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.
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
9.
Polytope compound
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A polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the analogs of polygonal compounds such as the hexagram. The outer vertices of a compound can be connected to form a convex polyhedron called the convex hull, the compound is a facetting of the convex hull. Another convex polyhedron is formed by the central space common to all members of the compound. This polyhedron can be used as the core for a set of stellations, a regular polyhedron compound can be defined as a compound which, like a regular polyhedron, is vertex-transitive, edge-transitive, and face-transitive. There are five regular compounds of polyhedra, best known is the compound of two tetrahedra, often called the stella octangula, a name given to it by Kepler. The vertices of the two tetrahedra define a cube and the intersection of the two an octahedron, which shares the same face-planes as the compound, thus it is a stellation of the octahedron, and in fact, the only finite stellation thereof. The stella octangula can also be regarded as a dual-regular compound, the compound of five tetrahedra comes in two enantiomorphic versions, which together make up the compound of 10 tetrahedra. Each of the compounds is self-dual, and the compound of 5 cubes is dual to the compound of 5 octahedra. There are five such compounds of the regular polyhedra, the tetrahedron is self-dual, so the dual compound of a tetrahedron with its dual polyhedron is also the regular Stella octangula. The cube-octahedron and dodecahedron-icosahedron dual compounds are the first stellations of the cuboctahedron and icosidodecahedron, the compound of the small stellated dodecahedron and great dodecahedron looks outwardly the same as the small stellated dodecahedron, because the great dodecahedron is completely contained inside. For this reason, the image shown above shows the small stellated dodecahedron in wireframe, in 1976 John Skilling published Uniform Compounds of Uniform Polyhedra which enumerated 75 compounds made from uniform polyhedra with rotational symmetry. This list includes the five regular compounds above, the 75 uniform compounds are listed in the Table below. Most are shown singularly colored by each polyhedron element, some chiral pairs of face groups are colored by symmetry of the faces within each polyhedron. If the definition of a polyhedron is generalised they are uniform. The section for entianomorphic pairs in Skillings list does not contain the compound of two great snub dodecicosidodecahedra, as the faces would coincide. Removing the coincident faces results in the compound of twenty octahedra, in 4-dimensions, there are a large number of regular compounds of regular polytopes. There are eighteen two-parameter families of regular tessellations of the Euclidean plane
10.
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
11.
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
12.
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
13.
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
14.
Stellation
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In geometry, stellation is the process of extending a polygon, polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. The new figure is a stellation of the original, the word stellation comes from the Latin stellātus, starred, which in turn comes from Latin stella, star. In 1619 Kepler defined stellation for polygons and polyhedra, as the process of extending edges or faces until they meet to form a new polygon or polyhedron and he stellated the regular dodecahedron to obtain two regular star polyhedra, the small stellated dodecahedron and great stellated dodecahedron. He also stellated the regular octahedron to obtain the stella octangula, stellating a regular polygon symmetrically creates a regular star polygon or polygonal compound. These polygons are characterised by the number of times m that the polygonal boundary winds around the centre of the figure, like all regular polygons, their vertices lie on a circle. M also corresponds to the number of vertices around the circle to get one end of a given edge to the other. A regular star polygon is represented by its Schläfli symbol, where n is the number of vertices, m is the used in sequencing the edges around it. Making m =1 gives the convex, if n and m do have a common divisor, then the figure is a regular compound. For example is the compound of two triangles or hexagram, while is a compound of two pentagrams. Some authors use the Schläfli symbol for such regular compounds, others regard the symbol as indicating a single path which is wound m times around n/m vertex points, such that one edge is superimposed upon another and each vertex point is visited m times. In this case a modified symbol may be used for the compound, a regular n-gon has /2 stellations if n is even, and /2 stellations if n is odd. Like the heptagon, the octagon also has two octagrammic stellations, one, being a star polygon, and the other, being the compound of two squares. A polyhedron is stellated by extending the edges or face planes of a polyhedron until they meet again to form a new polyhedron or compound, the interior of the new polyhedron is divided by the faces into a number of cells. The face planes of a polyhedron may divide space into many such cells, for a symmetrical polyhedron, these cells will fall into groups, or sets, of congruent cells - we say that the cells in such a congruent set are of the same type. A common method of finding stellations involves selecting one or more cell types and this can lead to a huge number of possible forms, so further criteria are often imposed to reduce the set to those stellations that are significant and unique in some way. A set of cells forming a layer around its core is called a shell. For a symmetrical polyhedron, a shell may be made up of one or more cell types, based on such ideas, several restrictive categories of interest have been identified. Adding successive shells to the core leads to the set of main-line stellations
15.
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
16.
Hexagon
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In geometry, a hexagon is a six sided polygon or 6-gon. The total of the angles of any hexagon is 720°. A regular hexagon has Schläfli symbol and can also be constructed as an equilateral triangle, t. A regular hexagon is defined as a hexagon that is both equilateral and equiangular and it is bicentric, meaning that it is both cyclic and tangential. The common length of the sides equals the radius of the circumscribed circle, all internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries and 6 reflection symmetries, the longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane, the cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral, the maximal diameter, D is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the circle, d, is twice the minimal radius or inradius. If a regular hexagon has successive vertices A, B, C, D, E, F, the regular hexagon has Dih6 symmetry, order 12. There are 3 dihedral subgroups, Dih3, Dih2, and Dih1, and 4 cyclic subgroups, Z6, Z3, Z2 and these symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order, r12 is full symmetry, and a1 is no symmetry. These two forms are duals of each other and have half the order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction and it can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites. G2 hexagons, with sides parallel are also called hexagonal parallelogons. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g6 subgroup has no degrees of freedom but can seen as directed edges. Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation, other hexagon shapes can tile the plane with different orientations
17.
