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 infinitely many points, a geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are systems that could be called finite geometries, attention is mostly paid to the finite projective. Finite geometries may be constructed via linear algebra, starting from vector spaces over a finite field, Finite geometries can also be defined purely axiomatically. Most common finite geometries are Galois geometries, since any finite projective space of three or greater is isomorphic to a projective space over a finite field. However, dimension two has affine and 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 plane geometry, affine and projective. In an affine plane, the sense of parallel lines applies. In a projective plane, by contrast, any two lines intersect at a point, so parallel lines do not exist. Both finite affine plane geometry and finite projective plane geometry may be described by fairly simple axioms. An affine plane geometry is a nonempty set X, along with a nonempty collection L of subsets of X, such that, For every two distinct points, there is exactly one line that contains both points. Playfairs axiom, Given a line ℓ and a point p not on ℓ, 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 affine plane contains only four points, it is called the affine plane of order 2. Since no three are collinear, any pair of points determines a line, and so this plane contains six lines. It corresponds to a tetrahedron where non-intersecting edges are considered parallel, or a square where not only opposite sides, but also diagonals are considered parallel. More generally, an affine plane of order n has n2 points and n2 + n lines, each line contains n points. The affine plane of order 3 is known as the Hesse configuration. A projective plane geometry is a nonempty set X, along with a nonempty collection L of subsets of X, such that, the intersection of any two distinct lines contains exactly one point
2.
Gino Fano
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Gino Fano was an Italian mathematician, best known as the founder of the finite geometry. He was born in Mantua, in Italy and died in Verona, Fano made various contributions on projective and algebraic geometry. His work in the foundations of geometry predates the similar, but more popular and he was the father of physicist Ugo Fano and mathematician Robert Fano and uncle to physicist and mathematician Giulio Racah. Fano was a writer in the area of finite projective spaces. In his article on proving the independence of his set of axioms for projective n-space, among other things, he considered the consequences of having a fourth harmonic point be equal to its conjugate. This leads to a configuration of seven points and seven lines contained in a finite space with 15 points,35 lines and 15 planes. All the planes in this space consist of seven points and seven lines and are now known as Fano planes, Fano went on to describe finite projective spaces of arbitrary dimension, in 1907 Gino Fano contributed two articles to Part III of Kleins encyclopedia. The first was a comparison of analytic geometry and synthetic geometry through their development in the 19th century. The second was on continuous groups in geometry and group theory as a principle in geometry. Collino, Alberto, Conte, Alberto, Verra, Alessandro, on the life and scientific work of Gino Fano. arXiv,1311.7177. The Search for Mathematical Roots 1870–1940, oConnor, John J. Robertson, Edmund F. Gino Fano, MacTutor History of Mathematics archive, University of St Andrews. Gino Fano at the Mathematics Genealogy Project
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, two lines intersect in a single point, but there are some pairs of lines that do not intersect. A projective plane can be thought of as a plane equipped with additional points at infinity where parallel lines intersect. Thus any two lines in a projective plane intersect in one and only one point. Renaissance artists, in developing the techniques of drawing in perspective, the archetypical example is the real projective plane, also known as the extended Euclidean plane. This example, in different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by PG, RP2. There are many projective planes, both infinite, such as the complex projective plane, and finite, such as the Fano plane. A projective plane is a 2-dimensional projective space, but 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 points and lines. Thus the expression point P is incident with l is used instead of either P is on l or l passes through P. To turn the ordinary Euclidean plane into a projective plane proceed as follows and that point is considered incident with each line of the class. Different parallel classes get different points and these points are called points at infinity. Add a new line which is considered incident with all the points at infinity and this line is called the line at infinity. The extended structure is a plane and is called the Extended Euclidean Plane or the real projective plane. The process outlined above, used to obtain it, is called projective completion or projectivization and this plane can also be constructed by starting from R3 viewed as a vector space, see below. The points of the Moulton plane are the points of the Euclidean plane, to create the Moulton plane from the Euclidean plane some of the lines are redefined. That is, some of their point sets will be changed, the Moulton plane has parallel classes of lines and is an affine plane. It can be projectivized, as in the example, to obtain the projective Moulton plane
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. The idea of a projective space relates to perspective, more precisely to the way an eye or a camera projects a 3D scene to a 2D image. All points that lie on a line, intersecting with the entrance pupil of the camera, are projected onto a common image point. In this case, the space is R3 with the camera entrance pupil at the origin. Projective spaces can be studied as a field in mathematics. Geometric objects, such as points, lines, or planes, 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 more consistent. For example, in the standard Euclidean geometry for the plane, in a projective representation of lines and points, however, such an intersection point exists even for parallel lines, and it can be computed in the same way as other intersection points. Other mathematical fields where projective spaces play a significant role are topology, the theory of Lie groups and algebraic groups, 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 exactly twice, say in P = and its antipodal point. P2 can also be described as the points on the sphere S2, for example, the point is identified with, etc. The usual way to write an element of the projective plane, the last formula goes under the name of homogeneous coordinates. In homogeneous coordinates, any point with z ≠0 is equivalent to, so there are two disjoint subsets of the projective plane, that consisting of the points = for z ≠0, and that consisting of the remaining points. The latter set can be subdivided similarly into two disjoint subsets, with points and, in the last case, x is necessarily nonzero, because the origin was not part of P2. This last point is equivalent to, geometrically, the first subset, which is isomorphic to R2, is in the image the yellow upper hemisphere, or equivalently the lower hemisphere. The second subset, isomorphic to R1, corresponds to the line, or, again. Finally we have the red point or the equivalent light red point and we thus have a disjoint decomposition P2 = R2 ⊔ R1 ⊔ point
5.
