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.
Dihedral group
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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
5.
Internal and external angles
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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°
6.
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
7.
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
8.
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
9.
Circumscribed circle
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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
10.
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
11.
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
12.
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
13.
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
14.
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
15.
Euclidean tilings by convex regular polygons
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Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his Harmonices Mundi and this means that, for every pair of flags, there is a symmetry operation mapping the first flag to the second. This is equivalent to the tiling being an edge-to-edge tiling by congruent regular polygons, There must be six equilateral triangles, four squares or three regular hexagons at a vertex, yielding the three regular tessellations. Vertex-transitivity means that for pair of vertices there is a symmetry operation mapping the first vertex to the second. Note that there are two mirror forms of 34.6 tiling, only one of which is shown in the following table. All other regular and semiregular tilings are achiral, though these yield the same set of tilings in the plane, in other spaces there are Archimedean tilings which are not uniform. Such periodic tilings may be classified by the number of orbits of vertices, edges and tiles. If there are k orbits of vertices, a tiling is known as k -uniform or k -isogonal, if there are t orbits of tiles, as t -isohedral, if there are e orbits of edges, as e -isotoxal. K-uniform tilings with the vertex figures can be further identified by their wallpaper group symmetry. 1-uniform tilings include 3 regular tilings, and 8 semiregular ones, There are 20 2-uniform tilings,61 3-uniform tilings,151 4-uniform tilings,332 5-uniform tilings and 673 6-uniform tilings. Each can be grouped by the m of distinct vertex figures. For edge-to-edge Euclidean tilings, the angles of the polygons meeting at a vertex must add to 360 degrees. A regular n -gon has internal angle 180 degrees, only eleven of these can occur in a uniform tiling of regular polygons, given in previous sections. In particular, if three polygons meet at a vertex and one has an odd number of sides, the two polygons must be the same. If they are not, they would have to alternate around the first polygon, vertex types are listed for each. If two tilings share the two vertex types, they are given subscripts 1,2. There are 61 3-uniform tilings of the Euclidean plane,39 are 3-Archimedean with 3 distinct vertex types, while 22 have 2 identical vertex types in different symmetry orbits. Chavey There are 151 4-uniform tilings of the Euclidean plane, Brian Galebachs search reproduced Krotenheerdts list of 33 4-uniform tilings with 4 distinct vertex types, as well as finding 85 of them with 3 vertex types, and 33 with 2 vertex types
16.
Compass-and-straightedge construction
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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
17.
Euclid's Elements
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Euclids Elements is a mathematical and geometric treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt circa 300 BC. It is a collection of definitions, postulates, propositions, the books cover Euclidean geometry and the ancient Greek version of elementary number theory. Elements is the second-oldest extant Greek mathematical treatise after Autolycus On the Moving Sphere and it has proven instrumental in the development of logic and modern science. According to Proclus, the element was used to describe a theorem that is all-pervading. The word element in the Greek language is the same as letter and this suggests that theorems in the Elements should be seen as standing in the same relation to geometry as letters to language. Euclids Elements has been referred to as the most successful and influential textbook ever written, for centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclids Elements was required of all students. Not until the 20th century, by which time its content was taught through other school textbooks. Scholars believe that the Elements is largely a collection of theorems proven by other mathematicians, the Elements may have been based on an earlier textbook by Hippocrates of Chios, who also may have originated the use of letters to refer to figures. This manuscript, the Heiberg manuscript, is from a Byzantine workshop around 900 and is the basis of modern editions, papyrus Oxyrhynchus 29 is a tiny fragment of an even older manuscript, but only contains the statement of one proposition. Although known to, for instance, Cicero, no record exists of the text having been translated into Latin prior to Boethius in the fifth or sixth century. The Arabs received the Elements from the Byzantines around 760, this version was translated into Arabic under Harun al Rashid circa 800, the Byzantine scholar Arethas commissioned the copying of one of the extant Greek manuscripts of Euclid in the late ninth century. Although known in Byzantium, the Elements was lost to Western Europe until about 1120, the first printed edition appeared in 1482, and since then it has been translated into many languages and published in about a thousand different editions. Theons Greek edition was recovered in 1533, in 1570, John Dee provided a widely respected Mathematical Preface, along with copious notes and supplementary material, to the first English edition by Henry Billingsley. Copies of the Greek text still exist, some of which can be found in the Vatican Library, the manuscripts available are of variable quality, and invariably incomplete. By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text, ancient texts which refer to the Elements itself, and to other mathematical theories that were current at the time it was written, are also important in this process. Such analyses are conducted by J. L. Heiberg and Sir Thomas Little Heath in their editions of the text, also of importance are the scholia, or annotations to the text. These additions, which distinguished themselves from the main text. The Elements is still considered a masterpiece in the application of logic to mathematics, in historical context, it has proven enormously influential in many areas of science
18.