Greek language
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Greek is an independent branch of the Indo-European family of languages, native to Greece and other parts of the Eastern Mediterranean. It has the longest documented history of any living language, spanning 34 centuries of written records and its writing system has been the Greek alphabet for the major part of its history, other systems, such as Linear B and the Cypriot syllabary, were used previously. The alphabet arose from the Phoenician script and was in turn the basis of the Latin, Cyrillic, Armenian, Coptic, Gothic and many other writing systems. Together with the Latin texts and traditions of the Roman world, during antiquity, Greek was a widely spoken lingua franca in the Mediterranean world and many places beyond. It would eventually become the official parlance of the Byzantine Empire, the language is spoken by at least 13.2 million people today in Greece, Cyprus, Italy, Albania, Turkey, and the Greek diaspora. Greek roots are used to coin new words for other languages, Greek. Greek has been spoken in the Balkan peninsula since around the 3rd millennium BC, the earliest written evidence is a Linear B clay tablet found in Messenia that dates to between 1450 and 1350 BC, making Greek the worlds oldest recorded living language. Among the Indo-European languages, its date of earliest written attestation is matched only by the now extinct Anatolian languages, the Greek language is conventionally divided into the following periods, Proto-Greek, the unrecorded but assumed last ancestor of all known varieties of Greek. The unity of Proto-Greek would have ended as Hellenic migrants entered the Greek peninsula sometime in the Neolithic era or the Bronze Age, Mycenaean Greek, the language of the Mycenaean civilisation. It is recorded in the Linear B script on tablets dating from the 15th century BC onwards, Ancient Greek, in its various dialects, the language of the Archaic and Classical periods of the ancient Greek civilisation. It was widely known throughout the Roman Empire, after the Roman conquest of Greece, an unofficial bilingualism of Greek and Latin was established in the city of Rome and Koine Greek became a first or second language in the Roman Empire. The origin of Christianity can also be traced through Koine Greek, Medieval Greek, also known as Byzantine Greek, the continuation of Koine Greek in Byzantine Greece, up to the demise of the Byzantine Empire in the 15th century. Much of the written Greek that was used as the language of the Byzantine Empire was an eclectic middle-ground variety based on the tradition of written Koine. Modern Greek, Stemming from Medieval Greek, Modern Greek usages can be traced in the Byzantine period and it is the language used by the modern Greeks, and, apart from Standard Modern Greek, there are several dialects of it. In the modern era, the Greek language entered a state of diglossia, the historical unity and continuing identity between the various stages of the Greek language is often emphasised. Greek speakers today still tend to regard literary works of ancient Greek as part of their own rather than a foreign language and it is also often stated that the historical changes have been relatively slight compared with some other languages. According to one estimation, Homeric Greek is probably closer to demotic than 12-century Middle English is to modern spoken English, Greek is spoken by about 13 million people, mainly in Greece, Albania and Cyprus, but also worldwide by the large Greek diaspora. Greek is the language of Greece, where it is spoken by almost the entire population
18.
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 ∨
19.
Compound of two tetrahedra
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In geometry, a compound of two tetrahedra is constructed by two overlapping tetrahedra, usually implied as regular tetrahedra. There is only one uniform polyhedral compound, the stellated octahedron and it has a regular octahedron core, and shares the same 8 vertices with the cube. There are lower symmetry variations on this compound, based on lower symmetry forms of the tetrahedron, a facetting of a rectangular cuboid, creating compounds of two tetragonal or two rhombic disphenoids, with a bipyramid or rhombic fusil cores. This is first in a set of compound of two antiprisms. A facetting of a trigonal trapezohedron creates a compound of two right triangular pyramids with a triangular antiprism core and this is first in a set of compounds of two pyramids positioned as point reflections of each other. If two regular tetrahedra are given the orientation on the 3-fold axis, a different compound is made, with D3h, symmetry. Other orientations can be chosen as 2 tetrahedra within the compound of five tetrahedra and compound of ten tetrahedra, Cundy, H. and Rollett, §3.10.8 in Mathematical Models, 3rd ed. Stradbroke, England, Tarquin Pub. pp. 139-141,1989. Weisstein, Eric W. Compound of two tetrahedra
20.