Projective Geometry
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Projective geometry is a topic of mathematics. It is the study of properties that are invariant with respect to projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than expressible by a transformation matrix and translations. The first issue for geometers is what kind of geometry is adequate for a novel situation, one source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See projective plane for the basics of geometry in two dimensions. While the ideas were available earlier, projective geometry was mainly a development of the 19th century and this included the theory of complex projective space, the coordinates used being complex numbers. Several major types of more abstract mathematics were based on projective geometry and it 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 many research subtopics, two examples of which are projective algebraic geometry and projective differential geometry. Projective geometry is an elementary form of geometry, meaning that it is not based on a concept of distance. In two dimensions it begins with the study of configurations of points and lines and that there is indeed some geometric interest in this sparse setting was first established by Desargues and others in their exploration of the principles of perspective art. In higher dimensional spaces there are considered hyperplanes, and other linear subspaces, Projective geometry can also be seen as a geometry of constructions with a straight-edge alone. Since projective geometry excludes compass constructions, there are no circles, no angles, no measurements, no parallels and it was realised that the theorems that do apply to projective geometry are simpler statements. For example, the different conic sections are all equivalent in projective geometry, during the early 19th century the work of Jean-Victor Poncelet, Lazare Carnot and others established projective geometry as an independent field of mathematics. Its rigorous foundations were addressed by Karl von Staudt and perfected by Italians Giuseppe Peano, Mario Pieri, Alessandro Padoa, after much work on the very large number of theorems in the subject, therefore, the basics of projective geometry became understood. The incidence structure and the cross-ratio are fundamental invariants under projective transformations, Projective geometry can be modeled by the affine plane plus a line at infinity and then treating that line as ordinary. An algebraic model for doing projective geometry in the style of geometry is given by homogeneous coordinates. In a foundational sense, projective geometry and ordered geometry are elementary since they involve a minimum of axioms and either can be used as the foundation for affine, Projective geometry is not ordered and so it is a distinct foundation for geometry
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, and subspaces, 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 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 theory, nonlinear mathematical models are sometimes approximated by linear models. The study of linear algebra first emerged from the study of determinants, determinants were used by Leibniz in 1693, and subsequently, Gabriel Cramer devised Cramers Rule for solving linear systems in 1750. Later, Gauss further developed the theory of solving linear systems by using Gaussian elimination, the study of matrix algebra first emerged in England in the mid-1800s. In 1844 Hermann Grassmann published his Theory of Extension which included foundational new topics of what is called linear algebra. In 1848, James Joseph Sylvester introduced the term matrix, which is Latin for womb, while studying compositions of linear transformations, Arthur Cayley was led to define matrix multiplication and inverses. Crucially, Cayley used a letter to denote a matrix. In 1882, Hüseyin Tevfik Pasha wrote the book titled Linear Algebra, the first modern and more precise definition of a vector space was introduced by Peano in 1888, by 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged. Linear algebra took its form in the first half of the twentieth century. The use of matrices in quantum mechanics, special relativity, the origin of many of these ideas is discussed in the articles on determinants and Gaussian elimination. Linear algebra first appeared in American graduate textbooks in the 1940s, following work by the School Mathematics Study Group, U. S. high schools asked 12th grade students to do matrix algebra, formerly reserved for college in the 1960s. In France during the 1960s, educators attempted to teach linear algebra through finite-dimensional vector spaces in the first year of secondary school and this was met with a backlash in the 1980s that removed linear algebra from the curriculum. To better suit 21st century applications, such as mining and uncertainty analysis
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 often simpler and 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. Since homogeneous coordinates are given to points at infinity, the number of coordinates required to allow this extension is one more than the dimension of the projective space being considered. For example, two coordinates are required to specify a point on the projective line and three homogeneous coordinates are required to specify a point in the projective plane. The real projective plane can be thought of as the Euclidean plane with additional points added, which are called points at infinity, and are considered to lie on a new line, the line at infinity. There is a point at infinity corresponding to each direction, informally defined as the limit of a point that moves in direction away from the origin. Parallel lines in the Euclidean plane are said to intersect at a point at infinity corresponding to their common direction, given a point on the Euclidean plane, 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 common, non-zero factor gives a new set of coordinates for the same point. In particular, is such a system of coordinates for the point. For example, the Cartesian point can be represented in coordinates as or. The original Cartesian coordinates are recovered by dividing the first two positions by the third, thus unlike Cartesian coordinates, a single point can be represented by infinitely many homogeneous coordinates. The equation of a line through the origin may be written nx + my =0 where n and 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 the limit, as t approaches infinity, in other words, as the point moves away from the origin, Z approaches 0 and the homogeneous coordinates of the point become. Thus we define as the coordinates of the point at infinity corresponding to the direction of the line nx + my =0. To summarize, Any point in the plane is represented by a triple, called the homogeneous coordinates or projective coordinates of the point. The point represented by a set of homogeneous coordinates is unchanged if the coordinates are multiplied by a common factor. Conversely, two sets of coordinates represent the same point if and only if one is obtained from the other by multiplying all the coordinates by the same non-zero constant. When Z is not 0 the point represented is the point in the Euclidean plane, when Z is 0 the point represented is a point at infinity
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. Scalars are often taken to be numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers. The operations of addition and scalar multiplication must satisfy certain requirements, called axioms. Euclidean vectors are an example of a vector space and they represent physical quantities such as forces, any two forces can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, Vector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces and these vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are commonly used. This is particularly the case of Banach spaces and Hilbert spaces, historically, the first ideas leading to vector spaces can be traced back as far as the 17th centurys analytic geometry, matrices, systems of linear equations, and Euclidean vectors. Today, vector spaces are applied throughout mathematics, science and engineering, furthermore, vector spaces furnish an abstract, 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 to more advanced notions in geometry and abstract algebra. The concept of space will first be explained by describing two particular examples, The first example of a vector space consists of arrows in a fixed plane. This is used in physics to describe forces or velocities, given any two such arrows, v and w, the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the sum of the two arrows and is denoted v + w, when a is negative, av is defined as the arrow pointing in the opposite direction, instead. Such a pair is written as, the sum of two such pairs and multiplication of a pair with a number is defined as follows, + = and a =. The first example above reduces to one if the arrows are represented by the pair of Cartesian coordinates of their end points. A vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below, elements of V are commonly called vectors. Elements of F are commonly called scalars, the second operation, called scalar multiplication takes any scalar a and any vector v and gives another vector av. In this article, vectors are represented in boldface to distinguish them from scalars
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, two lines intersect in a single point, but there are some pairs of lines that do not intersect. A projective plane can be thought of as a plane equipped with additional points at infinity where parallel lines intersect. Thus any two lines in a projective plane intersect in one and only one point. Renaissance artists, in developing the techniques of drawing in perspective, the archetypical example is the real projective plane, also known as the extended Euclidean plane. This example, in different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by PG, RP2. There are many projective planes, both infinite, such as the complex projective plane, and finite, such as the Fano plane. A projective plane is a 2-dimensional projective space, but 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 points and lines. Thus the expression point P is incident with l is used instead of either P is on l or l passes through P. To turn the ordinary Euclidean plane into a projective plane proceed as follows and that point is considered incident with each line of the class. Different parallel classes get different points and these points are called points at infinity. Add a new line which is considered incident with all the points at infinity and this line is called the line at infinity. The extended structure is a plane and is called the Extended Euclidean Plane or the real projective plane. The process outlined above, used to obtain it, is called projective completion or projectivization and this plane can also be constructed by starting from R3 viewed as a vector space, see below. The points of the Moulton plane are the points of the Euclidean plane, to create the Moulton plane from the Euclidean plane some of the lines are redefined. That is, some of their point sets will be changed, the Moulton plane has parallel classes of lines and is an affine plane. It can be projectivized, as in the example, to obtain the projective Moulton plane
