1.
Regular polygon
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In Euclidean geometry, a regular polygon is a polygon that is equiangular and equilateral. Regular polygons may be convex or star, in the limit, a sequence of regular polygons with an increasing number of sides becomes a circle, if the perimeter is fixed, or a regular apeirogon, if the edge length is fixed. These properties apply to all regular polygons, whether convex or star, a regular n-sided polygon has rotational symmetry of order n. All vertices of a regular polygon lie on a common circle and that is, a regular polygon is a cyclic polygon. Together with the property of equal-length sides, this implies that every regular polygon also has a circle or incircle that is tangent to every side at the midpoint. Thus a regular polygon is a tangential polygon, a regular n-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes. The symmetry group of a regular polygon is dihedral group Dn, D2, D3. It consists of the rotations in Cn, together with reflection symmetry in n axes that pass through the center, if n is even then half of these axes pass through two opposite vertices, and the other half through the midpoint of opposite sides. If n is odd then all pass through a vertex. All regular simple polygons are convex and those having the same number of sides are also similar. An n-sided convex regular polygon is denoted by its Schläfli symbol, for n <3 we have two degenerate cases, Monogon, degenerate in ordinary space. Digon, a line segment, degenerate in ordinary space. In certain contexts all the polygons considered will be regular, in such circumstances it is customary to drop the prefix regular. For instance, all the faces of uniform polyhedra must be regular, for n >2 the number of diagonals is n 2, i. e.0,2,5,9. for a triangle, square, pentagon, hexagon. The diagonals divide the polygon into 1,4,11,24, for a regular n-gon inscribed in a unit-radius circle, the product of the distances from a given vertex to all other vertices equals n. For a regular simple n-gon with circumradius R and distances di from a point in the plane to the vertices. For a regular n-gon, the sum of the distances from any interior point to the n sides is n times the apothem. This is a generalization of Vivianis theorem for the n=3 case, the sum of the perpendiculars from a regular n-gons vertices to any line tangent to the circumcircle equals n times the circumradius
2.
Edge (geometry)
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For edge in graph theory, see Edge In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a segment on the boundary. In a polyhedron or more generally a polytope, an edge is a segment where two faces meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal. In graph theory, an edge is an abstract object connecting two vertices, unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment. However, any polyhedron can be represented by its skeleton or edge-skeleton, conversely, the graphs that are skeletons of three-dimensional polyhedra can be characterized by Steinitzs theorem as being exactly the 3-vertex-connected planar graphs. Any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges and this equation is known as Eulers polyhedron formula. Thus the number of edges is 2 less than the sum of the numbers of vertices and faces, for example, a cube has 8 vertices and 6 faces, and hence 12 edges. In a polygon, two edges meet at each vertex, more generally, by Balinskis theorem, at least d edges meet at every vertex of a convex polytope. Similarly, in a polyhedron, exactly two faces meet at every edge, while in higher dimensional polytopes three or more two-dimensional faces meet at every edge. Thus, the edges of a polygon are its facets, the edges of a 3-dimensional convex polyhedron are its ridges, archived from the original on 4 February 2007
3.
Vertex (geometry)
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In geometry, a vertex is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. A vertex is a point of a polygon, polyhedron, or other higher-dimensional polytope. However, in theory, vertices may have fewer than two incident edges, which is usually not allowed for geometric vertices. However, a smooth approximation to a polygon will also have additional vertices. A polygon vertex xi of a simple polygon P is a principal polygon vertex if the diagonal intersects the boundary of P only at x and x, there are two types of principal vertices, ears and mouths. A principal vertex xi of a simple polygon P is called an ear if the diagonal that bridges xi lies entirely in P, according to the two ears theorem, every simple polygon has at least two ears. A principal vertex xi of a simple polygon P is called a mouth if the diagonal lies outside the boundary of P. Any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges and this equation is known as Eulers polyhedron formula. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces, for example, a cube has 12 edges and 6 faces, and hence 8 vertices
4.
Coxeter diagram
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In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors. It describes a kaleidoscopic construction, each node represents a mirror. An unlabeled branch implicitly represents order-3, each diagram represents a Coxeter group, and Coxeter groups are classified by their associated diagrams. Dynkin diagrams correspond to and are used to root systems. Branches of a Coxeter–Dynkin diagram are labeled with a number p. When p =2 the angle is 90° and the mirrors have no interaction, if a branch is unlabeled, it is assumed to have p =3, representing an angle of 60°. Two parallel mirrors have a branch marked with ∞, in principle, n mirrors can be represented by a complete graph in which all n /2 branches are drawn. In practice, nearly all interesting configurations of mirrors include a number of right angles, diagrams can be labeled by their graph structure. The first forms studied by Ludwig Schläfli are the orthoschemes which have linear graphs that generate regular polytopes, plagioschemes are simplices represented by branching graphs, and cycloschemes are simplices represented by cyclic graphs. Every Coxeter diagram has a corresponding Schläfli matrix with matrix elements ai, j = aj, as a matrix of cosines, it is also called a Gramian matrix after Jørgen Pedersen Gram. All Coxeter group Schläfli matrices are symmetric because their root vectors are normalized. It is related closely to the Cartan matrix, used in the similar but directed graph Dynkin diagrams in the cases of p =2,3,4, and 6. The determinant of the Schläfli matrix, called the Schläflian, and its sign determines whether the group is finite, affine and this rule is called Schläflis Criterion. The eigenvalues of the Schläfli matrix determines whether a Coxeter group is of type, affine type. The indefinite type is further subdivided, e. g. into hyperbolic. However, there are multiple non-equivalent definitions for hyperbolic Coxeter groups and we use the following definition, A Coxeter group with connected diagram is hyperbolic if it is neither of finite nor affine type, but every proper connected subdiagram is of finite or affine type. A hyperbolic Coxeter group is compact if all subgroups are finite, Finite and affine groups are also called elliptical and parabolic respectively. Hyperbolic groups are also called Lannér, after F. Lannér who enumerated the compact groups in 1950
5.
List of planar symmetry groups
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This article summarizes the classes of discrete symmetry groups of the Euclidean plane. The symmetry groups are named here by three naming schemes, International notation, orbifold notation, and Coxeter notation. There are three kinds of groups of the plane,2 rosette groups – 2D point groups 7 frieze groups – 2D line groups 17 wallpaper groups – 2D space groups. There are two families of discrete point groups, and they are specified with parameter n, which is the order of the group of the rotations in the group. The 7 frieze groups, the line groups, with a direction of periodicity are given with five notational names. The Schönflies notation is given as infinite limits of 7 dihedral groups, the yellow regions represent the infinite fundamental domain in each. The p1 and p2 groups, with no symmetry, are repeated in all classes. The related pure reflectional Coxeter group are given with all classes except oblique, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, Coxeter, Regular and Semi-Regular Polytopes III, Coxeter, H. S. M. & Moser, W. O. J. Generators and Relations for Discrete Groups. Johnson, Geometries and Transformations, Chapter 11, Finite symmetry groups Conways manuscript on Orbifold notation The 17 Wallpaper Groups
6.
Dihedral symmetry
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In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of groups, and they play an important role in group theory, geometry. The notation for the group of order n differs in geometry. In geometry, Dn or Dihn refers to the symmetries of the n-gon, in abstract algebra, Dn refers to the dihedral group of order n. The geometric convention is used in this article, a regular polygon with n sides has 2 n different symmetries, n rotational symmetries and n reflection symmetries. Usually, we take n ≥3 here. The associated rotations and reflections make up the dihedral group D n, if n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is even, there are n/2 axes of symmetry connecting the midpoints of opposite sides, in either case, there are n axes of symmetry and 2 n elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes, as with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, the following Cayley table shows the effect of composition in the group D3. R0 denotes the identity, r1 and r2 denote counterclockwise rotations by 120° and 240° respectively, for example, s2s1 = r1, because the reflection s1 followed by the reflection s2 results in a rotation of 120°. The order of elements denoting the composition is right to left, the composition operation is not commutative. In all cases, addition and subtraction of subscripts are to be performed using modular arithmetic with modulus n, if we center the regular polygon at the origin, then elements of the dihedral group act as linear transformations of the plane. This lets us represent elements of Dn as matrices, with composition being matrix multiplication and this is an example of a group representation. For example, the elements of the group D4 can be represented by the eight matrices. In general, the matrices for elements of Dn have the following form, rk is a rotation matrix, expressing a counterclockwise rotation through an angle of 2πk/n. Sk is a reflection across a line makes an angle of πk/n with the x-axis. Further equivalent definitions of Dn are, D1 is isomorphic to Z2, D2 is isomorphic to K4, the Klein four-group. D1 and D2 are exceptional in that, D1 and D2 are the only abelian dihedral groups, Dn is a subgroup of the symmetric group Sn for n ≥3
7.