Stellated octahedron
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The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula, a given to it by Johannes Kepler in 1609. It was depicted in Paciolis Divina Proportione,1509 and it is the simplest of five regular polyhedral compounds, and the only regular compound of two tetrahedra. It can also be seen as one of the stages in the construction of a 3D Koch Snowflake, the stellated octahedron can be constructed in several ways, It is a stellation of the regular octahedron, sharing the same face planes. It is also a regular compound, when constructed as the union of two regular tetrahedra. It can be obtained as an augmentation of the regular octahedron, in this construction it has the same topology as the convex Catalan solid, the triakis octahedron, which has much shorter pyramids. It is a facetting of the cube, sharing the vertex arrangement, a compound of two spherical tetrahedra can be constructed, as illustrated. The two tetrahedra of the view of the stellated octahedron are desmic, meaning that each edge of one tetrahedron crosses two opposite edges of the other tetrahedron. One of these two crossings is visible in the octahedron, the other crossing occurs at a point at infinity of the projective space. The same twelve tetrahedron vertices also form the points of Reyes configuration, the stella octangula numbers are figurate numbers that count the number of balls that can be arranged into the shape of a stellated octahedron. They are 0,1,14,51,124,245,426,679,1016,1449,1990, the stellated octahedron appears with several other polyhedra and polyhedral compounds in M. C. Eschers print Stars, and provides the form in Eschers Double Planetoid. Peter R. Cromwell, Polyhedra, Cambridge University Press Polyhedra H. S. M. Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8,3.6 The five regular compounds, pp. 47-50,6.2 Stellating the Platonic solids, pp. 96-104 Weisstein, Weisstein, Eric W. Compound of two tetrahedra
21.
Hanafi
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The Hanafi school is one of the four religious Sunni Islamic schools of jurisprudence. It is named after the scholar Abū Ḥanīfa an-Nu‘man ibn Thābit, the other major schools of Sharia in Sunni Islam are Maliki, Shafii and Hanbali. Hanafi is the fiqh with the largest number of followers among Sunni Muslims and it is predominant in the countries that were once part of the historic Ottoman Empire, Mughal Empire and Sultanates of Turkic rulers in the Indian subcontinent, northwest China and Central Asia. Thus, the Hanafi school came to be known as the Kufan or Iraqi school in earlier times. Ali and Abdullah, son of Masud formed much of the base of the school, as well as other such as Muhammad al-Baqir, Jafar al-Sadiq. Many jurists and historians had lived in Kufa including one of Abu Hanifas main teachers, in the early history of Islam, Hanafi doctrine was not fully compiled. The fiqh was fully compiled and documented in the 11th century, the Abbasids patronized the Hanafi school from the 10th century onwards. The Seljuk Turkish dynasties of 11th and 12th centuries, followed by Ottomans adopted Hanafi fiqh, the Turkic expansion spread Hanafi fiqh through Central Asia and into South Asia, with the establishment of Seljuk Empire, Timurid dynasty, Khanates and Delhi Sultanate. Nurit Tsafrir, The History of an Islamic School of Law, the Second Formation of Islamic Law, The Ḥanafī School in the Early Modern Ottoman Empire
22.
Judaism
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Judaism encompasses the religion, philosophy, culture and way of life of the Jewish people. Judaism is an ancient monotheistic Abrahamic religion, with the Torah as its text, and supplemental oral tradition represented by later texts such as the Midrash. Judaism is considered by religious Jews to be the expression of the relationship that God established with the Children of Israel. With between 14.5 and 17.4 million adherents worldwide, Judaism is the tenth-largest religion in the world, Judaism includes a wide corpus of texts, practices, theological positions, and forms of organization. Modern branches of Judaism such as Humanistic Judaism may be nontheistic, today, the largest Jewish religious movements are Orthodox Judaism, Conservative Judaism and Reform Judaism. Major sources of difference between groups are their approaches to Jewish law, the authority of the Rabbinic tradition. Orthodox Judaism maintains that the Torah and Jewish law are divine in origin, eternal and unalterable, Conservative and Reform Judaism are more liberal, with Conservative Judaism generally promoting a more traditional interpretation of Judaisms requirements than Reform Judaism. A typical Reform position is that Jewish law should be viewed as a set of guidelines rather than as a set of restrictions and obligations whose observance is required of all Jews. Historically, special courts enforced Jewish law, today, these still exist. Authority on theological and legal matters is not vested in any one person or organization, the history of Judaism spans more than 3,000 years. Judaism has its roots as a religion in the Middle East during the Bronze Age. Judaism is considered one of the oldest monotheistic religions, the Hebrews and Israelites were already referred to as Jews in later books of the Tanakh such as the Book of Esther, with the term Jews replacing the title Children of Israel. Judaisms texts, traditions and values strongly influenced later Abrahamic religions, including Christianity, Islam, many aspects of Judaism have also directly or indirectly influenced secular Western ethics and civil law. Jews are a group and include those born Jewish and converts to Judaism. In 2015, the world Jewish population was estimated at about 14.3 million, Judaism thus begins with ethical monotheism, the belief that God is one and is concerned with the actions of humankind. According to the Tanakh, God promised Abraham to make of his offspring a great nation, many generations later, he commanded the nation of Israel to love and worship only one God, that is, the Jewish nation is to reciprocate Gods concern for the world. He also commanded the Jewish people to one another, that is. These commandments are but two of a corpus of commandments and laws that constitute this covenant, which is the substance of Judaism
23.