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, and closely related to Desargues theorem, two triangles ABC and abc are said to be in perspective centrally if the lines Aa, Bb, and Cc meet in a common point. They are in perspective axially if the points of the corresponding triangle sides, X = AB ∩ ab, Y = AC ∩ ac, and Z = BC ∩ bc all lie on a common line. Desargues theorem in geometry states that two conditions are equivalent, if two triangles are in perspective centrally then they must also be in perspective axially, and vice versa. When this happens, the ten points and ten lines of the two together form an instance of the Desargues configuration. This construction is 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 called the complete pentahedron. The 5-cell or pentatope has five vertices, ten edges, ten triangular ridges, and five tetrahedral facets, each line intersects the hyperplane in a point, and each plane intersects the hyperplane in a line, these ten points and lines form an instance of the Desargues configuration. The Levi graph of the Desargues configuration, a graph having one vertex for each point or line in the configuration, is known as the Desargues graph, because of the symmetries and self-duality of the Desargues configuration, the Desargues graph is a symmetric graph. As a projective configuration, the Desargues configuration has the notation and its ten points can be viewed in a unique way as a pair of mutually inscribed pentagons, or as a self-inscribed decagon. There also exist eight other configurations that are not incidence-isomorphic to the Desargues configuration, in all of these configurations, each point has three other points that are not collinear with it. As with the Desargues configuration, the other depicted configuration can be viewed as a pair of mutually inscribed pentagons, barnes, John, Duality in three dimensions, Gems of Geometry, Springer, pp. 95–97, ISBN9783642309649 Coxeter, H. S. M. A memoir on the theory of mathematical form, Philosophical Transactions of the Royal Society of London,177, 1–70, doi,10. 1098/rstl.1886.0002 Weisstein, Eric W. Desargues Configuration
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 introduction of intuitive geometrical notions such as the length of a vector or the angle between two vectors. They also provide the means of defining orthogonality between vectors, inner product spaces generalize Euclidean spaces to vector spaces of any dimension, and are studied in functional analysis. An inner product induces a associated norm, thus an inner product space is also a normed vector space. A complete space with a product is called a Hilbert space. An space with a product is called a pre-Hilbert space, since its completion with respect to the norm induced by the inner product is a Hilbert space. Inner product spaces over the field of numbers are sometimes referred to as unitary spaces. In this article, the field of scalars denoted F is either the field of real numbers R or the field of complex numbers C, formally, an inner product space is a vector space V over the field F together with an inner product, i. e. Some authors, especially in physics and matrix algebra, prefer to define the inner product, then the first argument becomes conjugate linear, rather than the second. In those disciplines we would write the product ⟨ x, y ⟩ as ⟨ y | x ⟩, respectively y † x. Here the kets and columns are identified with the vectors of V and this reverse order is now occasionally followed in the more abstract literature, taking ⟨ x, y ⟩ to be conjugate linear in x rather than y. A few instead find a ground by recognizing both ⟨ ⋅, ⋅ ⟩ and ⟨ ⋅ | ⋅ ⟩ as distinct notations differing only in which argument is conjugate linear. There are various reasons why it is necessary to restrict the basefield to R and C in the definition. Briefly, the basefield has to contain an ordered subfield in order for non-negativity to make sense, the basefield has to have additional structure, such as a distinguished automorphism. More generally any quadratically closed subfield of R or C will suffice for this purpose, however in these cases when it is a proper subfield even finite-dimensional inner product spaces will fail to be metrically complete. In contrast all finite-dimensional inner product spaces over R or C, such as used in quantum computation, are automatically metrically complete. In some cases we need to consider non-negative semi-definite sesquilinear forms and this means that ⟨ x, x ⟩ is only required to be non-negative
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 binary code was designed to prevent spurious output from electromechanical switches. Today, Gray codes are used to facilitate error correction in digital communications such as digital terrestrial television. Bell Labs researcher Frank Gray introduced the term reflected binary code in his 1947 patent application and he derived the name from the fact that it may be built up from the conventional binary code by a sort of reflection process. The code was named after Gray by others who used it. Two different 1953 patent applications use Gray code as a name for the reflected binary code, one of those also lists minimum error code. A1954 patent application refers to the Bell Telephone Gray code, many devices indicate position by closing and opening switches. In the transition between the two 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. When the switches appear to be in position 001, the observer cannot tell if that is the real position 001, 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 property of a Gray code. In the standard Gray coding the least significant bit follows a pattern of 2 on,2 off, the next digit a pattern of 4 on,4 off. These codes are known as single-distance codes, reflecting the Hamming distance of 1 between adjacent codes. Reflected binary codes were applied to mathematical puzzles before they became known to engineers, martin Gardner wrote a popular account of the Gray code in his August 1972 Mathematical Games column in Scientific American. The French engineer Émile Baudot used Gray codes in telegraphy in 1878 and he received the French Legion of Honor medal for his work. The Gray code is attributed, incorrectly, to Elisha Gray. The method and apparatus were patented in 1953 and the name of Gray stuck to the codes. The PCM tube apparatus that Gray patented was made by Raymond W. Sears of Bell Labs, working with Gray and William M. Goodall, Gray codes are used in position encoders, in preference to straightforward binary encoding
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 precise definition, that an object is invariant to any of various transformations. Although these two meanings of symmetry can sometimes be told apart, they are related, so they are 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, an object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has symmetry if it can be translated without changing its overall shape. An object has symmetry if it can be simultaneously translated and rotated in three-dimensional space along a line known as a screw axis. An object has symmetry if it does not change shape when it is expanded or contracted. Fractals also exhibit a form of symmetry, where small portions of the fractal are similar in shape to large portions. Other symmetries include glide reflection symmetry and rotoreflection symmetry, a dyadic relation R is symmetric if and only if, whenever its true that Rab, its true that Rba. Thus, is the age as is symmetrical, for if Paul is the same age as Mary. Symmetric binary logical connectives are and, or, biconditional, nand, xor, the set of operations that preserve a given property of the object form a group. In general, every kind of structure in mathematics will have its own kind of symmetry, examples include even and odd functions in calculus, the symmetric group in abstract algebra, symmetric matrices in linear algebra, and the Galois group in Galois theory. In statistics, it appears as symmetric probability distributions, and as skewness, symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations. This concept has one of the most powerful tools of theoretical physics. See Noethers theorem, and also, Wigners classification, which says that the symmetries of the laws of physics determine the properties of the found in nature. Important symmetries in physics include continuous symmetries and discrete symmetries of spacetime, internal symmetries of particles, in biology, the notion of symmetry is mostly used explicitly to describe body shapes. Bilateral animals, including humans, are more or less symmetric with respect to the plane which divides the body into left
14.
Symmetry group
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In abstract algebra, the symmetry group of an object is the group of all transformations under which the object is invariant with composition as the group operation. For a space with a metric, it is a subgroup of the group of the space concerned. If not stated otherwise, this article considers symmetry groups in Euclidean geometry, the objects may be geometric figures, images, and patterns, such as a wallpaper pattern. The definition can be more precise by specifying what is meant by image or pattern. For symmetry of objects, one may also want to take their physical composition into account. The group of isometries of space induces an action on objects in it. The symmetry group is 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 figure invariant is called its symmetry group. The proper symmetry group of an object is equal to its symmetry group if. The proper symmetry group is then a subgroup of the orthogonal group SO. A discrete symmetry group is a group such that for every point of the space the set of images of the point under the isometries in the symmetry group is a discrete set. There are also continuous symmetry groups, which contain rotations of arbitrarily small angles or translations of arbitrarily small distances, the group of all symmetries of a sphere O is an example of this, and 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, two 3D figures have mirror symmetry, but with respect to different mirror planes. Two 3D figures have 3-fold rotational symmetry, but with respect to different axes, two 2D patterns have translational symmetry, each in one direction, the two translation vectors have the same length but a different direction. When considering isometry groups, one may restrict oneself to those where for all points the set of images under the isometries is topologically closed. This includes all discrete isometry groups and also involved in continuous symmetries. A figure with this group is non-drawable and up to arbitrarily fine detail homogeneous. The group generated by all translations, this group cannot be the group of a pattern, it would be homogeneous
15.
Projective linear group
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The projective special linear group, 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. Here SZ is the center of SL, and is identified with the group of nth roots of unity in K. If V is the vector space over a field F, namely V = Fn. 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 geometry, where the projective group acting on homogeneous coordinates is the underlying group of the geometry. Stated differently, the action of GL on V descends to an action of PGL on the projective space P. The projective linear groups therefore generalise the case PGL of Möbius transformations and this is reflected in the notation, PGL is the group associated to GL, and is the projective linear group of -dimensional projective space, not n-dimensional projective space. A related group is the group, which is defined axiomatically. A collineation is a map which sends collinear points to collinear points. Projective linear transforms are collineations, but in general not all collineations are projective linear transforms – PGL is in general a proper subgroup of the collineation group. Correspondingly, the quotient group PΓL/PGL = Gal corresponds to choices of linear structure, one may also define collineation groups for axiomatically defined projective spaces, where there is no natural notion of a projective linear transform. The elements of the linear group can be understood as tilting the plane along one of the axes, and then projecting to the original plane. Visually, this corresponds to standing at the origin, and turning ones angle of view, PGL sends collinear points to collinear points, but it is not the full collineation group, which is instead either PΓL or the full symmetric group for n =2. Every algebraic automorphism of a space is projective linear. The birational automorphisms form a group, the Cremona group. PGL acts faithfully on projective space, non-identity elements act non-trivially, concretely, the kernel of the action of GL on projective space is exactly the scalar maps, which are quotiented out in PGL. PGL acts 2-transitively on projective space, PGL acts sharply 3-transitively on the projective line
16.