Golden ratio
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In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship, expressed algebraically, for quantities a and b with a > b >0, a + b a = a b = def φ, where the Greek letter phi represents the golden ratio. Its value is, φ =1 +52 =1.6180339887 …, A001622 The golden ratio is also called the golden mean or golden section. Other names include extreme and mean ratio, medial section, divine proportion, divine section, golden proportion, golden cut, the golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts. The golden ratio has also used to analyze the proportions of natural objects as well as man-made systems such as financial markets. Two quantities a and b are said to be in the golden ratio φ if a + b a = a b = φ, one method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ, a + b a =1 + b a =1 +1 φ, multiplying by φ gives φ +1 = φ2 which can be rearranged to φ2 − φ −1 =0. First, the line segment A B ¯ is about doubled and then the semicircle with the radius A S ¯ around the point S is drawn, now the semicircle is drawn with the radius A B ¯ around the point B. The arising intersection point E corresponds 2 φ, next up, the perpendicular on the line segment A E ¯ from the point D will be establish. The subsequent parallel F S ¯ to the line segment C M ¯, produces, as it were and it is well recognizable, this triangle and the triangle M S C are similar to each other. The hypotenuse F S ¯ has due to the cathetuses S D ¯ =1 and D F ¯ =2 according the Pythagorean theorem, finally, the circle arc is drawn with the radius 5 around the point F. The golden ratio has been claimed to have held a fascination for at least 2,400 years. But the fascination with the Golden Ratio is not confined just to mathematicians, biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics. Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry, the division of a line into extreme and mean ratio is important in the geometry of regular pentagrams and pentagons. Euclid explains a construction for cutting a line in extreme and mean ratio, throughout the Elements, several propositions and their proofs employ the golden ratio. The golden ratio is explored in Luca Paciolis book De divina proportione, since the 20th century, the golden ratio has been represented by the Greek letter φ or less commonly by τ. Timeline according to Priya Hemenway, Phidias made the Parthenon statues that seem to embody the golden ratio, plato, in his Timaeus, describes five possible regular solids, some of which are related to the golden ratio
19.
Pentagon
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In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting, a self-intersecting regular pentagon is called a pentagram. A regular pentagon has Schläfli symbol and interior angles are 108°, a regular pentagon has five lines of reflectional symmetry, and rotational symmetry of order 5. The diagonals of a regular pentagon are in the golden ratio to its sides. The area of a regular convex pentagon with side length t is given by A = t 225 +1054 =5 t 2 tan 4 ≈1.720 t 2. A pentagram or pentangle is a regular star pentagon and its sides form the diagonals of a regular convex pentagon – in this arrangement the sides of the two pentagons are in the golden ratio. The area of any polygon is, A =12 P r where P is the perimeter of the polygon. Substituting the regular pentagons values for P and r gives the formula A =12 ×5 t × t tan 2 =5 t 2 tan 4 with side length t, like every regular convex polygon, the regular convex pentagon has an inscribed circle. The apothem, which is the r of the inscribed circle. Like every regular polygon, the regular convex pentagon has a circumscribed circle. For a regular pentagon with successive vertices A, B, C, D, E, the regular pentagon is constructible with compass and straightedge, as 5 is a Fermat prime. A variety of methods are known for constructing a regular pentagon, one method to construct a regular pentagon in a given circle is described by Richmond and further discussed in Cromwells Polyhedra. The top panel shows the construction used in Richmonds method to create the side of the inscribed pentagon, the circle defining the pentagon has unit radius. Its center is located at point C and a midpoint M is marked halfway along its radius and this point is joined to the periphery vertically above the center at point D. Angle CMD is bisected, and the bisector intersects the axis at point Q. A horizontal line through Q intersects the circle at point P, to determine the length of this side, the two right triangles DCM and QCM are depicted below the circle. Using Pythagoras theorem and two sides, the hypotenuse of the triangle is found as 5 /2
20.
