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Polygon
In elementary geometry, a polygon (/ˈpɒlɪɡɒn/) is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit. These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices (singular: vertex) or corners. The interior of the polygon is sometimes called its body. An n-gon is a polygon with n sides; for example, a triangle is a 3-gon. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The basic geometrical notion of a polygon has been adapted in various ways to suit particular purposes. Mathematicians are often concerned only with the bounding closed polygonal chain and with simple polygons which do not self-intersect, and they often define a polygon accordingly. A polygonal boundary may be allowed to intersect itself, creating star polygons and other self-intersecting polygons
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Absolute Value
In mathematics, the absolute value or modulus x of a real number x is the non-negative value of x without regard to its sign. Namely, x = x for a positive x, x = −x for a negative x (in which case −x is positive), and 0 = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero. Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces
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Hilbert Space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space
Hilbert space
is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz
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Complex Number
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x2 = −1, which is called an imaginary number because there is no real number that satisfies this equation. For the complex number a + bi, a is called the real part, and b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.[1][2] The complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number i.[3] This means that complex numbers can be added, subtracted, and multiplied, as polynomials in the variable i, with the rule i2 = −1 imposed. Furthermore, complex numbers can also be divided by nonzero complex numbers
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Symmetry Orbit
In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space. Common examples of spaces that groups act on are sets, vector spaces, and topological spaces. Actions of groups on vector spaces are called representations of the group. When there is a natural correspondence between the set of group elements and the set of space transformations, a group can be interpreted as acting on the space in a canonical way. For example, the symmetric group of a finite set consists of all bijective transformations of that set; thus, applying any element of the permutation group to an element of the set will produce another (not necessarily distinct) element of the set
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Greek Language
Greek (Modern Greek: ελληνικά [eliniˈka], elliniká, "Greek", ελληνική γλώσσα [eliniˈci ˈɣlosa] ( listen), ellinikí glóssa, "Greek language") is an independent branch of the Indo-European family of languages, native to Greece
Greece
and other parts of the Eastern Mediterranean
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Orthogonal (geometry)
In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms. Two elements u and v of a vector space with bilinear form B are orthogonal when B(u, v) = 0. Depending on the bilinear form, the vector space may contain nonzero self-orthogonal vectors. In the case of function spaces, families of orthogonal functions are used to form a basis. By extension, orthogonality is also used to refer to the separation of specific features of a system
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Radian
The radian (SI symbol rad) is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics. The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3 degrees (expansion at  A072097). The unit was formerly an SI supplementary unit, but this category was abolished in 1995 and the radian is now considered an SI derived unit.[1] Separately, the SI unit of solid angle measurement is the steradian. The radian is most commonly represented by the symbol rad.[2] An alternative symbol is c, the superscript letter c (for "circular measure"), the letter r, or a superscript R,[3] but these symbols are infrequently used as it can be easily mistaken for a degree symbol (°) or a radius (r)
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Degree (angle)
A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), is a measurement of a plane angle, defined so that a full rotation is 360 degrees. It is not an SI unit, as the SI unit
SI unit
of angular measure is the radian, but it is mentioned in the SI brochure as an accepted unit.[4] Because a full rotation equals 2π radians, one degree is equivalent to π/180 radians.Contents1 History 2 Subdivisions 3 Alternative units 4 See also 5 Notes 6 References 7 External linksHistory[edit] See also: DecansA circle with an equilateral chord (red). One sixtieth of this arc is a degree. Six such chords complete the circle.The original motivation for choosing the degree as a unit of rotations and angles is unknown
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Kepler–Poinsot Polyhedron
In geometry, a Kepler–Poinsot polyhedron
Kepler–Poinsot polyhedron
is any of four regular star polyhedra.[1] They may be obtained by stellating the regular convex dodecahedron and icosahedron, and differ from these in having regular pentagrammic faces or vertex figures.Contents1 Characteristics1.1 Non-convexity 1.2 Euler characteristic
Euler characteristic
χ 1.3 Duality 1.4 Summary2 Relationships among the regular polyhedra 3 History 4 Regular star polyhedra in art and culture 5 See also 6 References 7 External linksCharacteristics[edit] Non-convexity[edit] These figures have pentagrams (star pentagons) as faces or vertex figures. The small and great stellated dodecahedron have nonconvex regular pentagram faces
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Turn (geometry)
A turn is a unit of plane angle measurement equal to 2π radians, 360 degrees or 400 gradians. A turn is also referred to as a cycle (abbreviated cyc), revolution (abbreviated rev), complete rotation (abbreviated rot) or full circle. Subdivisions of a turn include half turns, quarter turns, centiturns, milliturns, points, etc.Contents1 Subdivision of turns 2 History 3 Unit conversion 4 Tau
Tau
proposals 5 Examples of use 6 Kinematics
Kinematics
of turns 7 See also 8 Notes and references 9 External linksSubdivision of turns[edit] A turn can be divided in 100 centiturns or 1000 milliturns, with each milliturn corresponding to an angle of 0.36°, which can also be written as 21′ 36″. A protractor divided in centiturns is normally called a percentage protractor. Binary fractions of a turn are also used. Sailors have traditionally divided a turn into 32 compass points
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Antiparallelogram
In geometry, an antiparallelogram is a quadrilateral having, like a parallelogram, two opposite pairs of equal-length sides, but in which the sides of one pair cross each other. The longer of the two pairs will always be the one that crosses. Antiparallelograms are also called contraparallelograms[1] or crossed parallelograms.[2] An antiparallelogram is a special case of a crossed quadrilateral, which has generally unequal edges.[3] A special form of the antiparallelogram is a crossed rectangle, in which two opposite edges are parallel.Contents1 Properties 2 In polyhedra 3 Four-bar linkages 4 Celestial mechanics 5 ReferencesProperties[edit] Every antiparallelogram has an axis of symmetry through its crossing point. Because of this symmetry, it has two pairs of equal angles as well as two pairs of equal sides.[2] Together with the kites and the isosceles trapezoids, antiparallelograms form one of three basic classes of quadrilaterals with a symmetry axis
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Orbit (dynamics)
In mathematics, in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. The orbit is a subset of the phase space and the set of all orbits is a partition of the phase space, that is, different orbits do not intersect in the phase space. Understanding the properties of orbits by using topological methods is one of the objectives of the modern theory of dynamical systems. For discrete-time dynamical systems, the orbits are sequences; for real dynamical systems, the orbits are curves; and for holomorphic dynamical systems, the orbits are Riemann surfaces.Contents1 Definition1.1 Real dynamical system 1.2 Discrete time dynamical system 1.3 General dynamical system 1.4 Notes2 Examples 3 Stability of orbits 4 See also 5 ReferencesDefinition[edit]Diagram showing the periodic orbit of a mass-spring system in simple harmonic motion
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Cartesian Coordinates
A Cartesian coordinate system
Cartesian coordinate system
is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis (plural axes) of the system, and the point where they meet is its origin, at ordered pair (0, 0)
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Centroid
In mathematics and physics, the centroid or geometric center of a plane figure is the arithmetic mean position of all the points in the shape. The definition extends to any object in n-dimensional space: its centroid is the mean position of all the points in all of the coordinate directions. Informally, it is the point at which a cutout of the shape could be perfectly balanced on the tip of a pin. While in geometry the term "barycenter" is a synonym for "centroid", in astrophysics and astronomy, barycenter is the center of mass of two or more bodies which are orbiting each other. In physics, the center of mass is the arithmetic mean of all points weighted by the local density or specific weight
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Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space
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