Projective special linear group
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The projective special linear group, 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. Here SZ is the center of SL, and is identified with the group of nth roots of unity in K. If V is the vector space over a field F, namely V = Fn. 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 geometry, where the projective group acting on homogeneous coordinates is the underlying group of the geometry. Stated differently, the action of GL on V descends to an action of PGL on the projective space P. The projective linear groups therefore generalise the case PGL of Möbius transformations and this is reflected in the notation, PGL is the group associated to GL, and is the projective linear group of -dimensional projective space, not n-dimensional projective space. A related group is the group, which is defined axiomatically. A collineation is a map which sends collinear points to collinear points. Projective linear transforms are collineations, but in general not all collineations are projective linear transforms – PGL is in general a proper subgroup of the collineation group. Correspondingly, the quotient group PΓL/PGL = Gal corresponds to choices of linear structure, one may also define collineation groups for axiomatically defined projective spaces, where there is no natural notion of a projective linear transform. The elements of the linear group can be understood as tilting the plane along one of the axes, and then projecting to the original plane. Visually, this corresponds to standing at the origin, and turning ones angle of view, PGL sends collinear points to collinear points, but it is not the full collineation group, which is instead either PΓL or the full symmetric group for n =2. Every algebraic automorphism of a space is projective linear. The birational automorphisms form a group, the Cremona group. PGL acts faithfully on projective space, non-identity elements act non-trivially, concretely, the kernel of the action of GL on projective space is exactly the scalar maps, which are quotiented out in PGL. PGL acts 2-transitively on projective space, PGL acts sharply 3-transitively on the projective line
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, and 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 nonabelian simple group after the alternating group A5 on five letters with 60 elements, the general linear group GL consists of all invertible 2×2 matrices over F7, the finite field with 7 elements. The subgroup SL consists of all matrices with unit determinant. Then PSL is defined to be the quotient group SL/ obtained by identifying I and −I, in this article, we let G denote any group isomorphic to PSL. G = PSL has 168 elements and this can be seen by counting the possible columns, there are 72−1 =48 possibilities for the first column, then 72−7 =42 possibilities for the second column. We must divide by 7−1 =6 to force the determinant equal to one and it is a general result that PSL is simple for n, q ≥2, unless = or. PSL is isomorphic to the symmetric group S3, and 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 classes and irreducible representations is 6. The sizes of conjugacy classes are 1,21,42,56,24,24, the dimensions of irreducible representations 1,3,3,6,7,8. Note that the classes 7A and 7B are exchanged by an automorphism, so the representatives from GL, the order of group is 168=3*7*8, this implies existence of Sylows subgroups of orders 3,7 and 8. It is easy to describe the first two, they are cyclic, since any group of order is cyclic. Any element of conjugacy class 3A56 generates Sylow 3-subgroup, any element from the conjugacy classes 7A24, 7B24 generates the Sylow 7-subgroup. The Sylow 2-subgroup is a group of order 8. It can be described as centralizer of any element from the conjugacy class 2A21, in the GL representation, a Sylow 2-subgroup consists of the upper triangular matrices. This group and its Sylow 2-subgroup provide a counter-example for various normal p-complement theorems for p =2, however, PSL is also isomorphic to PSL, the special linear group of 3×3 matrices over the field with 2 elements. The Fano plane can be used to describe multiplication of octonions, the Klein quartic is the projective variety over the complex numbers C defined by the quartic polynomial x3y + y3z + z3x =0. It is a compact Riemann surface of genus g =3 and this bound is due to the Hurwitz automorphisms theorem, which holds for all g>1
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. This forms a group, because the product of two matrices is again invertible, and 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 general linear group over R is the group of n×n invertible matrices of real numbers, and is denoted by GLn or GL. More generally, the linear group of degree n over any field F, or a ring R, is the set of n×n invertible matrices with entries from F. Typical notation is GLn or GL, or simply GL if the field is understood, more generally still, the general linear group of a vector space GL is the abstract automorphism group, not necessarily written as matrices. The special linear group, written SL or SLn, is the subgroup of GL consisting of matrices with a determinant of 1, the group GL and its subgroups are often called linear groups or matrix groups. These groups are important in the theory of representations, and also arise in the study of spatial symmetries and symmetries of vector spaces in general. The modular group may be realised as a quotient of the linear group SL. If n ≥2, then the group GL is not abelian, if V has finite dimension n, then GL and GL are isomorphic. The isomorphism is not canonical, it depends on a choice of basis in V, in a similar way, for a commutative ring R the group GL may be interpreted as the group of automorphisms of a free R-module M of rank n. One can also define GL for any R-module, but in general this is not isomorphic to GL, over a field F, a matrix is invertible if and only if its determinant is nonzero. Therefore, a definition of GL is as the group of matrices with nonzero determinant. Over a non-commutative ring R, determinants are not at all well behaved, in this case, GL may be defined as the unit group of the matrix ring M. The general linear group GL over the field of numbers is a real Lie group of dimension n2. To see this, note that the set of all n×n real matrices, Mn, the subset GL consists of those matrices whose determinant is non-zero. The determinant is a map, and hence GL is an open affine subvariety of Mn. The Lie algebra of GL, denoted g l n, consists of all n×n real matrices with the serving as the Lie bracket. As a manifold, GL is not connected but rather has two connected components, the matrices with positive determinant and the ones with negative determinant, the identity component, denoted by GL+, consists of the real n×n matrices with positive determinant
19.
Cycles and fixed points
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In mathematics, the cycles of a permutation π of a finite set S correspond bijectively to the orbits of the subgroup generated by π acting on S. These orbits are subsets of S that can be written as, 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. The size l of the orbit is called the length of the cycle, when l =1. A permutation is determined by giving an expression for each of its cycles, for example, let π = = be a permutation that maps 1 to 2,6 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 > j >1, s = s + s· There is only one way to construct a permutation of k elements with k cycles, Every cycle must have length 1 so every element must be a fixed point. Every cycle of length k may be written as permutation of the number 1 to k, There are k different ways to write a given cycle of length k, e. g. = = =. The value 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. We may choose one of the f permutations with k −1 elements and j fixed points and insert element k in an existing cycle of length >1 in front of one of the − j elements. We may choose one of the f permutations with k −1 elements and j +1 fixed points, =120 -120 +60 -20 +5 =45. Example, f =120 - =120 - =120 -76 =44, for every k >1, f = Example, f =4 × =4 ×11 =44 For every k >1, f = k. Example, f =120 × =120 × =120 × 44/120 =44 f ≈ k. / e where e is Eulers number ≈2.71828 Cyclic permutation Cycle notation Brualdi, Richard A
20.