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
21.
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
22.
Directed graph
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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+
23.
Star polygon
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In geometry, a star polygon is a type of non-convex polygon. Only the regular polygons have been studied in any depth. The first usage is included in polygrams which includes polygons like the pentagram, star polygon names combine a numeral prefix, such as penta-, with the Greek suffix -gram. The prefix is normally a Greek cardinal, but synonyms using other prefixes exist, for example, a nine-pointed polygon or enneagram is also known as a nonagram, using the ordinal nona from Latin. The -gram suffix derives from γραμμή meaning a line, alternatively for integers p and q, it can be considered as being constructed by connecting every qth point out of p points regularly spaced in a circular placement. A regular star polygon is denoted by its Schläfli symbol, where p and q are relatively prime, the symmetry group of is dihedral group Dn of order 2n, independent of k. A regular star polygon can also be obtained as a sequence of stellations of a regular core polygon. Regular star polygons were first studied systematically by Thomas Bradwardine, if p and q are not coprime, a degenerate polygon will result with coinciding vertices and edges. For example will appear as a triangle, but can be labeled with two sets of vertices 1-6 and this should be seen not as two overlapping triangles, but a double-winding of a single unicursal hexagon. For |n/d|, the vertices have an exterior angle, β. These polygons are often seen in tiling patterns, the parametric angle α can be chosen to match internal angles of neighboring polygons in a tessellation pattern. The interior of a polygon may be treated in different ways. Three such treatments are illustrated for a pentagram, branko Grunbaum and Geoffrey Shephard consider two of them, as regular star polygons and concave isogonal 2n-gons. These include, Where a side occurs, one side is treated as outside and this is shown in the left hand illustration and commonly occurs in computer vector graphics rendering. The number of times that the polygonal curve winds around a given region determines its density, the exterior is given a density of 0, and any region of density >0 is treated as internal. This is shown in the illustration and commonly occurs in the mathematical treatment of polyhedra. Where a line may be drawn between two sides, the region in which the line lies is treated as inside the figure and this is shown in the right hand illustration and commonly occurs when making a physical model. When the area of the polygon is calculated, each of these approaches yields a different answer, star polygons feature prominently in art and culture
24.
Polygram (geometry)
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A regular polygram can either be in a set of regular polygons or in a set of regular polygon compounds. The polygram names combine a numeral prefix, such as penta-, the prefix is normally a Greek cardinal, but synonyms using other prefixes exist. The -gram suffix derives from γραμμῆς meaning a line, a regular polygram, as a general regular polygon, is denoted by its Schläfli symbol, where p and q are relatively prime and q ≥2. For integers p and q, it can be considered as being constructed by connecting every qth point out of p points regularly spaced in a circular placement. In other cases where n and m have a factor, a polygram is interpreted as a lower polygon, with k = gcd. These figures are called regular compound polygons, list of regular polytopes and compounds#Stars Cromwell, P. Polyhedra, CUP, Hbk. P.175 Grünbaum, B. and G. C, shephard, Tilings and Patterns, New York, W. H. Freeman & Co. Polyhedra with Hollow Faces, Proc of NATO-ASI Conference on Polytopes, etc. ed T. Bisztriczky et al. John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Robert Lachlan, london, Macmillan,1893, p.83 polygrams. Branko Grünbaum, Metamorphoses of polygons, published in The Lighter Side of Mathematics, Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History
25.