Internal angle
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In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple polygon, regardless of whether it is convex or non-convex, a polygon has exactly one internal angle per vertex. If every internal angle of a polygon is less than 180°. In contrast, an angle is an angle formed by one side of a simple polygon. The sum of the angle and the external angle on the same vertex is 180°. The sum of all the angles of a simple polygon is 180° where n is the number of sides. The formula can be proved using induction and starting with a triangle for which the angle sum is 180°. The sum of the angles of any simple convex or non-convex polygon is 360°. The interior angle concept can be extended in a consistent way to crossed polygons such as star polygons by using the concept of directed angles, in other words, 360k° represents the sum of all the exterior angles. For example, for convex and concave polygons k =1, since the exterior angle sum is 360°
8.
Degree (angle)
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A degree, usually denoted by °, 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 of measure is the radian. Because a full rotation equals 2π radians, one degree is equivalent to π/180 radians, the original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year. Ancient astronomers noticed that the sun, which follows through the path over the course of the year. Some ancient calendars, such as the Persian calendar, used 360 days for a year, the use of a calendar with 360 days may be related to the use of sexagesimal numbers. The earliest trigonometry, used by the Babylonian astronomers and their Greek successors, was based on chords of a circle, a chord of length equal to the radius made a natural base quantity. One sixtieth of this, using their standard sexagesimal divisions, was a degree, Aristarchus of Samos and Hipparchus seem to have been among the first Greek scientists to exploit Babylonian astronomical knowledge and techniques systematically. Timocharis, Aristarchus, Aristillus, Archimedes, and Hipparchus were the first Greeks known to divide the circle in 360 degrees of 60 arc minutes, eratosthenes used a simpler sexagesimal system dividing a circle into 60 parts. Furthermore, it is divisible by every number from 1 to 10 except 7 and this property has many useful applications, such as dividing the world into 24 time zones, each of which is nominally 15° of longitude, to correlate with the established 24-hour day convention. Finally, it may be the case more than one of these factors has come into play. For many practical purposes, a degree is a small enough angle that whole degrees provide sufficient precision. When this is not the case, as in astronomy or for geographic coordinates, degree measurements may be written using decimal degrees, with the symbol behind the decimals. Alternatively, the sexagesimal unit subdivisions can be used. One degree is divided into 60 minutes, and one minute into 60 seconds, use of degrees-minutes-seconds is also called DMS notation. These subdivisions, also called the arcminute and arcsecond, are represented by a single and double prime. For example,40. 1875° = 40° 11′ 15″, or, using quotation mark characters, additional precision can be provided using decimals for the arcseconds component. The older system of thirds, fourths, etc. which continues the sexagesimal unit subdivision, was used by al-Kashi and other ancient astronomers, but is rarely used today
9.
Dual polygon
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In geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other. The dual of a polygon is an isotoxal polygon. For example, the rectangle and rhombus are duals, in a cyclic polygon, longer sides correspond to larger exterior angles in the dual, and shorter sides to smaller angles. Further, congruent sides in the original polygon yields congruent angles in the dual, for example, the dual of a highly acute isosceles triangle is an obtuse isosceles triangle. In the Dorman Luke construction, each face of a polyhedron is the dual polygon of the corresponding vertex figure. As an example of the duality of polygons we compare properties of the cyclic. This duality is perhaps more clear when comparing an isosceles trapezoid to a kite. The simplest qualitative construction of a polygon is a rectification operation. New edges are formed between these new vertices and that is, the polygon generated by applying it twice is in general not similar to the original polygon. As with dual polyhedra, one can take a circle and perform polar reciprocation in it. Combinatorially, one can define a polygon as a set of vertices, a set of edges, then the dual polygon is obtained by simply switching the vertices and edges. Thus for the triangle with vertices and edges, the triangle has vertices, and edges, where B connects AB & BC. This is not a particularly fruitful avenue, as combinatorially, there is a family of polygons, geometric duality of polygons is more varied. Dual curve Dual polyhedron Self-dual polygon Dual Polygon Applet by Don Hatch
10.
Convex polygon
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A convex polygon is a simple polygon in which no line segment between two points on the boundary ever goes outside the polygon. Equivalently, it is a polygon whose interior is a convex set. In a convex polygon, all angles are less than or equal to 180 degrees. A simple polygon which is not convex is called concave, the following properties of a simple polygon are all equivalent to convexity, Every internal angle is less than or equal to 180 degrees. Every point on line segment between two points inside or on the boundary of the polygon remains inside or on the boundary. The polygon is contained in a closed half-plane defined by each of its edges. For each edge, the points are all on the same side of the line that the edge defines. The angle at each vertex contains all vertices in its edges. The polygon is the hull of its edges. Additional properties of convex polygons include, The intersection of two convex polygons is a convex polygon, a convex polygon may br triangulated in linear time through a fan triangulation, consisting in adding diagonals from one vertex to all other vertices. Hellys theorem, For every collection of at least three convex polygons, if the intersection of three of them is nonempty, then the whole collection has a nonempty intersection. Krein–Milman theorem, A convex polygon is the hull of its vertices. Thus it is defined by the set of its vertices. Hyperplane separation theorem, Any two convex polygons with no points in common have a separator line, if the polygons are closed and at least one of them is compact, then there are even two parallel separator lines. Inscribed triangle property, Of all triangles contained in a convex polygon, inscribing triangle property, every convex polygon with area A can be inscribed in a triangle of area at most equal to 2A. Equality holds for a parallelogram.5 × Area ≤ Area ≤2 × Area, the mean width of a convex polygon is equal to its perimeter divided by pi. So its width is the diameter of a circle with the perimeter as the polygon. Every polygon inscribed in a circle, if not self-intersecting, is convex, however, not every convex polygon can be inscribed in a circle
11.
Cyclic polygon
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In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon. The center of circle is called the circumcenter and its radius is called the circumradius. A polygon which has a circle is called a cyclic polygon. All regular simple polygons, all isosceles trapezoids, all triangles, a related notion is the one of a minimum bounding circle, which is the smallest circle that completely contains the polygon within it. All triangles are cyclic, i. e. every triangle has a circumscribed circle and this can be proven on the grounds that the general equation for a circle with center and radius r in the Cartesian coordinate system is 2 +2 = r 2. Since this equation has three parameters only three points coordinate pairs are required to determine the equation of a circle, since a triangle is defined by its three vertices, and exactly three points are required to determine a circle, every triangle can be circumscribed. The circumcenter of a triangle can be constructed by drawing any two of the three perpendicular bisectors, the center is the point where the perpendicular bisectors intersect, and the radius is the length to any of the three vertices. This is because the circumcenter is equidistant from any pair of the triangles vertices, in coastal navigation, a triangles circumcircle is sometimes used as a way of obtaining a position line using a sextant when no compass is available. The horizontal angle between two landmarks defines the circumcircle upon which the observer lies, in the Euclidean plane, it is possible to give explicitly an equation of the circumcircle in terms of the Cartesian coordinates of the vertices of the inscribed triangle. Suppose that A = B = C = are the coordinates of points A, B, using the polarization identity, these equations reduce to the condition that the matrix has a nonzero kernel. Thus the circumcircle may alternatively be described as the locus of zeros of the determinant of this matrix, a similar approach allows one to deduce the equation of the circumsphere of a tetrahedron. A unit vector perpendicular to the containing the circle is given by n ^ = × | × |. An equation for the circumcircle in trilinear coordinates x, y, z is a/x + b/y + c/z =0, an equation for the circumcircle in barycentric coordinates x, y, z is a2/x + b2/y + c2/z =0. The isogonal conjugate of the circumcircle is the line at infinity, given in coordinates by ax + by + cz =0. Additionally, the circumcircle of a triangle embedded in d dimensions can be using a generalized method. Let A, B, and C be d-dimensional points, which form the vertices of a triangle and we start by transposing the system to place C at the origin, a = A − C, b = B − C. The circumcenter, p0, is given by p 0 = ×2 ∥ a × b ∥2 + C, the Cartesian coordinates of the circumcenter are U x =1 D U y =1 D with D =2. Without loss of generality this can be expressed in a form after translation of the vertex A to the origin of the Cartesian coordinate systems
12.