Transposition (mathematics)
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If S has k elements, the cycle is called a k-cycle. On the other hand, the permutation that sends 1 to 3,3 to 1,2 to 4 and 4 to 2 is not a cyclic permutation because it separately permutes the pairs, the set S is called the orbit of the cycle. Every permutation on finitely many elements can be decomposed into a collection of cycles on disjoint orbits, the cyclic parts of a permutation are cycles, thus the second example is composed of a 3-cycle and a 1-cycle and the third is composed of two 2-cycles. A permutation is called a cyclic permutation if and only if it has a single nontrivial cycle, for example, the permutation, written in two-line and also cycle notations, = =, is a six-cycle, its cycle 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 set, then of course the largest orbit. Let s 0 be any element of S, and put s i = σ i for any i ∈ Z, if S is finite, there is a minimal number k ≥1 for which s k = s 0. Then S =, and σ is the permutation defined by σ = s i +1 for 0 ≤ i < k and σ = x for any element of X ∖ S. The elements not fixed by σ can be pictured as s 0 ↦ s 1 ↦ s 2 ↦ ⋯ ↦ s k −1 ↦ s k = s 0, a cycle can be written using the compact cycle notation σ =. The length of a cycle is the number of elements of its largest orbit, a cycle of length k is also called a k-cycle. The orbit of a 1-cycle is called a point of the permutation. When cycle notation is used, the 1-cycles are often suppressed when no confusion will result, the number of k-cycles in the symmetric group Sn is given, for 1 ≤ k ≤ n, by the following equivalent formulas. K A k-cycle has signature k −1, a cycle with only two elements is called a transposition. For example the permutation π = that swaps 2 and 4, any permutation can be expressed as the composition of transpositions—formally, they are generators for the group. In fact, when the set being permuted is for some n, then any permutation can be expressed as a product of adjacent transpositions. This follows because an arbitrary transposition can be expressed as the product of adjacent transpositions, instead one may roll the elements keeping a where it is by executing the right factor first. This has moved z to the position of b, so after the first permutation, the transposition, executed thereafter, then addresses z by the index of b to swap what initially were a and z. In fact, the group is a Coxeter group, meaning that it is generated by elements of order 2
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. The origin of the quadrilateral is the two Latin words quadri, a variant of four, and latus, meaning side. Quadrilaterals are simple or complex, also called crossed, simple quadrilaterals are either convex or concave. The interior angles of a simple quadrilateral ABCD add up to 360 degrees of arc and this is a special case of the n-gon interior angle sum formula × 180°. All non-self-crossing quadrilaterals tile the plane by repeated rotation around the midpoints of their edges, any quadrilateral that is not self-intersecting is a simple quadrilateral. In a convex quadrilateral, all angles are less than 180°. Irregular quadrilateral or trapezium, no sides are parallel, trapezium or trapezoid, at least one pair of opposite sides are parallel. Isosceles trapezium or isosceles trapezoid, one pair of sides are parallel. Alternative definitions are a quadrilateral with an axis of symmetry bisecting one pair of opposite sides, parallelogram, a quadrilateral with two pairs of parallel sides. Equivalent conditions are that opposite sides are of length, that opposite angles are equal. In other words, parallelograms include all rhombi and all rhomboids, rhombus or rhomb, all four sides are of equal length. An equivalent condition is that the diagonals bisect each other. Rhomboid, a parallelogram in which adjacent sides are of unequal lengths, not all references agree, some define a rhomboid as a parallelogram which is not a rhombus. Rectangle, all four angles are right angles, an equivalent condition is that the diagonals bisect each other and are equal in length. Square, all four sides are of length, and all four angles are right angles. An equivalent condition is that opposite sides are parallel, that the diagonals bisect each other. A quadrilateral is a if and only if it is both a rhombus and a rectangle
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. More formally, a flag ψ of an n-polytope is a set such that Fi ≤ Fi+1, since, however, the minimal face F−1 and the maximal face Fn must be in every flag, they are often omitted from the list of faces, as a shorthand. These latter two are called improper faces, for example, a flag of a polyhedron comprises one vertex, one edge incident to that vertex, and one polygonal face incident to both, plus the two improper faces. A flag of a polyhedron is called a dart. A polytope may be regarded as if, and only if. This level of abstraction generalizes both the concept given above as 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 incidence geometry have the same size, this common value is the rank of the geometry
23.
Triangle
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A triangle is a polygon with three edges and three vertices. It is one of the shapes in geometry. A triangle with vertices A, B, and C is denoted △ A B C, in Euclidean geometry any three points, when non-collinear, determine a unique triangle and 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 all sides the same length. An equilateral triangle is also a 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 triangles isosceles triangles, the 45–45–90 right triangle, which appears in the tetrakis 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, a side can be marked with a pattern of ticks, short line segments in the form of tally marks, two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is no more than 3 ticks. Similarly, patterns of 1,2, or 3 concentric arcs inside the angles are used to indicate equal angles, triangles can also be classified according to their internal angles, measured here in degrees. A right triangle has one of its interior angles measuring 90°, the side opposite to the right angle is the hypotenuse, the longest side of the triangle. The other two sides are called the legs or catheti of the triangle, special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 +42 =52, in this situation,3,4, and 5 are a Pythagorean triple. The other one is a triangle that has 2 angles that each measure 45 degrees. Triangles that do not have an angle measuring 90° are called oblique triangles, a triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. If c is the length of the longest side, then a2 + b2 > c2, a triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side, then a2 + b2 < c2, a triangle with an interior angle of 180° is degenerate
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. An explicit quartic with twenty-eight real bitangents was first given by Plücker As Plücker showed, another quartic with 28 real bitangents can be formed by the locus of centers of ellipses with fixed axis lengths, tangent to two non-parallel lines. The Trott curve, another curve with 28 real bitangents, is the set of satisfying the degree four polynomial equation 144 −225 +350 x 2 y 2 +81 =0. These points form a quartic curve that has genus three and that has twenty-eight real bitangents. Like the examples of Plücker and of Blum and Guinand, the Trott curve has four separated ovals, the number for a curve of degree four. Additionally, each oval bounds a nonconvex region of the plane and has one bitangent spanning the nonconvex portion of its boundary, the dual curve to a quartic curve has 28 real ordinary double points, dual to the 28 bitangents of the primal curve. The 28 bitangents of a quartic may also be placed in correspondence with symbols of the form where a, b, c, d, e and f are all zero or one and where ad + be + cf =1. There are 64 choices for a, b, c, d, e and f, 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, the 28 bitangents of a quartic also correspond to pairs of the 56 lines on a degree-2 del Pezzo surface, and to the 28 odd theta characteristics. A quartic with 28 real bitangents, Canadian Mathematical Bulletin,7, 399–404, Cayley, Arthur, On the bitangents of a quartic, Salmons Higher Plane Curves, pp. 387–389. In The collected mathematical papers of Arthur Cayley, Andrew Russell Forsyth, gray, Jeremy, From the history of a simple group, The Mathematical Intelligencer,4, 59–67, doi,10. 1007/BF03023483, MR0672918. The Eightfold Way, MSRI Publications,35, Cambridge University Press, pp. 115–131, ISBN 0-521-66066-1, MR1722415. Manivel, L. Configurations of lines and models of Lie algebras, Journal of Algebra,304, 457–486, Plücker, J. Theorie der algebraischen Curven, gegrundet auf eine neue Behandlungsweise der analytischen Geometrie, Berlin, Adolph Marcus. Zur Theorie der Abelschen Funktionen für den Fall p =3, shioda, Tetsuji, Weierstrass transformations and cubic surfaces, Commentarii Mathematici Universitatis Sancti Pauli,44, 109–128, MR1336422. Trott, Michael, Applying GroebnerBasis to Three Problems in Geometry, Mathematica in Education and Research,6, 15–28
25.