Occult
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The occult is knowledge of the hidden. In common English usage, occult refers to knowledge of the paranormal, as opposed to knowledge of the measurable, the terms esoteric and arcane can also be used to describe the occult, in addition to their meanings unrelated to the supernatural. Occultism is the study of practices, including magic, alchemy, extra-sensory perception, astrology, spiritualism, religion. Alchemy was common among important seventeenth-century scientists, such as Isaac Newton, Newton was even accused of introducing occult agencies into natural science when he postulated gravity as a force capable of acting over vast distances. By the eighteenth century these unorthodox religious and philosophical concerns were well-defined as occult, inasmuch as they lay on the outermost fringe of accepted forms of knowledge and they were, however, preserved by antiquarians and mystics. Occult science is the research into or formulation of occult concepts in a manner that resembles the way natural science researches or describes phenomena. In his 1871 book Primitive Culture, the anthropologist Edward Tylor used the term occult science as a synonym for magic, Occult qualities are properties that have no known rational explanation, in the Middle Ages, for example, magnetism was considered an occult quality. Newtons contemporaries severely criticized his theory that gravity was effected through action at a distance, some religions and sects enthusiastically embrace occultism as an integral esoteric aspect of mystical religious experience. This attitude is common within Wicca and many other modern pagan religions, some other religious denominations disapprove of occultism in most or all forms. They may view the occult as being anything supernatural or paranormal which is not achieved by or through God, monistic in contrast to Christian dualistic beliefs of a separation between body and spirit, Gnostic i. e. Walker, Benjamin. Encyclopedia of the Occult, the Esoteric and the Supernatural, harold W. Percival, Joined the Theosophical Society in 1892. Blavatsky, Occultism versus the Occult Arts, Lucifer, May 1888 Bardon, true to His Ways, Purity & Safety in Christian Spiritual Practice, ISBN 1-932124-61-6. ISBN 1-57863-150-5 Forshaw, Peter, The Occult Middle Ages, in Christopher Partridge, The Occult World, London, Routledge,2014 Gettings, Fred, Vision of the Occult, ISBN 0-7126-1438-9 Kontou, Tatiana – Willburn, Sarah. The Ashgate Research Companion to Nineteenth-Century Spiritualism and the Occult, ISBN 978-0-7546-6912-8 Martin, W. Rische, J. Rische, K. & VanGordon, K. W. B. Eerdmans Publishing Co.201 p. N. B, the scope of this study also embraces the occult. ISBN 0-8028-0262-1 Partridge, Christopher, The Occult World, London, the Tree of Life, An Illustrated Study in Magic. Newton, Isaac, Observations upon the Prophecies of Daniel, Observations upon the Prophecies of Daniel, and the Apocalypse of St. John by Sir Isaac Newton Rogers, L. W. Hints to Young Students of Occultism. Albany, NY, The Theosophical Book Company, joseph H. Peterson, Twilit Grotto, Archives of Western Esoterica Occult Science and Philosophy of the Renaissance
24.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
25.
Root system
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In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory. Finally, root systems are important for their own sake, as in graph theory. As a first example, consider the six vectors in 2-dimensional Euclidean space, R2, as shown in the image at the right and these vectors span the whole space. If you consider the line perpendicular to any root, say β, then the reflection of R2 in that line sends any other root, say α, moreover, the root to which it is sent equals α + nβ, where n is an integer. These six vectors satisfy the definition, and therefore they form a root system. Let V be a finite-dimensional Euclidean vector space, with the standard Euclidean inner product denoted by, in this context, a root system that also satisfies the integrality condition is known as a crystallographic root system. Other authors omit condition 2, then they call root systems satisfying condition 2 reduced, in this article, all root systems are assumed to be reduced and crystallographic. In view of property 3, the integrality condition is equivalent to stating that β, Note that the operator ⟨ ⋅, ⋅ ⟩, Φ × Φ → Z defined by property 4 is not an inner product. It is not necessarily symmetric and is only in the first argument. The rank of a root system Φ is the dimension of V, two root systems may be combined by regarding the Euclidean spaces they span as mutually orthogonal subspaces of a common Euclidean space. A root system which does not arise from such a combination, such as the systems A2, B2, and G2 pictured to the right, is said to be irreducible. Two root systems and are called if there is an invertible linear transformation E1 → E2 which sends Φ1 to Φ2 such that for each pair of roots. The group of isometries of V generated by reflections through hyperplanes associated to the roots of Φ is called the Weyl group of Φ, as it acts faithfully on the finite set Φ, the Weyl group is always finite. The root lattice of a root system Φ is the Z-submodule of V generated by Φ, there is only one root system of rank 1, consisting of two nonzero vectors. This root system is called A1, in rank 2 there are four possibilities, corresponding to σ α = β + n α, where n =0,1,2,3. Whenever Φ is a system in V, and U is a subspace of V spanned by Ψ = Φ ∩ U. Thus, the exhaustive list of four systems of rank 2 shows the geometric possibilities for any two roots chosen from a root system of arbitrary rank
26.
Lie group
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In mathematics, a Lie group /ˈliː/ is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie, who laid the foundations of the theory of transformation groups. The term groupes de Lie first appeared in French in 1893 in the thesis of Lie’s student Arthur Tresse, an extension of Galois theory to the case of continuous symmetry groups was one of Lies principal motivations. Lie groups are smooth manifolds and as such can be studied using differential calculus. Lie groups play an role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various geometries by specifying an appropriate transformation group that leaves certain geometric properties invariant and this idea later led to the notion of a G-structure, where G is a Lie group of local symmetries of a manifold. On a global level, whenever a Lie group acts on an object, such as a Riemannian or a symplectic manifold. The presence of continuous symmetries expressed via a Lie group action on a manifold places strong constraints on its geometry, Linear actions of Lie groups are especially important, and are studied in representation theory. This insight opened new possibilities in pure algebra, by providing a uniform construction for most finite simple groups, a real Lie group is a group that is also a finite-dimensional real smooth manifold, in which the group operations of multiplication and inversion are smooth maps. Smoothness of the group multiplication μ, G × G → G μ = x y means that μ is a mapping of the product manifold G×G into G. These two requirements can be combined to the requirement that the mapping ↦ x −1 y be a smooth mapping of the product manifold into G. The 2×2 real invertible matrices form a group under multiplication, denoted by GL or by GL2 and this is a four-dimensional noncompact real Lie group. This group is disconnected, it has two connected components corresponding to the positive and negative values of the determinant, the rotation matrices form a subgroup of GL, denoted by SO. It is a Lie group in its own right, specifically, using the rotation angle φ as a parameter, this group can be parametrized as follows, SO =. Addition of the angles corresponds to multiplication of the elements of SO, thus both multiplication and inversion are differentiable maps. The orthogonal group also forms an example of a Lie group. All of the examples of Lie groups fall within the class of classical groups. Hilberts fifth problem asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples, if the underlying manifold is allowed to be infinite-dimensional, then one arrives at the notion of an infinite-dimensional Lie group
27.