Equilateral polygon
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In geometry, three or more than three straight lines make a polygon and an equilateral polygon is a polygon which has all sides of the same length. Except in the case, it need not be equiangular. If the number of sides is at least five, an equilateral polygon need not be a convex polygon, all regular polygons and isotoxal polygons are equilateral. An equilateral triangle is a triangle with 60° internal angles. An equilateral quadrilateral is called a rhombus, an isotoxal polygon described by an angle α and it includes the square as a special case. A convex equilateral pentagon can be described by two angles α and β, which determine the other angles. Concave equilateral pentagons exist, as do concave equilateral polygons with any number of sides. An equilateral polygon which is cyclic is a regular polygon, a tangential polygon is equilateral if and only if the alternate angles are equal. Thus if the number of n is odd, a tangential polygon is equilateral if. The principal diagonals of a hexagon each divide the hexagon into quadrilaterals, in any convex equilateral hexagon with common side a, there exists a principal diagonal d1 such that d 1 a ≤2 and a principal diagonal d2 such that d 2 a >3. Triambi are equilateral hexagons with trigonal symmetry, Equilateral triangle With interactive animation A Property of Equiangular Polygons, a discussion of Vivianis theorem at Cut-the-knot
13.
Isogonal figure
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In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are equivalent. That implies that each vertex is surrounded by the kinds of face in the same or reverse order. Technically, we say that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope is transitive on its vertices, all vertices of a finite n-dimensional isogonal figure exist on an -sphere. The term isogonal has long used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as symmetry groups, all regular polygons, apeirogons and regular star polygons are isogonal. The dual of a polygon is an isotoxal polygon. Some even-sided polygons and apeirogons which alternate two edge lengths, for example a rectangle, are isogonal, all planar isogonal 2n-gons have dihedral symmetry with reflection lines across the mid-edge points. An isogonal polyhedron and 2D tiling has a kind of vertex. An isogonal polyhedron with all faces is also a uniform polyhedron. Geometrically distorted variations of uniform polyhedra and tilings can also be given the vertex configuration, isogonal polyhedra and 2D tilings may be further classified, Regular if it is also isohedral and isotoxal, this implies that every face is the same kind of regular polygon. Quasi-regular if it is also isotoxal but not isohedral, semi-regular if every face is a regular polygon but it is not isohedral or isotoxal. Uniform if every face is a polygon, i. e. it is regular, quasiregular or semi-regular. Noble if it is also isohedral and these definitions can be extended to higher-dimensional polytopes and tessellations. Most generally, all uniform polytopes are isogonal, for example, the dual of an isogonal polytope is called an isotope which is transitive on its facets. A polytope or tiling may be called if its vertices form k transitivity classes. A more restrictive term, k-uniform is defined as a figure constructed only from regular polygons. They can be represented visually with colors by different uniform colorings, edge-transitive Face-transitive Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p.369 Transitivity Grünbaum, Branko, Shephard, G. C
14.
Isotoxal figure
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In geometry, a polytope, or a tiling, is isotoxal or edge-transitive if its symmetries act transitively on its edges. The term isotoxal is derived from the Greek τοξον meaning arc, an isotoxal polygon is an equilateral polygon, but not all equilateral polygons are isotoxal. The duals of isotoxal polygons are isogonal polygons, in general, an isotoxal 2n-gon will have Dn dihedral symmetry. A rhombus is a polygon with D2 symmetry. All regular polygons are isotoxal, having double the symmetry order. A regular 2n-gon is a polygon and can be marked with alternately colored vertices. An isotoxal polyhedron or tiling must be either isogonal or isohedral or both, regular polyhedra are isohedral, isogonal and isotoxal. Quasiregular polyhedra are isogonal and isotoxal, but not isohedral, their duals are isohedral and isotoxal, not every polyhedron or 2-dimensional tessellation constructed from regular polygons is isotoxal. An isotoxal polyhedron has the dihedral angle for all edges. There are nine convex isotoxal polyhedra formed from the Platonic solids,8 formed by the Kepler–Poinsot polyhedra, cS1 maint, Multiple names, authors list Coxeter, Harold Scott MacDonald, Longuet-Higgins, M. S. Miller, J. C. P. Uniform polyhedra, Philosophical Transactions of the Royal Society of London, mathematical and Physical Sciences,246, 401–450, doi,10. 1098/rsta.1954.0003, ISSN 0080-4614, JSTOR91532, MR0062446
15.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
16.
Polygon
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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 vertices or corners. The interior of the polygon is called its body. An n-gon is a polygon with n sides, for example, 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 these and other generalizations of polygons are described below. The word polygon derives from the Greek adjective πολύς much, many and it has been suggested that γόνυ knee may be the origin of “gon”. Polygons are primarily classified by the number of sides, Polygons may be characterized by their convexity or type of non-convexity, Convex, any line drawn through the polygon meets its boundary exactly twice. As a consequence, all its interior angles are less than 180°, equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. Non-convex, a line may be found which meets its boundary more than twice, equivalently, there exists a line segment between two boundary points that passes outside the polygon. Simple, the boundary of the polygon does not cross itself, there is at least one interior angle greater than 180°. Star-shaped, the interior is visible from at least one point. The polygon must be simple, and may be convex or concave, self-intersecting, the boundary of the polygon crosses itself. Branko Grünbaum calls these coptic, though this term does not seem to be widely used, star polygon, a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped, equiangular, all corner angles are equal. Cyclic, all lie on a single circle, called the circumcircle. Isogonal or vertex-transitive, all lie within the same symmetry orbit. The polygon is cyclic and equiangular
17.
Compass and straightedge
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The idealized ruler, known as a straightedge, is assumed to be infinite in length, and has no markings on it and only one edge. The compass is assumed to collapse when lifted from the page, more formally, the only permissible constructions are those granted by Euclids first three postulates. It turns out to be the case that every point constructible using straightedge, the ancient Greek mathematicians first conceived compass-and-straightedge constructions, and a number of ancient problems in plane geometry impose this restriction. The ancient Greeks developed many constructions, but in cases were unable to do so. Gauss showed that some polygons are constructible but that most are not, some of the most famous straightedge-and-compass problems were proven impossible by Pierre Wantzel in 1837, using the mathematical theory of fields. In spite of existing proofs of impossibility, some persist in trying to solve these problems, in terms of algebra, a length is constructible if and only if it represents a constructible number, and an angle is constructible if and only if its cosine is a constructible number. A number is constructible if and only if it can be using the four basic arithmetic operations. Circles can only be starting from two given points, the centre and a point on the circle. The compass may or may not collapse when its not drawing a circle, the straightedge is infinitely long, but it has no markings on it and has only one straight edge, unlike ordinary rulers. It can only be used to draw a segment between two points or to extend an existing segment. The modern compass generally does not collapse and several modern constructions use this feature and it would appear that the modern compass is a more powerful instrument than the ancient collapsing compass. However, by Proposition 2 of Book 1 of Euclids Elements, although the proposition is correct, its proofs have a long and checkered history. Eyeballing it and getting close does not count as a solution and that is, it must have a finite number of steps, and not be the limit of ever closer approximations. One of the purposes of Greek mathematics was to find exact constructions for various lengths, for example. The Greeks could not find constructions for these three problems, among others, Squaring the circle, Drawing a square the same area as a given circle, doubling the cube, Drawing a cube with twice the volume of a given cube. Trisecting the angle, Dividing a given angle into three smaller angles all of the same size, for 2000 years people tried to find constructions within the limits set above, and failed. All three have now been proven under mathematical rules to be generally impossible, the ancient Greek mathematicians first attempted compass-and-straightedge constructions, and they discovered how to construct sums, differences, products, ratios, and square roots of given lengths. They could also construct half of an angle, a square whose area is twice that of another square, a square having the same area as a given polygon
18.