Hexagon
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In geometry, a hexagon is a six sided polygon or 6-gon. The total of the angles of any hexagon is 720°. A regular hexagon has Schläfli symbol and can also be constructed as an equilateral triangle, t. A regular hexagon is defined as a hexagon that is both equilateral and equiangular and it is bicentric, meaning that it is both cyclic and tangential. The common length of the sides equals the radius of the circumscribed circle, all internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries and 6 reflection symmetries, the longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane, the cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral, the maximal diameter, D is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the circle, d, is twice the minimal radius or inradius. If a regular hexagon has successive vertices A, B, C, D, E, F, the regular hexagon has Dih6 symmetry, order 12. There are 3 dihedral subgroups, Dih3, Dih2, and Dih1, and 4 cyclic subgroups, Z6, Z3, Z2 and these symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order, r12 is full symmetry, and a1 is no symmetry. These two forms are duals of each other and have half the order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction and it can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites. G2 hexagons, with sides parallel are also called hexagonal parallelogons. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g6 subgroup has no degrees of freedom but can seen as directed edges. Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation, other hexagon shapes can tile the plane with different orientations
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 order theory, an order is not modeled as a binary relation. One does not say that east is more clockwise than west, instead, a cyclic order is defined as a ternary relation, meaning after a, one reaches b before c. A ternary relation is called an order if it is cyclic, asymmetric, transitive. Dropping the total requirement results in a cyclic order. A set with an order is called a cyclically ordered set or simply a cycle. In a finite cycle, each element has a next element, There are also continuously variable cycles with infinitely many elements, such as the oriented unit circle in the plane. Cyclic orders are related to the more familiar linear orders. Any linear order can be bent into a circle, and any order can be cut at a point. These operations, along with the constructions of intervals and covering maps. Cycles have more symmetries than linear orders, and they naturally occur as residues of linear structures. A cyclic order on a set X with n elements is like an arrangement of X on a clock face, for an n-hour clock. Each element x in X has an element and a previous element. There are a few 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. A cycle with n elements is also a Zn-torsor, a set with a transitive action by a finite cyclic group. Another formulation is to make X into the standard directed cycle graph on n vertices and it can be instinctive to use cyclic orders for symmetric functions, for example as in xy + yz + zx where writing the final monomial as xz would distract from the pattern. A substantial use of cyclic orders is in the determination of the classes of free groups. A cyclic order on a set X can be determined by an order on X
27.
Group (mathematics)
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In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, namely closure and it allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in areas within and outside mathematics makes them a central organizing principle of contemporary mathematics. Groups share a kinship with the notion of symmetry. The concept of a group arose from the study of polynomial equations, after contributions from other fields such as number theory and 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 smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. A theory has developed for finite groups, which culminated with the classification of finite simple groups. Since the mid-1980s, geometric group theory, which studies finitely generated groups as objects, has become a particularly active area in group theory. One of the most familiar groups is the set of integers Z which consists of the numbers, −4, −3, −2, −1,0,1,2,3,4. The following properties of integer addition serve as a model for the group axioms given in the definition below. For any two integers a and b, the sum a + b is also an integer and that is, addition of integers always yields an integer. This property is known as closure under addition, for all integers a, b and c, + c = a +. Expressed in words, adding a to b first, and then adding the result to c gives the final result as adding a to the sum of b and c. If a is any integer, then 0 + a = a +0 = a, zero is called the identity element of addition because adding it to any integer returns the same integer. For every integer a, there is a b such that a + b = b + a =0. The integer b is called the element of the integer a and is denoted −a. The integers, together with the operation +, form a mathematical object belonging to a class sharing similar structural aspects. To appropriately understand these structures as a collective, the abstract definition is developed
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, and 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 nonabelian simple group after the alternating group A5 on five letters with 60 elements, the general linear group GL consists of all invertible 2×2 matrices over F7, the finite field with 7 elements. The subgroup SL consists of all matrices with unit determinant. Then PSL is defined to be the quotient group SL/ obtained by identifying I and −I, in this article, we let G denote any group isomorphic to PSL. G = PSL has 168 elements and this can be seen by counting the possible columns, there are 72−1 =48 possibilities for the first column, then 72−7 =42 possibilities for the second column. We must divide by 7−1 =6 to force the determinant equal to one and it is a general result that PSL is simple for n, q ≥2, unless = or. PSL is isomorphic to the symmetric group S3, and 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 classes and irreducible representations is 6. The sizes of conjugacy classes are 1,21,42,56,24,24, the dimensions of irreducible representations 1,3,3,6,7,8. Note that the classes 7A and 7B are exchanged by an automorphism, so the representatives from GL, the order of group is 168=3*7*8, this implies existence of Sylows subgroups of orders 3,7 and 8. It is easy to describe the first two, they are cyclic, since any group of order is cyclic. Any element of conjugacy class 3A56 generates Sylow 3-subgroup, any element from the conjugacy classes 7A24, 7B24 generates the Sylow 7-subgroup. The Sylow 2-subgroup is a group of order 8. It can be described as centralizer of any element from the conjugacy class 2A21, in the GL representation, a Sylow 2-subgroup consists of the upper triangular matrices. This group and its Sylow 2-subgroup provide a counter-example for various normal p-complement theorems for p =2, however, PSL is also isomorphic to PSL, the special linear group of 3×3 matrices over the field with 2 elements. The Fano plane can be used to describe multiplication of octonions, the Klein quartic is the projective variety over the complex numbers C defined by the quartic polynomial x3y + y3z + z3x =0. It is a compact Riemann surface of genus g =3 and this bound is due to the Hurwitz automorphisms theorem, which holds for all g>1
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. There are many equivalent ways to define a matroid, the most significant being in terms of independent sets, bases, circuits, closed sets or flats, closure operators, and rank functions. Matroid theory borrows extensively from the terminology of linear algebra and graph theory, Matroids have found applications in geometry, topology, combinatorial optimization, network theory and coding theory. There are many equivalent ways to define a matroid. In terms of independence, a finite matroid M is a pair, alternatively, at least one subset of E is independent, i. e. Every subset of an independent set is independent, i. e. for each A ′ ⊂ A ⊂ E and this is sometimes called the hereditary property. If A and B are two independent sets of I and A has more elements than B, then there exists x ∈ A ∖ B such that B ∪ is in I and 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 that is not independent is called dependent. A maximal independent set—that is, 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 —that is, 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. For instance, one may define a matroid M to be a pair, if A and B are distinct members of B and a ∈ A ∖ B, then there exists an element b ∈ B ∖ A such that A ∖ ∪ ∈ B. It follows from the basis exchange property that no member of B can be a subset of another. It is a result of matroid theory, directly analogous to a similar theorem of bases in linear algebra. This number is called the rank of M, if M is a matroid on E, and A is a subset of E, then a matroid on A can be defined by considering a subset of A to be independent if and only if it is independent in M. This allows us to talk about submatroids and about the rank of any subset of E, the rank of a subset A is given by the rank function r of the matroid, which has the following properties, The value of the rank function is always a non-negative integer. For any subset A of E, r ≤ | A |, for any two subsets A and B of E, r + r ≤ r + r. That is, the rank is a submodular function, for any set A and element x, r ≤ r ≤ r +1. From the first of two inequalities it follows more generally that, if A ⊂ B ⊂ E, then r ≤ r ≤ r