Equilateral triangle
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In geometry, an equilateral triangle is a triangle in which all three sides are equal. In the familiar Euclidean geometry, equilateral triangles are also equiangular and they are regular polygons, and can therefore also be referred to as regular triangles. Thus these are properties that are unique to equilateral triangles, the three medians have equal lengths. The three angle bisectors have equal lengths, every triangle center of an equilateral triangle coincides with its centroid, which implies that the equilateral triangle is the only triangle with no Euler line connecting some of the centers. For some pairs of triangle centers, the fact that they coincide is enough to ensure that the triangle is equilateral, in particular, A triangle is equilateral if any two of the circumcenter, incenter, centroid, or orthocenter coincide. It is also equilateral if its circumcenter coincides with the Nagel point, for any triangle, the three medians partition the triangle into six smaller triangles. A triangle is equilateral if and only if any three of the triangles have either the same perimeter or the same inradius. A triangle is equilateral if and only if the circumcenters of any three of the triangles have the same distance from the centroid. Morleys trisector theorem states that, in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, a version of the isoperimetric inequality for triangles states that the triangle of greatest area among all those with a given perimeter is equilateral. That is, PA, PB, and PC satisfy the inequality that any two of them sum to at least as great as the third. By Eulers inequality, the triangle has the smallest ratio R/r of the circumradius to the inradius of any triangle, specifically. The triangle of largest area of all those inscribed in a circle is equilateral. The ratio of the area of the incircle to the area of an equilateral triangle, the ratio of the area to the square of the perimeter of an equilateral triangle,1123, is larger than that for any other triangle. If a segment splits an equilateral triangle into two regions with equal perimeters and with areas A1 and A2, then 79 ≤ A1 A2 ≤97, in no other triangle is there a point for which this ratio is as small as 2. For any point P in the plane, with p, q, and t from the vertices A, B. For any point P on the circle of an equilateral triangle, with distances p, q. There are numerous triangle inequalities that hold with equality if and only if the triangle is equilateral, an equilateral triangle is the most symmetrical triangle, having 3 lines of reflection and rotational symmetry of order 3 about its center. Its symmetry group is the group of order 6 D3
26.
Pentagram
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A pentagram is the shape of a five-pointed star drawn with five straight strokes. The word pentagram comes from the Greek word πεντάγραμμον, from πέντε, five + γραμμή, the word pentacle is sometimes used synonymously with pentagram The word pentalpha is a learned modern revival of a post-classical Greek name of the shape. The pentagram is the simplest regular star polygon, the pentagram contains ten points and fifteen line segments. It is represented by the Schläfli symbol, like a regular pentagon, and a regular pentagon with a pentagram constructed inside it, the regular pentagram has as its symmetry group the dihedral group of order 10. The pentagram can be constructed by connecting alternate vertices of a pentagon and it can also be constructed as a stellation of a pentagon, by extending the edges of a pentagon until the lines intersect. Each intersection of edges sections the edges in the golden ratio, also, the ratio of the length of the shorter segment to the segment bounded by the two intersecting edges is φ. As the four-color illustration shows, r e d g r e e n = g r e e n b l u e = b l u e m a g e n t a = φ. The pentagram includes ten isosceles triangles, five acute and five obtuse isosceles triangles, in all of them, the ratio of the longer side to the shorter side is φ. The acute triangles are golden triangles, the obtuse isosceles triangle highlighted via the colored lines in the illustration is a golden gnomon. The pentagram of Venus is the apparent path of the planet Venus as observed from Earth, the tips of the five loops at the center of the figure have the same geometric relationship to one another as the five vertices, or points, of a pentagram. Groups of five intersections of curves, equidistant from the center, have the same geometric relationship. In early monumental Sumerian script, or cuneiform, a pentagram glyph served as a logogram for the word ub, meaning corner, angle, nook, the word Pentemychos was the title of the cosmogony of Pherecydes of Syros. Here, the five corners are where the seeds of Chronos are placed within the Earth in order for the cosmos to appear. The pentangle plays an important symbolic role in the 14th-century English poem Sir Gawain, heinrich Cornelius Agrippa and others perpetuated the popularity of the pentagram as a magic symbol, attributing the five neoplatonic elements to the five points, in typical Renaissance fashion. By the mid-19th century a distinction had developed amongst occultists regarding the pentagrams orientation. With a single point upwards it depicted spirit presiding over the four elements of matter, however, the influential writer Eliphas Levi called it evil whenever the symbol appeared the other way up. It is the goat of lust attacking the heavens with its horns and it is the sign of antagonism and fatality. It is the goat of lust attacking the heavens with its horns, faust, The pentagram thy peace doth mar
27.
