isomorphism if and only if it is bijective. In topology, where the morphisms are continuous functions, isomorphisms are also called homeomorphisms or functions. In mathematical analysis, where the morphisms are differentiable functions, isomorphisms are also called diffeomorphisms. A canonical isomorphism is a canonical map, an isomorphism. Two objects are said to be canonically isomorphic if there is a canonical isomorphism between them. Isomorphisms are formalized using theory. Let R + let R be the additive group of real numbers. The exponential function exp: R → R + satisfies exp = for all x, y ∈ R, so it too is a homomorphism. Since log is a homomorphism that has an inverse, also a homomorphism, log is an isomorphism of groups. Because log is an isomorphism, it translates multiplication of real numbers into addition of real numbers. This facility makes it possible to multiply real numbers using a slide rule with a logarithmic scale. Consider the integers from 0 to 5 with addition modulo 6.
Isomorphism
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The group of fifth roots of unity under multiplication is isomorphic to the group of rotations of the regular pentagon under composition.
36.
Incidence structure
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In mathematics, an abstract system consisting of two types of objects and a single relationship between these types of objects is called an incidence structure. What is left is the structure of the Euclidean plane. Even in a geometric setting, incidence structures are not limited to lines; higher-dimensional objects can be used. The study of finite structures is sometimes called finite geometry. The elements of I are called flags. In some common situations L may be a set of subsets of P in which incidence I will be containment. Incidence structures of this type are called set-theoretic. This example also shows that while the geometric language of lines is used, the object types need not be these geometric objects. An structure is uniform if each line is incident with the same number of points. Each of these examples, except the second, is uniform with three points per line. Any graph is a uniform structure with two points per line. Incidence structures are seldom studied in their full generality; it is typical to study incidence structures that satisfy some additional axioms. It follows immediately from the definition: C ∗ ∗ = C. This is an abstract version of projective duality. A structure C, isomorphic to its dual C∗ is called self-dual.
Incidence structure
37.
Projective geometry
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Projective geometry is a topic of mathematics. Projective is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, a selective set of basic geometric concepts. The first issue for geometers is what kind of geometry is adequate for a situation. One source for projective geometry was indeed the theory of perspective. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a drawing. See projective plane for the basics of projective geometry in two dimensions. While the ideas were available earlier, projective geometry was mainly a development of the 19th century. This included the theory of complex space, the coordinates used being complex numbers. Major types of more abstract mathematics were based on projective geometry. Projective was also a subject with a large number of practitioners for its own sake, as synthetic geometry. Another topic that developed from axiomatic studies of projective geometry is finite geometry. The topic of projective geometry is itself now divided into two examples of which are projective algebraic geometry and projective differential geometry. Projective geometry is an non-metrical form of geometry, meaning that it is not based on a concept of distance. In two dimensions Projective begins with the study of configurations of lines.
Projective geometry
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Growth measure and the polar vortices. Based on the work of Lawrence Edwards
Projective geometry
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Projecting a sphere to a plane.
Projective geometry
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Forms
38.
Projective configuration
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They were popularized by Hilbert and Cohn-Vossen's 1932 book Anschauliche Geometrie, reprinted in English. That is, the girth of the corresponding graph must be at least six. These numbers necessarily satisfy the equation γ = ℓ π as this product is the number of point-line incidences. Configurations having the same symbol, say, need not be isomorphic as incidence structures. For instance, there exist three different configurations: two less notable configurations. In p = ℓ and consequently, γ = π. The notation is often condensed to avoid repetition. For example abbreviates to. Notable projective configurations include the following:, the simplest possible configuration, consisting of a incident to a line. Often excluded as being trivial. , the triangle. Each of its three sides meets two of its three vertices, versa. More generally any polygon of n sides forms a configuration of type and, complete quadrilateral respectively. , the Fano plane. This configuration can not be constructed in the Euclidean plane.
Projective configuration
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Configurations (4 3 6 2) (a complete quadrangle, at left) and (6 2 4 3) (a complete quadrilateral, at right).
39.