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. There are many equivalent ways to define a matroid, the most significant being in terms of independent sets, bases, circuits, closed sets or flats, closure operators, and rank functions. Matroid theory borrows extensively from the terminology of linear algebra and graph theory, Matroids have found applications in geometry, topology, combinatorial optimization, network theory and coding theory. There are many equivalent ways to define a matroid. In terms of independence, a finite matroid M is a pair, alternatively, at least one subset of E is independent, i. e. Every subset of an independent set is independent, i. e. for each A ′ ⊂ A ⊂ E and this is sometimes called the hereditary property. If A and B are two independent sets of I and A has more elements than B, then there exists x ∈ A ∖ B such that B ∪ is in I and 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 that is not independent is called dependent. A maximal independent set—that is, 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 —that is, 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. For instance, one may define a matroid M to be a pair, if A and B are distinct members of B and a ∈ A ∖ B, then there exists an element b ∈ B ∖ A such that A ∖ ∪ ∈ B. It follows from the basis exchange property that no member of B can be a subset of another. It is a result of matroid theory, directly analogous to a similar theorem of bases in linear algebra. This number is called the rank of M, if M is a matroid on E, and A is a subset of E, then a matroid on A can be defined by considering a subset of A to be independent if and only if it is independent in M. This allows us to talk about submatroids and about the rank of any subset of E, the rank of a subset A is given by the rank function r of the matroid, which has the following properties, The value of the rank function is always a non-negative integer. For any subset A of E, r ≤ | A |, for any two subsets A and B of E, r + r ≤ r + r. That is, the rank is a submodular function, for any set A and element x, r ≤ r ≤ r +1. From the first of two inequalities it follows more generally that, if A ⊂ B ⊂ E, then r ≤ r ≤ r
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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 that is both graphic and co-graphic is called a planar matroid, these are exactly the graphic matroids formed from planar graphs. If G is a graph, and F is the family of sets of edges that form forests in G. Thus, F forms the independent sets of a matroid, called the graphic matroid of G or M, more generally, a matroid is called graphic whenever it is isomorphic to the graphic matroid of a graph, regardless of whether its elements are themselves edges in a graph. The bases of a graphic matroid M are the forests of G. The corank of the matroid is known as the circuit rank or 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. In the lattice of flats of this matroid, there is an order relation x ≤ y whenever the corresponding to flat x is a refinement of the partition corresponding to flat y. Thus, the lattice of flats of the 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, the graphic matroid of a graph G can be defined as the column matroid of any oriented incidence matrix of G. Such a matrix has one row for each vertex, and one column for each edge, the column 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 sum to zero, conversely, if a set of edges forms a forest, then by repeatedly removing leaves from this forest it can be shown by induction that the corresponding set of columns is 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, therefore, graphic matroids form a subset of the regular matroids, matroids that have representations over all possible fields. Graphic matroids are connected if and only if the graph is both connected and 2-vertex-connected. The first three of these are the forbidden minors for the regular matroids, and the duals of M and M are regular, if a matroid is graphic, its dual cannot contain the duals of these five forbidden minors. Thus, the dual must also be regular, and cannot contain as minors the two graphic matroids M and M, if G is planar, the dual of M is the graphic matroid of the dual graph of G. While G may have multiple dual graphs, their graphic matroids are all isomorphic, a minimum weight basis of a graphic matroid is a minimum spanning tree
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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 and this definition is relatively modern, generalizing the classical definition of Steiner systems which in addition required that k = t +1. An S was called a Steiner triple system, while an S was called a Steiner quadruple system, with the generalization of the definition, this naming system is no longer strictly adhered to. A long-standing problem in design theory was if any nontrivial Steiner systems have t ≥6 and this was solved in the affirmative by Peter Keevash in 2014. A finite affine plane of order q, with the lines as blocks, is an S, an affine plane of order q can be obtained from a projective plane of the same order by removing one block and all of the points in that block from the projective plane. Choosing different blocks to remove in this way can lead to non-isomorphic affine planes, an S is called a Steiner triple system, and its blocks are called triples. It is common to see the abbreviation STS for a Steiner triple system of order n, the number of triples through a point is /2 and hence the total number of triples is n/6. This shows that n must be of the form 6k+1 or 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 projective plane of order 2 is an STS and the plane of order 3 is an STS. Up to isomorphism, the STS and STS are unique, there are two STSs,80 STSs, and 11,084,874,829 STSs. We can define a multiplication on the set S using the Steiner triple system by setting aa = a for all a in S and this makes S an idempotent, commutative quasigroup. It has the property that ab = c implies bc = a. Conversely, any quasigroup with these properties arises from a Steiner triple system, commutative idempotent quasigroups satisfying this additional property are called Steiner quasigroups. An S is called a Steiner quadruple system, a necessary and sufficient condition for the existence of an S is that n ≡2 or 4. The abbreviation SQS is often used for these systems, up to isomorphism, SQS and SQS are unique, there are 4 SQSs and 1,054,163 SQSs. An S is called a Steiner quintuple system, a necessary condition for the existence of such a system is that n ≡3 or 5 which comes from considerations that apply to all the classical Steiner systems. An additional necessary condition is that n ≢4, which comes from the fact that the number of blocks must be an integer, There is a unique Steiner quintuple system of order 11, but none of order 15 or order 17. Systems are known for orders 23,35,47,71,83,107,131,167 and 243, the smallest order for which the existence is not known is 21
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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 identity element is called a loop. There are at least two structurally equivalent formal definitions of quasigroup, one defines a quasigroup as a set with one binary operation, and the other, from universal algebra, defines a quasigroup as having three primitive operations. The homomorphic image of a quasigroup defined with a binary operation, however. We begin with the first definition, a quasigroup is a set, Q, with a binary operation, ∗, obeying the Latin square property. This states that, for each a and b in Q, the uniqueness requirement can be replaced by the requirement that the magma be cancellative. The unique solutions to these equations are written x = a \ b and y = b / a, the operations \ and / are called, respectively, left and right division. The empty set equipped with the empty binary operation satisfies this definition of a quasigroup, some authors accept the empty quasigroup but others explicitly exclude it. Algebraic structures axiomatized solely by identities are called varieties, many standard results in universal algebra hold only for varieties. Quasigroups are varieties if left and right division are taken as primitive, a quasigroup is a type algebra satisfying the identities, y = x ∗, y = x \, y = ∗ x, y = / x. In other words, Multiplication and division in order, one after the other. Hence if is a quasigroup according to the first definition, then is the same quasigroup in the sense of universal algebra. A loop is a quasigroup with an identity element, that is and it follows that the identity element, e, is unique, and that every element of Q has a unique left and right inverse. Since the presence of an identity element is essential, a loop cannot be empty. e, a loop that is associative is a group. A group can have a non-associative pique isotope, but it cannot have a nonassociative loop isotope, there are also some weaker associativity-like properties which have been given special names. A Bol loop is a loop that either, x ∗ = ∗ z for each x, y and z in Q. A loop that is both a left and right Bol loop is a Moufang loop, a narrower class that is a total symmetric quasigroup in which all conjugates coincide as one operation, xy = x / y = x \ y. Another way to define totally symmetric quasigroup is as a quasigroup which additionally is commutative
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Octonion
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In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold O. There are three lower-dimensional normed division algebras over the reals, the real numbers R themselves, the complex numbers C, 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 form of associativity. Octonions are not as known as the quaternions and complex numbers. Despite this, they have interesting properties and are related to a number of exceptional structures in mathematics. Additionally, octonions have applications in such as string theory, special relativity. The octonions were discovered in 1843 by John T. Graves, the octonions were discovered independently by Cayley and are sometimes referred to as Cayley numbers or the Cayley algebra. Hamilton described the 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. Every octonion is a linear combination of the unit octonions. Addition and subtraction of octonions is done by adding and subtracting corresponding terms and hence their coefficients, 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. The above definition though is not unique, but is one of 480 possible definitions for octonion multiplication with e0 =1. The others can be obtained by permuting and changing the signs of the basis elements. The 480 different algebras are isomorphic, and there is rarely a need to consider which particular multiplication rule is used. Each of these 480 definitions is invariant up to signs under some 7-cycle of the points, a common choice is to use the definition invariant under the 7-cycle with e1e2 = e4 as it is particularly easy to remember the multiplication. A variation of this sometimes used is to label the elements of the basis by the elements ∞,0,1,2,6, of the projective line over the finite field of order 7. The multiplication is given by e∞ =1 and e1e2 = e4. These are the nonzero codewords of the quadratic residue code of length 7 over the field of 2 elements