Pierpont prime
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A Pierpont prime is a prime number of the form 2 u 3 v +1 for some nonnegative integers u and v. That is, they are the prime numbers p for which p −1 is 3-smooth. They are named after the mathematician James Pierpont, who introduced them in the study of regular polygons that can be constructed using conic sections. It is possible to prove that if v =0 and u >0, then u must be a power of 2, if v is positive then u must also be positive, and the Pierpont prime is of the form 6k +1. Empirically, the Pierpont primes do not seem to be rare or sparsely distributed. There are 36 Pierpont primes less than 106,59 less than 109,151 less than 1020, there are few restrictions from algebraic factorisations on the Pierpont primes, so there are no requirements like the Mersenne prime condition that the exponent must be prime. As there are Θ numbers of the form in this range. Andrew M. Gleason made this explicit, conjecturing there are infinitely many Pierpont primes. According to Gleasons conjecture there are Θ Pierpont primes smaller than N, when 2 u >3 v, the primality of 2 u 3 v +1 can be tested by Proths theorem. As part of the ongoing search for factors of Fermat numbers. The following table gives values of m, k, and n such that k ⋅2 n +1 divides 22 m +1, the left-hand side is a Pierpont prime when k is a power of 3, the right-hand side is a Fermat number. As of 2017, the largest known Pierpont prime is 3 ×210829346 +1, whose primality was discovered by Sai Yik Tang, in the mathematics of paper folding, the Huzita axioms define six of the seven types of fold possible. It has been shown that these folds are sufficient to allow the construction of the points that solve any cubic equation. It follows that they allow any regular polygon of N sides to be formed, as long as N >3 and of the form 2m3nρ and this is the same class of regular polygons as those that can be constructed with a compass, straightedge, and angle-trisector. Regular polygons which can be constructed with compass and straightedge are the special case where n =0 and ρ is a product of distinct Fermat primes, themselves a subset of Pierpont primes. In 1895, James Pierpont studied the same class of regular polygons, Pierpont generalized compass and straightedge constructions in a different way, by adding the ability to draw conic sections whose coefficients come from previously constructed points. As he showed, the regular N-gons that can be constructed with these operations are the ones such that the totient of N is 3-smooth. Since the totient of a prime is formed by subtracting one from it, however, Pierpont did not describe the form of the composite numbers with 3-smooth totients. As Gleason later showed, these numbers are exactly the ones of the form 2m3nρ given above, the smallest prime that is not a Pierpont prime is 11, therefore, the hendecagon is the smallest regular polygon that cannot be constructed with compass, straightedge and angle trisector
19.
Angle trisection
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Angle trisection is a classical problem of compass and straightedge constructions of ancient Greek mathematics. It concerns construction of an equal to one third of a given arbitrary angle. The problem as stated is generally impossible to solve, as proved by Pierre Wantzel in 1837, however, although there is no way to trisect an angle in general with just a compass and a straightedge, some special angles can be trisected. For example, it is straightforward to trisect a right angle. It is possible to trisect an angle by using tools other than straightedge. For example, neusis construction, also known to ancient Greeks, involves simultaneous sliding and rotation of a marked straightedge, other techniques were developed by mathematicians over the centuries. Because it is defined in terms, but complex to prove unsolvable. These solutions often involve mistaken interpretations of the rules, or are simply incorrect, three problems proved elusive, specifically, trisecting the angle, doubling the cube, and squaring the circle. The problem of angle trisection reads, Construct an angle equal to one-third of an arbitrary angle. Pierre Wantzel published a proof of the impossibility of trisecting an arbitrary angle in 1837. Wantzels proof, restated in modern terminology, uses the algebra of field extensions. However Wantzel published these results earlier than Galois and did not use the connection between field extensions and groups that is the subject of Galois theory itself. The problem of constructing an angle of a given measure θ is equivalent to constructing two segments such that the ratio of their length is cos θ. From a solution to one of two problems, one may pass to a solution of the other by a compass and straightedge construction. The triple-angle formula gives an expression relating the cosines of the angle and its trisection. It follows that, given a segment that is defined to have unit length and this equivalence reduces the original geometric problem to a purely algebraic problem. Every irrational number which is constructible in a step from some given numbers is a root of a polynomial of degree 2 with coefficients in the field generated by these numbers. Therefore, any number which is constructible by a sequence of steps is a root of a polynomial whose degree is a power of two
20.
Neusis construction
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The neusis is a geometric construction method that was used in antiquity by Greek mathematicians. The neusis construction consists of fitting a line element of length in between two given lines, in such a way that the line element, or its extension. That is, one end of the element has to lie on l. A neusis construction might be performed by means of a neusis ruler, in the figure one end of the ruler is marked with a yellow eye with crosshairs, this is the origin of the scale division on the ruler. A second marking on the ruler indicates the distance a from the origin, the yellow eye is moved along line l, until the blue eye coincides with line m. The position of the element thus found is shown in the figure as a dark blue bar. Point P is called the pole of the neusis, line l the directrix, or guiding line, length a is called the diastema. Neuseis have been important because they provide a means to solve geometric problems that are not solvable by means of compass. Examples are the trisection of any angle in three parts, the doubling of the cube, and the construction of a regular heptagon, nonagon. Mathematicians such as Archimedes of Syracuse and Pappus of Alexandria freely used neuseis, Sir Isaac Newton followed their line of thought, nevertheless, gradually the technique dropped out of use. Modified by the recent finding by Benjamin and Snyder that the regular hendecagon is neusis-constructible, T. L. Heath, the historian of mathematics, has suggested that the Greek mathematician Oenopides was the first to put compass-and-straightedge constructions above neuseis. One hundred years after him Euclid too shunned neuseis in his influential textbook. The next attack on the neusis came when, from the fourth century BC, under its influence a hierarchy of three classes of geometrical constructions was developed. In the end the use of neusis was deemed acceptable only when the two other, higher categories of constructions did not offer a solution, Neusis became a kind of last resort that was invoked only when all other, more respectable, methods had failed. Using neusis where other methods might have been used was branded by the late Greek mathematician Pappus of Alexandria as a not inconsiderable error. R. Boeker, Neusis, in, Paulys Realencyclopädie der Classischen Altertumswissenschaft, the most comprehensive survey, however, the author sometimes has rather curious opinions. T. L. Heath, A history of Greek Mathematics, H. G. Zeuthen, Die Lehre von den Kegelschnitten im Altertum. MathWorld page Angle Trisection by Paper Folding
21.
Unit circle
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In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, the circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted S1, the generalization to higher dimensions is the unit sphere, if is a point on the unit circles circumference, then | x | and | y | are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation x 2 + y 2 =1. The interior of the circle is called the open unit disk. One may also use other notions of distance to define other unit circles, such as the Riemannian circle, see the article on mathematical norms for additional examples. The unit circle can be considered as the complex numbers. In quantum mechanics, this is referred to as phase factor, the equation x2 + y2 =1 gives the relation cos 2 + sin 2 =1. The unit circle also demonstrates that sine and cosine are periodic functions, triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius OA from the origin to a point P on the circle such that an angle t with 0 < t < π/2 is formed with the positive arm of the x-axis. Now consider a point Q and line segments PQ ⊥ OQ, the result is a right triangle △OPQ with ∠QOP = t. Because PQ has length y1, OQ length x1, and OA length 1, sin = y1 and cos = x1. Having established these equivalences, take another radius OR from the origin to a point R on the circle such that the same angle t is formed with the arm of the x-axis. Now consider a point S and line segments RS ⊥ OS, the result is a right triangle △ORS with ∠SOR = t. It can hence be seen that, because ∠ROQ = π − t, R is at in the way that P is at. The conclusion is that, since is the same as and is the same as, it is true that sin = sin and it may be inferred in a similar manner that tan = −tan, since tan = y1/x1 and tan = y1/−x1. A simple demonstration of the above can be seen in the equality sin = sin = 1/√2, when working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than π/2. However, when defined with the circle, these functions produce meaningful values for any real-valued angle measure – even those greater than 2π
22.