Petrie polygon
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In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every consecutive sides belong to one of the facets. The Petrie polygon of a polygon is the regular polygon itself. For every regular polytope there exists an orthogonal projection onto a plane such that one Petrie polygon becomes a regular polygon with the remainder of the interior to it. The plane in question is the Coxeter plane of the group of the polygon. These polygons and projected graphs are useful in visualizing symmetric structure of the regular polytopes. John Flinders Petrie was the son of Egyptologist Flinders Petrie. He was born in 1907 and as a schoolboy showed remarkable promise of mathematical ability, in periods of intense concentration he could answer questions about complicated four-dimensional objects by visualizing them. He first noted the importance of the skew polygons which appear on the surface of regular polyhedra. When my incredulity had begun to subside, he described them to me, one consisting of squares, six at each vertex, in 1938 Petrie collaborated with Coxeter, Patrick du Val, and H. T. Flather to produce The Fifty-Nine Icosahedra for publication, realizing the geometric facility of the skew polygons used by Petrie, Coxeter named them after his friend when he wrote Regular Polytopes. In 1972, a few months after his retirement, Petrie was killed by a car attempting to cross a motorway near his home in Surrey. The idea of Petrie polygons was later extended to semiregular polytopes, the Petrie polygon of the regular polyhedron has h sides, where h+2=24/. The regular duals, and, are contained within the same projected Petrie polygon, three of the Kepler–Poinsot polyhedra have hexagonal, and decagrammic, petrie polygons. The Petrie polygon projections are most useful for visualization of polytopes of dimension four and this table represents Petrie polygon projections of 3 regular families, and the exceptional Lie group En which generate semiregular and uniform polytopes for dimensions 4 to 8. Coxeter, H. S. M. Regular Polytopes, 3rd ed, Section 4.3 Flags and Orthoschemes, Section 11.3 Petrie polygons Ball, W. W. R. and H. S. M. Coxeter Mathematical Recreations and Essays, 13th ed. The Beauty of Geometry, Twelve Essays, Dover Publications LCCN 99-35678 Peter McMullen, Egon Schulte Abstract Regular Polytopes, ISBN 0-521-81496-0 Steinberg, Robert, ON THE NUMBER OF SIDES OF A PETRIE POLYGON Weisstein, Eric W. Petrie polygon. Weisstein, Eric W. Cross polytope graphs, Weisstein, Eric W. Gosset graph 3_21
28.
Projection (linear algebra)
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In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. That is, whenever P is applied twice to any value, though abstract, this definition of projection formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on an object by examining the effect of the projection on points in the object. For example, the function maps the point in three-dimensional space R3 to the point is an orthogonal projection onto the x–y plane. This function is represented by the matrix P =, the action of this matrix on an arbitrary vector is P =. To see that P is indeed a projection, i. e. P = P2, a simple example of a non-orthogonal projection is P =. Via matrix multiplication, one sees that P2 = = = P. proving that P is indeed a projection, the projection P is orthogonal if and only if α =0. Let W be a finite dimensional space and P be a projection on W. Suppose the subspaces U and V are the range and kernel of P respectively, then P has the following properties, By definition, P is idempotent. P is the identity operator I on U ∀ x ∈ U, P x = x and we have a direct sum W = U ⊕ V. Every vector x ∈ W may be decomposed uniquely as x = u + v with u = P x and v = x − P x = x, the range and kernel of a projection are complementary, as are P and Q = I − P. The operator Q is also a projection and the range and kernel of P become the kernel and range of Q and we say P is a projection along V onto U and Q is a projection along U onto V. In infinite dimensional spaces, the spectrum of a projection is contained in as −1 =1 λ I +1 λ P. Only 0 or 1 can be an eigenvalue of a projection, the corresponding eigenspaces are the kernel and range of the projection. Decomposition of a space into direct sums is not unique in general. Therefore, given a subspace V, there may be many projections whose range is V, if a projection is nontrivial it has minimal polynomial x 2 − x = x, which factors into distinct roots, and thus P is diagonalizable. The product of projections is not, in general, a projection, if projections commute, then their product is a projection. When the vector space W has a product and is complete the concept of orthogonality can be used
29.