John Baez
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John Carlos Baez is an American mathematical physicist and a professor of mathematics at the University of California, Riverside in Riverside, California. He is known on spin foams in loop quantum gravity. For some time, his research had focused to physics and other things. It now has a following in its new form, the blog "Azimuth". This Week's Finds anticipated the concept of a personal weblog. Additionally, Baez is known as the author of the crackpot index. Baez was born in California. He graduated from Princeton University in Princeton, New Jersey, in mathematics in 1982. In 1986, he graduated from the Massachusetts Institute of Technology in Cambridge, Massachusetts, under the direction of Irving Segal. After a post-doctoral period at Yale University in Connecticut, he has been teaching -- since 1989 -- at UC Riverside. Baez is also co-founder of a group blog concerning higher category theory and its applications, as well as its philosophical repercussions. The list of blog authors has extended since. The n-Café community is associated with the nLab wiki and forum, which now run independently of n-Café. It is hosted at Austin's official website. Albert Baez, interested him in physics as a child.
John Baez
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John C. Baez (August 2009)
40.
International Standard Serial Number
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An International Standard Serial Number is an eight-digit serial number used to uniquely identify a serial publication. The ISSN is especially helpful in distinguishing between serials with the same title. ISSN are used in cataloging, interlibrary loans, other practices in connection with serial literature. The ISSN system was first published as ISO 3297 in 1975. 46/SC 9 is responsible for maintaining the standard. When a serial with the same content is published in more than one media type, a different ISSN is assigned to each media type. For example, many serials are published both in electronic media. The ISSN system refers as print ISSN and electronic ISSN, respectively. The format of the ISSN is an eight code, divided by a hyphen into two four-digit numbers. As an number, it can be represented by the first seven digits. The last digit, which may be 0-9 or an X, is a check digit. The ISSN of the journal Hearing Research, for example, is 0378-5955, where the final 5 is the digit, C = 5. For calculations, an upper X in the check digit position indicates a check digit of 10. To calculate the sum of all eight digits of the ISSN multiplied by its position in the number, counting from the right. The modulus 11 of the sum must be 0.
International Standard Serial Number
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ISSN encoded in an EAN-13 barcode with sequence variant 0 and issue number 5
41.
PlanetMath
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PlanetMath is a free, collaborative, online mathematics encyclopedia. The emphasis is on rigour, openness, pedagogy, real-time content, also community of about 24,000 people with various maths interests. Intended to be comprehensive, the project is currently hosted by the University of Waterloo. The site is owned by a nonprofit corporation, "PlanetMath.org, Ltd". The main PlanetMath focus is on encyclopedic some forum discussions. In addition, the project hosts data about books, research-level papers. A system for semi-private messaging among users is also in place. Developing software recommendations for improved content authoring and editorial functions. PlanetMath content is licensed under the copyleft Creative Commons Attribution/Share-Alike License. The software running PlanetMath runs on Linux and the web server Apache. It has been released under the free BSD License. As of March 2013 PlanethMath has retired Noösphere and runs now on a software called Planetary, which itself was implemented with Drupal. Encyclopedic bibliographic materials related to physics, mathematics and mathematical physics are developed by PlanetPhysics. The site, launched in 2005, uses a significantly different moderation model with emphasis on current research in physics and peer review.
PlanetMath
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PlanetMath
42.
Fano plane
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The Fano plane can be constructed as the projective plane over the finite field with two elements. One can similarly construct projective planes with the Fano plane being the smallest. The lines of the Fano plane may also be given homogeneous coordinates, again using non-zero triples of binary digits. The lines can be classified into three types. In the remaining 111, each binary code has exactly two nonzero bits. It consists of 168 different permutations. The group is made up of 6 conjugacy classes. Then D is on the same line as A and B. A maps to B, B to C, C to D. Then D is on the same line as A and C. See Fano plane collineations for a complete list. Hence, by the Pólya theorem, the number of inequivalent colorings of the Fano plane with n colors is: 1 168. The Fano plane contains the following numbers of different types. There are 24 symmetries fixing any point. There are 24 symmetries fixing any line.
Fano plane
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The Fano plane