Prime number
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n
23.
Cyclic group
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In algebra, a cyclic group or monogenous group is a group that is generated by a single element. Each element can be written as a power of g in multiplicative notation and this element g is called a generator of the group. Every infinite cyclic group is isomorphic to the group of Z. Every finite cyclic group of n is isomorphic to the additive group of Z/nZ. Every cyclic group is a group, and every finitely generated abelian group is a direct product of cyclic groups. A group G is called if there exists an element g in G such that G = ⟨g⟩ =. Since any group generated by an element in a group is a subgroup of that group, for example, if G = is a group of order 6, then g6 = g0, and G is cyclic. In fact, G is essentially the same as the set with addition modulo 6, for example,1 +2 ≡3 corresponds to g1 · g2 = g3, and 2 +5 ≡1 corresponds to g2 · g5 = g7 = g1, and so on. One can use the isomorphism χ defined by χ = i, the name cyclic may be misleading, it is possible to generate infinitely many elements and not form any literal cycles, that is, every gn is distinct. A group generated in this way is called a cyclic group. The French mathematicians known as Nicolas Bourbaki referred to a group as a monogenous group. The set of integers, with the operation of addition, forms a group and it is an infinite cyclic group, because all integers can be written as a finite sum or difference of copies of the number 1. In this group,1 and −1 are the only generators, every infinite cyclic group is isomorphic to this group. For every positive n, the set of integers modulo n, again with the operation of addition, forms a finite cyclic group. An element g is a generator of this group if g is relatively prime to n, thus, the number of different generators is φ, where φ is the Euler totient function, the function that counts the number of numbers modulo n that are relatively prime to n. Every finite cyclic group is isomorphic to a group Z/n, where n is the order of the group, the integer and modular addition operations, used to define the cyclic groups, are both the addition operations of commutative rings, also denoted Z and Z/n. If p is a prime, then Z/p is a finite field, every field with p elements is isomorphic to this one. For every positive n, the subset of the integers modulo n that are relatively prime to n, with the operation of multiplication
24.
John Horton Conway
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John Horton Conway FRS is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has also contributed to many branches of mathematics, notably the invention of the cellular automaton called the Game of Life. Conway is currently Professor Emeritus of Mathematics at Princeton University in New Jersey, Conway was born in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at an early age, his mother has recalled that he could recite the powers of two when he was four years old. By the age of eleven his ambition was to become a mathematician, after leaving secondary school, Conway entered Gonville and Caius College, Cambridge to study mathematics. Conway, who was a terribly introverted adolescent in school, interpreted his admission to Cambridge as an opportunity to transform himself into a new person and he was awarded his Bachelor of Arts degree in 1959 and began to undertake research in number theory supervised by Harold Davenport. Having solved the problem posed by Davenport on writing numbers as the sums of fifth powers. It appears that his interest in games began during his years studying the Cambridge Mathematical Tripos and he was awarded his doctorate in 1964 and was appointed as College Fellow and Lecturer in Mathematics at the University of Cambridge. After leaving Cambridge in 1986, he took up the appointment to the John von Neumann Chair of Mathematics at Princeton University, Conway is especially known for the invention of the Game of Life, one of the early examples of a cellular automaton. His initial experiments in that field were done with pen and paper, since the game was introduced by Martin Gardner in Scientific American in 1970, it has spawned hundreds of computer programs, web sites, and articles. It is a staple of recreational mathematics, there is an extensive wiki devoted to curating and cataloging the various aspects of the game. From the earliest days it has been a favorite in computer labs, at times Conway has said he hates the game of life–largely because it has come to overshadow some of the other deeper and more important things he has done. Nevertheless, the game did help launch a new branch of mathematics, the Game of Life is now known to be Turing complete. Conways career is intertwined with mathematics popularizer and Scientific American columnist Martin Gardner, when Gardner featured Conways Game of Life in his Mathematical Games column in October 1970, it became the most widely read of all his columns and made Conway an instant celebrity. Gardner and Conway had first corresponded in the late 1950s, for instance, he discussed Conways game of Sprouts, Hackenbush, and his angel and devil problem. In the September 1976 column he reviewed Conways book On Numbers and Games, Conway is widely known for his contributions to combinatorial game theory, a theory of partisan games. This he developed with Elwyn Berlekamp and Richard Guy, and with them also co-authored the book Winning Ways for your Mathematical Plays and he also wrote the book On Numbers and Games which lays out the mathematical foundations of CGT. He is also one of the inventors of sprouts, as well as philosophers football and he developed detailed analyses of many other games and puzzles, such as the Soma cube, peg solitaire, and Conways soldiers
25.
Directed edge
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In mathematics, and more specifically in graph theory, a directed graph is a graph that is a set of vertices connected by edges, where the edges have a direction associated with them. It differs from an ordinary or undirected graph, in that the latter is defined in terms of unordered pairs of vertices, more specifically, these entities are addressed as directed multigraphs. On the other hand, the definition allows a directed graph to have loops. More specifically, directed graphs without loops are addressed as directed graphs. Symmetric directed graphs are directed graphs where all edges are bidirected, simple directed graphs are directed graphs that have no loops and no multiple arrows with same source and target nodes. As already introduced, in case of arrows the entity is usually addressed as directed multigraph. Some authors describe digraphs with loops as loop-digraphs. Complete directed graphs are directed graphs where each pair of vertices is joined by a symmetric pair of directed arrows. It follows that a complete digraph is symmetric, oriented graphs are directed graphs having no bidirected edges. It follows that a graph is an oriented graph iff it hasnt any 2-cycle. Tournaments are oriented graphs obtained by choosing a direction for each edge in undirected complete graphs. Directed acyclic graphs are directed graphs with no directed cycles, multitrees are DAGs in which no two directed paths from a single starting vertex meet back at the same ending vertex. Oriented trees or polytrees are DAGs formed by orienting the edges of undirected acyclic graphs, rooted trees are oriented trees in which all edges of the underlying undirected tree are directed away from the roots. Rooted directed graphs are digraphs in which a vertex has been distinguished as the root, control flow graphs are rooted digraphs used in computer science as a representation of the paths that might be traversed through a program during its execution. Signal-flow graphs are directed graphs in which nodes represent system variables and branches represent functional connections between pairs of nodes, flow graphs are digraphs associated with a set of linear algebraic or differential equations. State diagrams are directed multigraphs that represent finite state machines, representations of a quiver label its vertices with vector spaces and its edges compatibly with linear transformations between them, and transform via natural transformations. If a path leads from x to y, then y is said to be a successor of x and reachable from x, the arrow is called the inverted arrow of. The adjacency matrix of a graph is unique up to identical permutation of rows. Another matrix representation for a graph is its incidence matrix. For a vertex, the number of head ends adjacent to a vertex is called the indegree of the vertex, the indegree of v is denoted deg− and its outdegree is denoted deg+
26.