G2 (mathematics)
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In mathematics, G2 is the name of three simple Lie groups, their Lie algebras g 2, as well as some algebraic groups. They are the smallest of the five exceptional simple Lie groups, G2 has rank 2 and dimension 14. It has two representations, with dimension 7 and 14. The Lie algebra g 2, being the smallest exceptional simple Lie algebra, was the first of these to be discovered in the attempt to classify simple Lie algebras. On May 23,1887, Wilhelm Killing wrote a letter to Friedrich Engel saying that he had found a 14-dimensional simple Lie algebra, in the same year, in the same journal, Engel noticed the same thing. Later it was discovered that the 2-dimensional distribution is related to a ball rolling on another ball. The space of configurations of the ball is 5-dimensional, with a 2-dimensional distribution that describes motions of the ball where it rolls without slipping or twisting. In 1900, Engel discovered that a generic antisymmetric trilinear form on a 7-dimensional complex vector space is preserved by a group isomorphic to the form of G2. In 1908 Cartan mentioned that the group of the octonions is a 14-dimensional simple Lie group. In 1914 he stated that this is the real form of G2. In older books and papers, G2 is sometimes denoted by E2, there are 3 simple real Lie algebras associated with this root system, The underlying real Lie algebra of the complex Lie algebra G2 has dimension 28. It has complex conjugation as an automorphism and is simply connected. The maximal compact subgroup of its associated group is the form of G2. The Lie algebra of the form is 14-dimensional. The associated Lie group has no outer automorphisms, no center, the Lie algebra of the non-compact form has dimension 14. The associated simple Lie group has fundamental group of order 2 and its maximal compact subgroup is SU × SU/. It has a double cover that is simply connected. The Dynkin diagram for G2 is given by and its Cartan matrix is, Although they span a 2-dimensional space, as drawn, it is much more symmetric to consider them as vectors in a 2-dimensional subspace of a three-dimensional space
28.
Cube
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In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube is the only regular hexahedron and is one of the five Platonic solids and it has 6 faces,12 edges, and 8 vertices. The cube is also a square parallelepiped, an equilateral cuboid and it is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. The cube is dual to the octahedron and it has cubical or octahedral symmetry. The cube has four special orthogonal projections, centered, on a vertex, edges, face, the first and third correspond to the A2 and B2 Coxeter planes. The cube can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. In analytic geometry, a surface with center and edge length of 2a is the locus of all points such that max = a. For a cube of length a, As the volume of a cube is the third power of its sides a × a × a, third powers are called cubes, by analogy with squares. A cube has the largest volume among cuboids with a surface area. Also, a cube has the largest volume among cuboids with the same linear size. They were unable to solve this problem, and in 1837 Pierre Wantzel proved it to be impossible because the root of 2 is not a constructible number. The cube has three uniform colorings, named by the colors of the faces around each vertex,111,112,123. The cube has three classes of symmetry, which can be represented by coloring the faces. The highest octahedral symmetry Oh has all the faces the same color, the dihedral symmetry D4h comes from the cube being a prism, with all four sides being the same color. The lowest symmetry D2h is also a symmetry, with sides alternating colors. Each symmetry form has a different Wythoff symbol, a cube has eleven nets, that is, there are eleven ways to flatten a hollow cube by cutting seven edges. To color the cube so that no two adjacent faces have the color, one would need at least three colors
29.