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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 source and target coincide. For most algebraic structures, including groups and rings, a homomorphism is an isomorphism if, in topology, where the morphisms are continuous functions, isomorphisms are also called homeomorphisms or bicontinuous functions. In mathematical analysis, where the morphisms are functions, isomorphisms are also called diffeomorphisms. A canonical isomorphism is a map that is an isomorphism. Two objects are said to be isomorphic if there is a canonical isomorphism between them. Isomorphisms are formalized using category theory, let R + be the multiplicative group of positive real numbers, and let R be the additive group of real numbers. The logarithm function log, R + → R satisfies log = log x + log y for all x, y ∈ R +, so it is a group homomorphism. The exponential function exp, R → R + satisfies exp = for all x, y ∈ R, the identities log exp x = x and exp log y = y show that log and exp are inverses of each other. Since log is a homomorphism that has an inverse that is also a homomorphism, because log is an isomorphism, it translates multiplication of positive real numbers into addition of real numbers. This facility makes it possible to real numbers using a ruler. Consider the group, the integers from 0 to 5 with addition modulo 6 and these structures are isomorphic under addition, if you identify them using the following scheme, ↦0 ↦1 ↦2 ↦3 ↦4 ↦5 or in general ↦ mod 6. For example, + =, which translates in the system as 1 +3 =4. Even though these two groups look different in that the sets contain different elements, they are indeed isomorphic, more generally, the direct product of two cyclic groups Z m and Z n is isomorphic to if and only if m and n are coprime. For example, R is an ordering ≤ and S an ordering ⊑, such an isomorphism is called an order isomorphism or an isotone isomorphism. If X = Y, then this is a relation-preserving automorphism, in a concrete category, such as the category of topological spaces or categories of algebraic objects like groups, rings, and modules, an isomorphism must be bijective on the underlying sets. In algebraic categories, an isomorphism is the same as a homomorphism which is bijective on underlying sets, in abstract algebra, two basic isomorphisms are defined, Group isomorphism, an isomorphism between groups Ring isomorphism, an isomorphism between rings. Just as the automorphisms of an algebraic structure form a group, letting a particular isomorphism identify the two structures turns this heap into a group
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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 setting, incidence structures are not limited to just points and lines. The study of structures is sometimes called finite geometry. An incidence structure is a triple where P is a set whose elements are called points, L is a disjoint set whose elements are called lines, the elements of I are called flags. If is in I then it was typical to say that point p lies on line l or that the line l passes through point p, in some common situations L may be a set of subsets of P in which case incidence I will be containment. Incidence structures of type are called set-theoretic. This is not always the case, for example, if P is a set of vectors and L a set of square matrices and this example also shows that while the geometric language of points and lines is used, the object types need not be these geometric objects. An incidence structure is uniform if each line is incident with the number of points. Each of these examples, except the second, is uniform with three points per line, Any graph is a uniform incidence structure with two points per line. For these examples, the vertices of the form the point set, the edges of the graph form the line set. Incidence structures are studied in their full generality, it is typical to study incidence structures that satisfy some additional axioms. For instance, a linear space is an incidence structure that satisfies. If the first axiom above is replaced by the stronger, Any two distinct points are incident with one common line, the incidence structure is called a linear space. If we interchange the role of points and lines in C = we obtain the structure, C∗ =. It follows immediately from the definition that, C∗∗ = C and this is an abstract version of projective duality. A structure C that is isomorphic to its dual C∗ is called self-dual, the Fano plane above is a self-dual incidence structure. Incidence structures use a geometric terminology, but in graph theoretic terms they are called hypergraphs and they are also known as a set system or family of sets in a general context
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Projective geometry
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Projective geometry is a topic of mathematics. It is the study of properties that are invariant with respect to projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than expressible by a transformation matrix and translations. The first issue for geometers is what kind of geometry is adequate for a novel situation, one source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See projective plane for the basics of geometry in two dimensions. While the ideas were available earlier, projective geometry was mainly a development of the 19th century and this included the theory of complex projective space, the coordinates used being complex numbers. Several major types of more abstract mathematics were based on projective geometry and it 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 many research subtopics, two examples of which are projective algebraic geometry and projective differential geometry. Projective geometry is an elementary form of geometry, meaning that it is not based on a concept of distance. In two dimensions it begins with the study of configurations of points and lines and that there is indeed some geometric interest in this sparse setting was first established by Desargues and others in their exploration of the principles of perspective art. In higher dimensional spaces there are considered hyperplanes, and other linear subspaces, Projective geometry can also be seen as a geometry of constructions with a straight-edge alone. Since projective geometry excludes compass constructions, there are no circles, no angles, no measurements, no parallels and it was realised that the theorems that do apply to projective geometry are simpler statements. For example, the different conic sections are all equivalent in projective geometry, during the early 19th century the work of Jean-Victor Poncelet, Lazare Carnot and others established projective geometry as an independent field of mathematics. Its rigorous foundations were addressed by Karl von Staudt and perfected by Italians Giuseppe Peano, Mario Pieri, Alessandro Padoa, after much work on the very large number of theorems in the subject, therefore, the basics of projective geometry became understood. The incidence structure and the cross-ratio are fundamental invariants under projective transformations, Projective geometry can be modeled by the affine plane plus a line at infinity and then treating that line as ordinary. An algebraic model for doing projective geometry in the style of geometry is given by homogeneous coordinates. In a foundational sense, projective geometry and ordered geometry are elementary since they involve a minimum of axioms and either can be used as the foundation for affine, Projective geometry is not ordered and so it is a distinct foundation for geometry
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Projective configuration
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Ernst Steinitz wrote his dissertation on the subject in 1894, and they were popularized by Hilbert and Cohn-Vossens 1932 book Anschauliche Geometrie, reprinted in English. Configurations may be studied either as concrete sets of points and lines in a geometry, such as the Euclidean or projective planes. That is, the girth of the bipartite graph must be at least six. A configuration in the plane is denoted by, where p is the number of points, ℓ the number of lines, γ the number of lines per point and these numbers necessarily satisfy the equation p γ = ℓ π as this product is the number of point-line incidences. Configurations having the symbol, say, need not be isomorphic as incidence structures. For instance, there exist three different configurations, the Pappus configuration and two less notable configurations, in some configurations, p = ℓ and consequently, γ = π. These are called symmetric or balanced configurations and the notation is often condensed to avoid repetition, notable projective configurations include the following, the simplest possible configuration, consisting of a point incident to a line. Each of its three sides meets two of its three vertices, and vice versa, more generally any polygon of n sides forms a configuration of type and, the complete quadrangle and complete quadrilateral respectively. This configuration exists as an incidence geometry, but cannot be constructed in the Euclidean plane. This configuration describes two quadrilaterals that are inscribed and circumscribed in each other. It cannot be constructed in Euclidean plane geometry but the equations defining it have nontrivial solutions in complex numbers, the Hesse configuration of nine inflection points of a cubic curve in the complex projective plane and the twelve lines determined by pairs of these points. The Gray configuration, the Klein configuration, the projective dual of a configuration is a configuration in which the roles of point and line are exchanged. Types of configurations therefore come in pairs, except when taking the dual results in an isomorphic configuration. These exceptions are called self-dual configurations and in such cases p = l, the number of nonisomorphic configurations of type, starting at n =7, is given by the sequence 1,1,3,10,31,229,2036,21399,245342. These numbers count configurations as abstract structures, regardless of realizability. As Gropp discusses, nine of the ten configurations, and all of the and configurations, are realizable in the Euclidean plane, but for each n ≥16 there is at least one nonrealizable configuration. Gropp also points out an error in this sequence, an 1895 paper attempted to list all configurations, and found 228 of them. There are several techniques for constructing configurations, generally starting from known configurations, some of the simplest of these techniques construct symmetric configurations