Canadian dollar
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The Canadian dollar is the currency of Canada. It is abbreviated with the dollar sign $, or sometimes Can$ or C$ to distinguish it from other dollar-denominated currencies and it is divided into 100 cents. Canadas dollar is the fifth most held reserve currency in the world, accounting for approximately 2% of all global reserves, behind only the U. S. dollar, the euro, the yen and the pound sterling. In 1841, the Province of Canada adopted a new system based on the Halifax rating, the new Canadian pound was equal to four US dollars, making one pound sterling equal to 1 pound,4 shillings, and 4 pence Canadian. Thus, the new Canadian pound was worth 16 shillings and 5.3 pence sterling, the 1850s was a decade of wrangling over whether to adopt a sterling monetary system or a decimal monetary system based on the US dollar. In 1851, the Legislative Council and the Legislative Assembly passed an act for the purposes of introducing a sterling unit in conjunction with decimal fractional coinage. The idea was that the coins would correspond to exact amounts in relation to the U. S. dollar fractional coinage. This gold standard was introduced with the gold sovereign being legal tender at £1 = US$ 4.86 2⁄3, no coinage was provided for under the 1853 act. Sterling coinage was legal tender and all other silver coins were demonetized. The British government in principle allowed for a decimal coinage but nevertheless held out the hope that a unit would be chosen under the name of royal. However, in 1857, the decision was made to introduce a decimal coinage into the Province of Canada in conjunction with the U. S. dollar unit, in 1859, Canadian colonial postage stamps were issued with decimal denominations for the first time. In 1860, the colonies of New Brunswick and Nova Scotia followed the colony of Canada in adopting a system based on the U. S. dollar unit. In the following year, Canadian colonial postage stamps were issued with the shown in dollars. The U. S. dollar was created in 1792 on the basis of the weight of a selection of worn Spanish dollars. As such, the Spanish dollar was slightly more than the U. S. dollar, and likewise. In 1867, the colonies of Canada, New Brunswick, and Nova Scotia were united in a called the Dominion of Canada. In 1871, Prince Edward Island went decimal within the U. S. dollar unit, however, the currency of Prince Edward Island was absorbed into the Canadian system shortly afterwards, when Prince Edward island joined the Dominion of Canada in 1873. The Canadian Parliament passed the Uniform Currency Act in April 1871, tying up loose ends as to the currencies of the various provinces, the gold standard was temporarily abandoned during the First World War and definitively abolished on April 10,1933
27.
Loonie
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The Canadian one dollar coin, commonly called the loonie, is a gold-coloured one-dollar coin introduced in 1987. It bears images of a loon, a bird which is common and well known in Canada, on the reverse. It is produced by the Royal Canadian Mint at its facility in Winnipeg, the coins outline is an 11-sided curve of constant width. Its diameter of 26.5 mm and its 11-sidedness matched that of the already-circulating Susan B, Anthony dollar in the United States, and its thickness of 1.95 mm was a close match to the latters 2.0 mm. Its gold colour differed from the silver-coloured Anthony dollar, however, other coins using a curve of constant width include the 7-sided British twenty pence and fifty pence coins. The coin has become the symbol of the Canadian dollar, media often discuss the rate at which the loonie is trading against other currencies, the nickname loonie became so widely recognized that in 2006 the Royal Canadian Mint secured the rights to it. When the Canadian two-dollar coin was introduced in 1996, it was in turn nicknamed the toonie, Canada first minted a silver dollar coin in 1935 to celebrate the 25th anniversary of George Vs reign as king. The voyageur dollar, so named because it featured an Indian and a French voyageur paddling a canoe on the reverse, was minted in silver until 1967, after which it was composed primarily of nickel. The coins did not see wide circulation, mainly due to their size and weight, the nickel version weighed 15.6 grams and was 32.1 millimetres in diameter, and was itself smaller than the silver version. By 1982, the Royal Canadian Mint had begun work on a new composition for the coin that it hoped would lead to increased circulation. At the same time, vending machine operators and transit systems were lobbying the Government of Canada to replace the dollar banknotes with wider circulating coins, a Commons committee recommended in 1985 that the dollar bill be eliminated despite a lack of evidence that Canadians would support the move. The government argued that it would save between $175 and $250 million over 20 years by switching from bills that had a lifespan of less than a year to coins that would last two decades. The government announced on March 25,1986, that the new coin would be launched the following year as a replacement for the dollar bill. It was expected to cost $31.8 million to produce the first 300 million coins, from the proceeds, a total of $60 million over five years was dedicated toward funding the 1988 Winter Olympics in Calgary. The failure of the Susan B, Anthony dollar coin in the United States had been considered and it was believed Americans refused to support the coin due to its similarity to their quarter coin and its lack of esthetic appeal. In announcing the new Canadian dollar coin, the government stated it would be the overall size as the Susan B. Anthony coin – slightly larger than a quarter – to allow for compatibility with American manufactured vending machines and it was also found to be the third time that the Mint had lost master dies within five years. The Mint also disagreed with the Royal Canadian Mounted Polices contention that the dies were simply lost in transit, fearing the possibility of counterfeiting, the government approved a new design for the reverse, replacing the voyageur with a Robert-Ralph Carmichael design of a common loon floating in water
28.
Indian rupee
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The Indian rupee, is the official currency of the Republic of India. The rupee is subdivided into 100 paise, though as of 2011,25 paise is no more a legal tender, the issuance of the currency is controlled by the Reserve Bank of India. The Reserve Bank manages currency in India and derives its role in management on the basis of the Reserve Bank of India Act,1934. The rupee is named after the coin, rupiya, first issued by Sultan Sher Shah Suri in the 16th century. In 2010, a new symbol ₹, was officially adopted and it was derived from the combination of the Devanagari consonant र and the Latin capital letter R without its vertical bar. The parallel lines at the top are said to make an allusion to the tricolour Indian flag, the first series of coins with the new rupee symbol started in circulation on 8 July 2011. On 8 November 2016 the Government of India announced the demonetisation of ₹500 and ₹1000 banknotes with effect from midnight of the same day, a newly redesigned series of ₹500 banknote, in addition to a new denomination of ₹2000 banknote is in circulation since 10 November 2016. The new redesigned series is expected to be enlarged with banknotes in the denominations of ₹1000, ₹100. The word rupee was derived from the Sanskrit word रूप्यकम् or rupaya, the modern Indian rupee has a direct lineage from the rupiya, the silver coin, issued by Sher Shah Suri, continued by the Mughal rulers. Rūpa means to form or shape, example, Rūpyarūpa, rūpya — wrought silver, however, in the region of Bengal, the term taka has always been used to refer to currency. In the 14th century, Ibn Battuta noticed that people in the Bengal Sultanate referred to gold, today, the currency of Bangladesh is officially known as taka. The word taka in Bengali is also used generically to mean any money, currency. Thus, colloquially, a person speaking in Bengali may use taka to refer to money regardless of what currency it is denominated in, thus, in the states of West Bengal and Tripura the Indian rupee is officially known টাকা. Whereas, in the states of Assam and Odisha, the Indian rupee is known by names derived from the Sanskrit word ṭaṅka, টকা in Assamese. During his five-year rule from 1540 to 1545, Sultan Sher Shah Suri issued a coin of silver, weighing 178 grains, the silver coin remained in use during the Mughal period, Maratha era as well as in British India. Among the earliest issues of paper rupees include, the Bank of Hindustan, the General Bank of Bengal and Bihar, historically, the rupee was a silver coin. This had severe consequences in the century when the strongest economies in the world were on the gold standard. The discovery of large quantities of silver in the United States and several European colonies resulted in a decline in the value of relative to gold
29.