Mandala
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A mandala is a spiritual and ritual symbol in Hinduism and Buddhism, representing the universe. In common use, mandala has become a term for any diagram, chart or geometric pattern that represents the cosmos metaphysically or symbolically. The basic form of most mandalas is a square with four gates containing a circle with a center point, each gate is in the general shape of a T. Mandalas often exhibit radial balance. The term appears in the Rigveda as the name of the sections of the work, a yantra is a two- or three-dimensional geometric composition used in sadhanas, puja or meditative rituals. It is considered to represent the abode of the deity, each yantra is unique and calls the deity into the presence of the practitioner through the elaborate symbolic geometric designs. Yantras are not representations, but are lived, experiential, nondual realities, as Khanna describes, Despite its cosmic meanings a yantra is a reality lived. The Rajamandala was formulated by the Indian author Kautilya in his work on politics and it describes circles of friendly and enemy states surrounding the kings state. In historical, social and political sense, the mandala is also employed to denote traditional Southeast Asian political formations. It was adopted by 20th century Western historians from ancient Indian political discourse as a means of avoiding the term state in the conventional sense, empires such as Bagan, Ayutthaya, Champa, Khmer, Srivijaya and Majapahit are known as mandala in this sense. The mandala can be found in the form of the stupa and in the Atanatiya Sutta in the Digha Nikaya, Mandalas are traditionally found in large amounts in Buddhist Monasteries all over the world. One can also buy Mandalas and Thankas/Pauva in places like Thamel, in the Tibetan branch of Vajrayana Buddhism, mandalas have been developed into sandpainting. They are also a key part of Anuttarayoga Tantra meditation practices, the mandala can be shown to represent in visual form the core essence of the Vajrayana teachings. The mind is a microcosm representing various divine powers at work in the universe, the mandala represents the nature of the Pure Land, Enlightened mind. A mandala can also represent the universe, which is traditionally depicted with Mount Meru as the axis mundi in the center. In the mandala, the circle of fire usually symbolises wisdom. Inside these rings lie the walls of the palace itself, specifically a place populated by deities. One well-known type of mandala is the mandala of the Five Buddhas, such Buddhas are depicted depending on the school of Buddhism, and even the specific purpose of the mandala. A common mandala of this type is that of the Five Wisdom Buddhas, when paired with another mandala depicting the Five Wisdom Kings, this forms the Mandala of the Two Realms
30.
India
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India, officially the Republic of India, is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and it is bounded by the Indian Ocean on the south, the Arabian Sea on the southwest, and the Bay of Bengal on the southeast. It shares land borders with Pakistan to the west, China, Nepal, and Bhutan to the northeast, in the Indian Ocean, India is in the vicinity of Sri Lanka and the Maldives. Indias Andaman and Nicobar Islands share a border with Thailand. The Indian subcontinent was home to the urban Indus Valley Civilisation of the 3rd millennium BCE, in the following millennium, the oldest scriptures associated with Hinduism began to be composed. Social stratification, based on caste, emerged in the first millennium BCE, early political consolidations took place under the Maurya and Gupta empires, the later peninsular Middle Kingdoms influenced cultures as far as southeast Asia. In the medieval era, Judaism, Zoroastrianism, Christianity, and Islam arrived, much of the north fell to the Delhi sultanate, the south was united under the Vijayanagara Empire. The economy expanded in the 17th century in the Mughal empire, in the mid-18th century, the subcontinent came under British East India Company rule, and in the mid-19th under British crown rule. A nationalist movement emerged in the late 19th century, which later, under Mahatma Gandhi, was noted for nonviolent resistance, in 2015, the Indian economy was the worlds seventh largest by nominal GDP and third largest by purchasing power parity. Following market-based economic reforms in 1991, India became one of the major economies and is considered a newly industrialised country. However, it continues to face the challenges of poverty, corruption, malnutrition, a nuclear weapons state and regional power, it has the third largest standing army in the world and ranks sixth in military expenditure among nations. India is a constitutional republic governed under a parliamentary system. It is a pluralistic, multilingual and multi-ethnic society and is home to a diversity of wildlife in a variety of protected habitats. The name India is derived from Indus, which originates from the Old Persian word Hindu, the latter term stems from the Sanskrit word Sindhu, which was the historical local appellation for the Indus River. The ancient Greeks referred to the Indians as Indoi, which translates as The people of the Indus, the geographical term Bharat, which is recognised by the Constitution of India as an official name for the country, is used by many Indian languages in its variations. Scholars believe it to be named after the Vedic tribe of Bharatas in the second millennium B. C. E and it is also traditionally associated with the rule of the legendary emperor Bharata. Gaṇarājya is the Sanskrit/Hindi term for republic dating back to the ancient times, hindustan is a Persian name for India dating back to the 3rd century B. C. E. It was introduced into India by the Mughals and widely used since then and its meaning varied, referring to a region that encompassed northern India and Pakistan or India in its entirety
31.
Hindu
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Hindu refers to any person who regards themselves as culturally, ethnically, or religiously adhering to aspects of Hinduism. It has historically used as a geographical, cultural, or religious identifier for people indigenous to South Asia. The historical meaning of the term Hindu has evolved with time, by the 16th century, the term began to refer to residents of India who were not Turks or Muslims. The historical development of Hindu self-identity within the Indian population, in a religious or cultural sense, is unclear, competing theories state that Hindu identity developed in the British colonial era, or that it developed post-8th century CE after the Islamic invasion and medieval Hindu-Muslim wars. A sense of Hindu identity and the term Hindu appears in texts dated between the 13th and 18th century in Sanskrit and regional languages. The 14th- and 18th-century Indian poets such as Vidyapati, Kabir and Eknath used the phrase Hindu dharma, the Christian friar Sebastiao Manrique used the term Hindu in religious context in 1649. In the 18th century, the European merchants and colonists began to refer to the followers of Indian religions collectively as Hindus, in contrast to Mohamedans for Mughals, scholars state that the custom of distinguishing between Hindus, Buddhists, Jains and Sikhs is a modern phenomenon. Hindoo is a spelling variant, whose use today may be considered derogatory. At more than 1.03 billion, Hindus are the third largest group after Christians. The vast majority of Hindus, approximately 966 million, live in India, according to Indias 2011 census. After India, the next 9 countries with the largest Hindu populations are, in decreasing order, Nepal, Bangladesh, Indonesia, Pakistan, Sri Lanka, United States, Malaysia, United Kingdom and Myanmar. These together accounted for 99% of the worlds Hindu population, the word Hindu is derived from the Indo-Aryan and Sanskrit word Sindhu, which means a large body of water, covering river, ocean. It was used as the name of the Indus river and also referred to its tributaries, the Punjab region, called Sapta Sindhava in the Vedas, is called Hapta Hindu in Zend Avesta. The 6th-century BCE inscription of Darius I mentions the province of Hidush, the people of India were referred to as Hinduvān and hindavī was used as the adjective for Indian in the 8th century text Chachnama. The term Hindu in these ancient records is an ethno-geographical term, the Arabic equivalent Al-Hind likewise referred to the country of India. Among the earliest known records of Hindu with connotations of religion may be in the 7th-century CE Chinese text Record of the Western Regions by the Buddhist scholar Xuanzang, Xuanzang uses the transliterated term In-tu whose connotation overflows in the religious according to Arvind Sharma. The Hindu community occurs as the amorphous Other of the Muslim community in the court chronicles, wilfred Cantwell Smith notes that Hindu retained its geographical reference initially, Indian, indigenous, local, virtually native. Slowly, the Indian groups themselves started using the term, differentiating themselves, the poet Vidyapatis poem Kirtilata contrasts the cultures of Hindus and Turks in a city and concludes The Hindus and the Turks live close together, Each makes fun of the others religion
32.