14-simplex
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In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. Specifically, a k-simplex is a polytope which is the convex hull of its k +1 vertices. More formally, suppose the k +1 points u 0, …, u k ∈ R k are affinely independent, then, the simplex determined by them is the set of points C =. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, a single point may be considered a 0-simplex, and a line segment may be considered a 1-simplex. A simplex may be defined as the smallest convex set containing the given vertices, a regular simplex is a simplex that is also a regular polytope. A regular n-simplex may be constructed from a regular -simplex by connecting a new vertex to all original vertices by the edge length. In topology and combinatorics, it is common to “glue together” simplices to form a simplicial complex, the associated combinatorial structure is called an abstract simplicial complex, in which context the word “simplex” simply means any finite set of vertices. A 1-simplex is a line segment, the convex hull of any nonempty subset of the n+1 points that define an n-simplex is called a face of the simplex. In particular, the hull of a subset of size m+1 is an m-simplex. The 0-faces are called the vertices, the 1-faces are called the edges, the -faces are called the facets, in general, the number of m-faces is equal to the binomial coefficient. Consequently, the number of m-faces of an n-simplex may be found in column of row of Pascals triangle, a simplex A is a coface of a simplex B if B is a face of A. Face and facet can have different meanings when describing types of simplices in a simplicial complex, see simplical complex for more detail. The regular simplex family is the first of three regular polytope families, labeled by Coxeter as αn, the two being the cross-polytope family, labeled as βn, and the hypercubes, labeled as γn. A fourth family, the infinite tessellation of hypercubes, he labeled as δn, an -simplex can be constructed as a join of an n-simplex and a point. An -simplex can be constructed as a join of an m-simplex, the two simplices are oriented to be completely normal from each other, with translation in a direction orthogonal to both of them. A 1-simplex is a joint of two points, ∨ =2, a general 2-simplex is the join of 3 points, ∨∨. An isosceles triangle is the join of a 1-simplex and a point, a general 3-simplex is the join of 4 points, ∨∨∨. A 3-simplex with mirror symmetry can be expressed as the join of an edge and 2 points, a 3-simplex with triangular symmetry can be expressed as the join of an equilateral triangle and 1 point,3. ∨ or ∨
30.
Great 120-cell
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In geometry, the great 120-cell or great polydodecahedron is a regular star 4-polytope with Schläfli symbol. It is one of 10 regular Schläfli-Hess polytopes and it has the same edge arrangement as the 600-cell, icosahedral 120-cell as well as the same face arrangement as the grand 120-cell. H. S. M. Coxeter, Regular Polytopes, 3rd, john H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Klitzing, Richard. 4D uniform polytopes o3o5o5/2x - sishi, Regular polychora Discussion on names Reguläre Polytope The Regular Star Polychora
31.
Grand stellated 120-cell
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In geometry, the grand stellated 120-cell or grand stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol. It is one of 10 regular Schläfli-Hess polytopes and it has the same edge arrangement as the grand 600-cell, icosahedral 120-cell, and the same face arrangement as the great stellated 120-cell. H. S. M. Coxeter, Regular Polytopes, 3rd, john H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Klitzing, Richard. 4D uniform polytopes x5/2o5o5/2o - gashi, Regular polychora Discussion on names Reguläre Polytope The Regular Star Polychora
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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
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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
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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