Reuleaux hendecagon
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A Reuleaux triangle is a shape formed from the intersection of three circular disks, each having its center on the boundary of the other two. It is a curve of constant width, the simplest and best known such curve other than the circle itself, Constant width means that the separation of every two parallel supporting lines is the same, independent of their orientation. Because all its diameters are the same, the Reuleaux triangle is one answer to the question Other than a circle, Reuleaux triangles have also been called spherical triangles, but that term more properly refers to triangles on the curved surface of a sphere. Among constant-width shapes with a width, the Reuleaux triangle has the minimum area. By several numerical measures it is the farthest from being centrally symmetric and it provides the largest constant-width shape avoiding the points of an integer lattice, and is closely related to the shape of the quadrilateral maximizing the ratio of perimeter to diameter. It can perform a rotation within a square while at all times touching all four sides of the square. However, although it covers most of the square in this process, it fails to cover a small fraction of the squares area. Because of this property of rotating within a square, the Reuleaux triangle is sometimes known as the Reuleaux rotor. The Reuleaux triangle is the first of a sequence of Reuleaux polygons, some of these curves have been used as the shapes of coins. Alternatively, the surface of revolution of the Reuleaux triangle also has constant width, the Reuleaux triangle may be constructed either directly from three circles, or by rounding the sides of an equilateral triangle. The three-circle construction may be performed with a compass alone, not even needing a straightedge, by the Mohr–Mascheroni theorem the same is true more generally of any compass-and-straightedge construction, but the construction for the Reuleaux triangle is particularly simple. The first step is to mark two points of the plane, and use the compass to draw a circle centered at one of the marked points. Next, one draws a circle, of the same radius, centered at the other marked point. Finally, one draws a circle, again of the same radius. The central region in the arrangement of three circles will be a Reuleaux triangle. Alternatively, a Reuleaux triangle may be constructed from an equilateral triangle T by drawing three arcs of circles, each centered at one vertex of T and connecting the two vertices. Or, equivalently, it may be constructed as the intersection of three disks centered at the vertices of T, with equal to the side length of T. In any pair of parallel supporting lines, one of the two lines will necessarily touch the triangle at one of its vertices, the other supporting line may touch the triangle at any point on the opposite arc, and their distance equals the radius of this arc
30.
Susan B. Anthony dollar
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Anthony dollar was a United States dollar coin minted from 1979 to 1981, when the series was suspended due to poor public acceptance, and again in 1999. Finally, a round planchet with an inner border was chosen for the smaller dollar. Several proposals were submitted, and social reformer Susan B, Anthony was selected as the design subject. The reverse design of the Eisenhower dollar was kept, both sides of the coin were designed by Frank Gasparro, the Chief Engraver of the United States Mint. The Mint struck 500 million coins in anticipation of public demand. Anthony dollar was poorly received, in part because of confusion caused by its similarity in size, despite its poor reception, the coins began seeing use in vending machines and mass transit systems, gradually depleting the surplus. In 1997, Congress passed a law authorizing the mintage of a new gold-colored one dollar coin depicting Sacagawea, to fill the gap, the Susan B. Anthony dollar was struck again in 1999, the series was retired the following year, special coins for sale to collectors were struck in proof finish through the run of the Susan B. Anthony dollar, and some minting varieties are valuable to collectors, however, most circulation strikes remained in government stockpiles for years after minting, so many are available in uncirculated grades, and the premium over face value is minimal. In the early 1960s, as the price of silver rose, no silver dollars had been minted in the United States since 1935, and a shortage developed in the Western United States, especially in areas in which gambling was common. As a result, Congress voted to authorize production of 45 million new silver Peace dollars on August 3,1964. However, the move drew strong condemnation from critics who believed that the issuance of the coins was influenced by special interests, and that they would be quickly removed from circulation. The dollars were melted, and the Coinage Act of 1965, enacted on July 23,1965. President Richard Nixon signed the bill into law on December 31,1970, both the obverse and reverse designs were created by Frank Gasparro, the Chief Engraver of the United States Mint. The coin, known as the Eisenhower dollar, proved unpopular with the public, in 1976, the Research Triangle Institute conducted a survey of United States coinage. Among other things, they recommended the dollar, which also saw little use, be entirely eliminated from production. Their report read in part, A conveniently-sized dollar coin would significantly broaden the capabilities of consumers for cash transactions, members of the automatic merchandising industry have expressed a strong interest in a smaller dollar, indicating their willingness to adapt their machinery to its use. Numismatic historian David L. Ganz suggested that Eisenhower, a Republican, was chosen as a means of balancing the half dollar, the Mint began preparation for the reduced-diameter dollar coin in 1976
31.
Hendecagram
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In geometry, a hendecagram is a star polygon that has eleven vertices. The name hendecagram combines a Greek numeral prefix, hendeca-, with the Greek suffix -gram, the hendeca- prefix derives from Greek ἕνδεκα meaning eleven. The -gram suffix derives from γραμμῆς meaning a line and these same four forms can also be considered as stellations of a regular hendecagon. As with all odd regular polygons and star polygons whose orders are not products of distinct Fermat primes, however, Hilton & Pedersen describe folding patterns for making the hendecagrams, and out of strips of paper. Prisms over the hendecagrams and may be used to approximate the shape of DNA molecules, fort Wood, now the base of the Statue of Liberty in New York City, is a star fort in the form of an irregular 11-point star. The Topkapı Scroll contains images of an 11-pointed star Girih form used in Islamic art, the star in this scroll is not one of the regular forms of the hendecagram, but instead uses lines that connect the vertices of a hendecagon to nearly-opposite midpoints of the hendecagons edges. 11-pointed star Girih patterns are used on the exterior of the Momine Khatun Mausoleum. An 11-point star-shaped cross-section was used in the Space Shuttle Solid Rocket Booster, hendecagrammic prism Weisstein, Eric W. Polygram
32.
10-simplex
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In geometry, a 10-simplex is a self-dual regular 10-polytope. Its dihedral angle is cos−1, or approximately 84. 26° and it can also be called a hendecaxennon, or hendeca-10-tope, as an 11-facetted polytope in 10-dimensions. The name hendecaxennon is derived from hendeca for 11 facets in Greek and -xenn, having 9-dimensional facets and this construction is based on facets of the 11-orthoplex. The 2-skeleton of the 10-simplex is topologically related to the 11-cell abstract regular polychoron which has the same 11 vertices,55 edges, but only 1/3 the faces. Coxeter, Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p.296, Table I, Regular Polytopes, three regular polytopes in n-dimensions H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973, p.296, Table I, Regular Polytopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. 10D uniform polytopes x3o3o3o3o3o3o3o3o3o - ux, Polytopes of Various Dimensions Multi-dimensional Glossary
33.
American Mathematical Monthly
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The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by the Mathematical Association of America, the American Mathematical Monthly is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content, in this the American Mathematical Monthly fulfills a different role from that of typical mathematical research journals. The American Mathematical Monthly is the most widely read journal in the world according to records on JSTOR. Since 1997, the journal has been available online at the Mathematical Association of Americas website, the MAA gives the Lester R. Ford Awards annually to authors of articles of expository excellence published in the American Mathematical Monthly
34.
JSTOR
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JSTOR is a digital library founded in 1995. Originally containing digitized back issues of journals, it now also includes books and primary sources. It provides full-text searches of almost 2,000 journals, more than 8,000 institutions in more than 160 countries have access to JSTOR, most access is by subscription, but some older public domain content is freely available to anyone. William G. Bowen, president of Princeton University from 1972 to 1988, JSTOR originally was conceived as a solution to one of the problems faced by libraries, especially research and university libraries, due to the increasing number of academic journals in existence. Most libraries found it prohibitively expensive in terms of cost and space to maintain a collection of journals. By digitizing many journal titles, JSTOR allowed libraries to outsource the storage of journals with the confidence that they would remain available long-term, online access and full-text search ability improved access dramatically. Bowen initially considered using CD-ROMs for distribution, JSTOR was initiated in 1995 at seven different library sites, and originally encompassed ten economics and history journals. JSTOR access improved based on feedback from its sites. Special software was put in place to make pictures and graphs clear, with the success of this limited project, Bowen and Kevin Guthrie, then-president of JSTOR, wanted to expand the number of participating journals. They met with representatives of the Royal Society of London and an agreement was made to digitize the Philosophical Transactions of the Royal Society dating from its beginning in 1665, the work of adding these volumes to JSTOR was completed by December 2000. The Andrew W. Mellon Foundation funded JSTOR initially, until January 2009 JSTOR operated as an independent, self-sustaining nonprofit organization with offices in New York City and in Ann Arbor, Michigan. JSTOR content is provided by more than 900 publishers, the database contains more than 1,900 journal titles, in more than 50 disciplines. Each object is identified by an integer value, starting at 1. In addition to the site, the JSTOR labs group operates an open service that allows access to the contents of the archives for the purposes of corpus analysis at its Data for Research service. This site offers a facility with graphical indication of the article coverage. Users may create focused sets of articles and then request a dataset containing word and n-gram frequencies and they are notified when the dataset is ready and may download it in either XML or CSV formats. The service does not offer full-text, although academics may request that from JSTOR, JSTOR Plant Science is available in addition to the main site. The materials on JSTOR Plant Science are contributed through the Global Plants Initiative and are only to JSTOR
35.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
36.