Nara-Narayana
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Nara-Narayana is a Hindu deity pair. Nara-Narayana is the avatar of the God Vishnu on earth. In the concept of Nara-Narayana, the human soul Nara is the companion of the Divine Narayana. The Hindu epic Mahabharata identifies the God Krishna with Narayana and Arjuna - the chief hero of the epic - with Nara, the legend of Nara-Narayana is also told in the scripture Bhagavata Purana. Hindus believe that the pair dwells at Badrinath, where their most important temple stands, the Nara-Narayana pair is frequently worshipped in temples of the Swaminarayan Faith. Nara means human, and Narayana refers to the name of the deity, in epic poetry, they are the sons of Dharma by Murti or Ahimsa and emanations of Vishnu, Arjuna being identified with Nara, and Krishna with Narayana. - Mahabharata, Harivamsa and Purana. Narayana is Vishnu while Nara is Shesha Nara-Narayana are depicted jointly or separately in images, when depicted separately, Nara is portrayed with two hands and wearing deer skin while Narayana is shown on the right in the usual form of Vishnu. Sometimes, both of them are depicted identical to each other and they are depicted four-armed holding a mace, a discus, a conch and a lotus, resembling Vishnu. Arjuna and Krishna are often referred to as Nara-Narayana in the Mahabharata and are considered part incarnations of Nara and Narayana respectively, in a previous life, the duo were born as the sages Nara and Narayana, and who performed great penances at the holy spot of Badrinath. Nara and Narayana were the Fourth Avatar of Lord Vishnu, the twins were sons of Dharma, the son of Brahma and his wife Murti or Ahimsa. They live at Badrika performing severe austerities and meditation for the welfare of the world and these two inseparable sages took avatars on earth for the welfare of mankind. Legend has it that once Lord Shiva tried to bring the fame of Nara and Narayana before the entire world. To do that, he hurled his own potent weapon Paashupathastra at the meditating rishis, the power of their meditation was so intense that the astra lost its power before them. Lord Shiva stated that this happened since the duo were jnanis of the first order constantly in the state of Nirvikalpa Samadhi, the Bhagavata Purana tells the story of the birth of Urvashi from the sages Nara-Narayana. Once, sages Nara-Narayana were meditating in the shrine of Badrinath situated in the Himalayas. Their penances and austerities alarmed the gods, so Indra, the King of Devas, sent Kamadeva, Vasanta and apsaras to inspire them with passion, the sage Narayana took a flower and placed it on his thigh. Immediately there sprung from it a beautiful nymph whose charms far excelled those of the celestial nymphs, Narayana sent this nymph to Indra with them, and from her having been produced from the thigh of the sage, she was called Urvashi. In Badrinath Temples sanctorium, to the far side of the stone image of Badri-Vishala, are the images of Nara. Also, the Nara and Narayana peaks tower over Badrinath, in Vanaparvan, Krishna says to Arjuna, O invincible one, you are Nara and I am Hari Narayana, and we, the sages Nara-Narayana, have come to this world at the proper time
33.