Carl Friedrich Gauss
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Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick, in the Duchy of Brunswick-Wolfenbüttel, as the son of poor working-class parents. Gauss later solved this puzzle about his birthdate in the context of finding the date of Easter and he was christened and confirmed in a church near the school he attended as a child. A contested story relates that, when he was eight, he figured out how to add up all the numbers from 1 to 100, there are many other anecdotes about his precocity while a toddler, and he made his first ground-breaking mathematical discoveries while still a teenager. He completed Disquisitiones Arithmeticae, his opus, in 1798 at the age of 21. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day, while at university, Gauss independently rediscovered several important theorems. Gauss was so pleased by this result that he requested that a regular heptadecagon be inscribed on his tombstone, the stonemason declined, stating that the difficult construction would essentially look like a circle. The year 1796 was most productive for both Gauss and number theory and he discovered a construction of the heptadecagon on 30 March. He further advanced modular arithmetic, greatly simplifying manipulations in number theory, on 8 April he became the first to prove the quadratic reciprocity law. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic, the prime number theorem, conjectured on 31 May, gives a good understanding of how the prime numbers are distributed among the integers. Gauss also discovered that every integer is representable as a sum of at most three triangular numbers on 10 July and then jotted down in his diary the note, ΕΥΡΗΚΑ. On October 1 he published a result on the number of solutions of polynomials with coefficients in finite fields, in 1831 Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber, leading to new knowledge in magnetism and the discovery of Kirchhoffs circuit laws in electricity. It was during this time that he formulated his namesake law and they constructed the first electromechanical telegraph in 1833, which connected the observatory with the institute for physics in Göttingen. In 1840, Gauss published his influential Dioptrische Untersuchungen, in which he gave the first systematic analysis on the formation of images under a paraxial approximation. Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its cardinal points and he derived the Gaussian lens formula. In 1845, he became associated member of the Royal Institute of the Netherlands, in 1854, Gauss selected the topic for Bernhard Riemanns Habilitationvortrag, Über die Hypothesen, welche der Geometrie zu Grunde liegen. On the way home from Riemanns lecture, Weber reported that Gauss was full of praise, Gauss died in Göttingen, on 23 February 1855 and is interred in the Albani Cemetery there. Two individuals gave eulogies at his funeral, Gausss son-in-law Heinrich Ewald and Wolfgang Sartorius von Waltershausen and his brain was preserved and was studied by Rudolf Wagner who found its mass to be 1,492 grams and the cerebral area equal to 219,588 square millimeters. Highly developed convolutions were also found, which in the early 20th century were suggested as the explanation of his genius, Gauss was a Lutheran Protestant, a member of the St. Albans Evangelical Lutheran church in Göttingen
37.
Power of two
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In mathematics, a power of two means a number of the form 2n where n is an integer, i. e. the result of exponentiation with number two as the base and integer n as the exponent. In a context where only integers are considered, n is restricted to values, so we have 1,2. Because two is the base of the numeral system, powers of two are common in computer science. Written in binary, a power of two always has the form 100…000 or 0. 00…001, just like a power of ten in the decimal system, a word, interpreted as an unsigned integer, can represent values from 0 to 2n −1 inclusively. Corresponding signed integer values can be positive, negative and zero, either way, one less than a power of two is often the upper bound of an integer in binary computers. As a consequence, numbers of this show up frequently in computer software. For example, in the original Legend of Zelda the main character was limited to carrying 255 rupees at any time. Powers of two are used to measure computer memory. A byte is now considered eight bits (an octet, resulting in the possibility of 256 values, the prefix kilo, in conjunction with byte, may be, and has traditionally been, used, to mean 1,024. However, in general, the term kilo has been used in the International System of Units to mean 1,000, binary prefixes have been standardized, such as kibi meaning 1,024. Nearly all processor registers have sizes that are powers of two,32 or 64 being most common, powers of two occur in a range of other places as well. For many disk drives, at least one of the size, number of sectors per track. The logical block size is almost always a power of two. Numbers that are not powers of two occur in a number of situations, such as video resolutions, but they are often the sum or product of two or three powers of two, or powers of two minus one. For example,640 =512 +128 =128 ×5, put another way, they have fairly regular bit patterns. A prime number that is one less than a power of two is called a Mersenne prime, for example, the prime number 31 is a Mersenne prime because it is 1 less than 32. Similarly, a number that is one more than a positive power of two is called a Fermat prime—the exponent itself is a power of two. A fraction that has a power of two as its denominator is called a dyadic rational, the numbers that can be represented as sums of consecutive positive integers are called polite numbers, they are exactly the numbers that are not powers of two
38.
List of polygons
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In geometry, a polygon /ˈpɒlɪɡɒn/ is traditionally a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain or circuit. These segments are called its edges or sides, and the points where two edges meet are the vertices or corners. The word polygon comes from Late Latin polygōnum, from Greek πολύγωνον, noun use of neuter of πολύγωνος, individual polygons are named according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e. g. pentagon, dodecagon. The triangle, quadrilateral and nonagon are exceptions, although the regular forms trigon, tetragon, polygons are primarily named by prefixes from Greek numbers. To construct the name of a polygon with more than 20 and less than 100 edges, the kai connector is not included by some authors. Extending the system up to 999 is expressed with these prefixes, platonic solid Dice NAMING POLYGONS Benjamin Franklin Finkel, A Mathematical Solution Book Containing Systematic Solutions to Many of the Most Difficult Problems,1888
39.
Monogon
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In geometry a monogon is a polygon with one edge and one vertex. Since a monogon has only one side and only one vertex, in Euclidean geometry a monogon is a degenerate polygon because its endpoints must coincide, unlike any Euclidean line segment. Most definitions of a polygon in Euclidean geometry do not admit the monogon, in spherical geometry, a monogon can be constructed as a vertex on a great circle. This forms a dihedron, with two hemispherical monogonal faces which share one 360° edge and one vertex and its dual, a hosohedron, has two antipodal vertices at the poles, one 360 degree lune face, and one edge between the two vertices. Digon Herbert Busemann, The geometry of geodesics, new York, Academic Press,1955 Coxeter, H. S. M, Regular Polytopes
40.
Digon
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In geometry, a digon is a polygon with two sides and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved, a regular digon has both angles equal and both sides equal and is represented by Schläfli symbol. It may be constructed on a sphere as a pair of 180 degree arcs connecting antipodal points, the digon is the simplest abstract polytope of rank 2. A truncated digon, t is a square, an alternated digon, h is a monogon. A straight-sided digon is regular even though it is degenerate, because its two edges are the length and its two angles are equal. As such, the regular digon is a constructible polygon, some definitions of a polygon do not consider the digon to be a proper polygon because of its degeneracy in the Euclidean case. A digon as a face of a polyhedron is degenerate because it is a degenerate polygon, but sometimes it can have a useful topological existence in transforming polyhedra. A spherical lune is a digon whose two vertices are antipodal points on the sphere, a spherical polyhedron constructed from such digons is called a hosohedron. The digon is an important construct in the theory of networks such as graphs. Topological equivalences may be established using a process of reduction to a set of polygons. The digon represents a stage in the simplification where it can be removed and substituted by a line segment. The cyclic groups may be obtained as rotation symmetries of polygons, monogon Demihypercube Herbert Busemann, The geometry of geodesics. New York, Academic Press,1955 Coxeter, Regular Polytopes, Dover Publications Inc,1973 ISBN 0-486-61480-8 Weisstein, a. B. Ivanov, Digon, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 Media related to Digons at Wikimedia Commons
41.
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