Meditation
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The word meditation carries different meanings in different contexts. Meditation has been practiced since antiquity as a component of religious traditions. Meditation often involves an internal effort to self-regulate the mind in some way, Meditation is often used to clear the mind and ease many health concerns, such as high blood pressure, depression, and anxiety. It may be sitting, or in an active way—for instance. Prayer beads or other objects are commonly used during meditation in order to keep track of or remind the practitioner about some aspect of that training. The term meditation can refer to the state itself, as well as to practices or techniques employed to cultivate the state, Meditation may also involve repeating a mantra and closing the eyes. The mantra is chosen based on its suitability to the individual meditator, Meditation has a calming effect and directs awareness inward until pure awareness is achieved, described as being awake inside without being aware of anything except awareness itself. In brief, there are dozens of styles of meditation practice. The English meditation is derived from the Latin meditatio, from a verb meditari, meaning to think, contemplate, devise, in the Old Testament, hāgâ means to sigh or murmur, and also, to meditate. When the Hebrew Bible was translated into Greek, hāgâ became the Greek melete, the Latin Bible then translated hāgâ/melete into meditatio. The use of the term meditatio as part of a formal, the term meditation in English may also refer to practices from Islamic Sufism, or other traditions such as Jewish Kabbalah and Christian Hesychasm. An edited book about meditation published in 2003, for example, included contributions by authors describing Hindu, Buddhist, Taoist, Jewish, Christian. Christian, Judaic, and Islamic forms of meditation are typically devotional, scriptural or thematic, the history of meditation is intimately bound up with the religious context within which it was practiced. Some of the earliest references to meditation are found in the Hindu Vedas of Nepal, wilson translates the most famous Vedic mantra Gayatri thus, We meditate on that desirable light of the divine Savitri, who influences our pious rites. Around the 6th to 5th centuries BC, other forms of meditation developed via Confucianism and Taoism in China as well as Hinduism, Jainism, and early Buddhism in Nepal and India. In the west, by 20 BC Philo of Alexandria had written on some form of exercises involving attention and concentration. The Pāli Canon, which dates to 1st century BC considers Indian Buddhist meditation as a step towards liberation, by the time Buddhism was spreading in China, the Vimalakirti Sutra which dates to 100 AD included a number of passages on meditation, clearly pointing to Zen. The Silk Road transmission of Buddhism introduced meditation to other Asian countries, returning from China around 1227, Dōgen wrote the instructions for zazen
34.
Moksha
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Moksha, also called vimoksha, vimukti and mukti, is a term in Hinduism and Hindu philosophy which refers to various forms of emancipation, liberation, and release. In its soteriological and eschatological senses, it refers to freedom from saṃsāra, in its epistemological and psychological senses, moksha refers to freedom from ignorance, self-realization and self-knowledge. In Hindu traditions, moksha is a concept and included as one of the four aspects and goals of human life. Together, these four aims of life are called Puruṣārtha in Hinduism, the concept of moksha is found in Jainism, Buddhism and Hinduism. In some schools of Indian religions, moksha is considered equivalent to and used interchangeably with terms such as vimoksha, vimukti, kaivalya, apavarga, mukti, nihsreyasa. However, terms such as moksha and nirvana differ and mean different states between various schools of Hinduism, Buddhism and Jainism, the term nirvana is more common in Buddhism, while moksha is more prevalent in Hinduism. Moksha is derived from the root Sanskrit, मुच्, muc, in Vedas and early Upanishads, the word Sanskrit, मुच्यते, mucyate appears, which means to be set free or release - such as of a horse from its harness. The definition and meaning of moksha varies between schools of Indian religions. Moksha means freedom, liberation, from what and how is where the schools differ, Moksha is also a concept that means liberation from rebirth or saṃsāra. This liberation can be attained while one is on earth, or eschatologically, some Indian traditions have emphasized liberation on concrete, ethical action within the world. This liberation is a transformation that permits one to see the truth. For example, Vivekachudamani - an ancient book on moksha, explains one of many steps on the path to moksha, as. Samsara originated with religious movements in the first millennium BCE and these movements such as Buddhism, Jainism and new schools within Hinduism, saw human life as bondage to a repeated process of rebirth. This bondage to repeated rebirth and life, each subject to injury, disease. By release from this cycle, the involved in this cycle also ended. This release was called moksha, nirvana, kaivalya, mukti, in earliest Vedic literature, heaven and hell sufficed soteriological curiosities. The rebirth idea ultimately flowered into the ideas of saṃsāra, or transmigration - where one’s balance sheet of karma determined one’s rebirth, along with this idea of saṃsāra, the ancient scholars developed the concept of moksha, as a state that released a person from the saṃsāra cycle. Moksha release in eschatological sense in these ancient literature of Hinduism, suggests van Buitenen, comes from self-knowledge, the meaning of moksha in epistemological and psychological sense has been variously explained by scholars
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Nirvana
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Nirvāṇa literally means blown out, as in an oil lamp. The term nirvana is most commonly associated with Buddhism, and represents its ultimate state of soteriological release, in Indian religions, nirvana is synonymous with moksha and mukti. All Indian religions assert it to be a state of perfect quietude, freedom, highest happiness along with it being the liberation from samsara, however, Buddhist and non-Buddhist traditions describe these terms for liberation differently. In the Buddhist context, nirvana refers to realization of non-self and emptiness, in Hindu philosophy, it is the union of or the realization of the identity of Atman with Brahman, depending on the Hindu tradition. In Jainism, it is also the goal, it represents the release of a soul from karmic bondage. The word nirvāṇa, states Steven Collins, is from the verbal root √vā blow in the form of past participle vāna blown, hence the original meaning of the word is blown out, extinguished. Sandhi changes the spelling, the v of vāna causes nis to become nir, the term nirvana in the soteriological sense of blown out, extinguished state of liberation does not appear in the Vedas nor in the pre-Buddhist Upanishads. According to Collins, the Buddhists seem to have been the first to call it nirvana, nirvāṇa is a term found in the texts of all major Indian religions – Buddhism, Hinduism, Jainism and Sikhism. It refers to the peace of mind that is acquired with moksha, liberation from samsara, or release from a state of suffering. The hope for life after death started with notions of going to the worlds of the Fathers or Ancestors and/or the world of the Gods or Heaven. The earliest layers of Vedic text incorporate the concept of life, followed by an afterlife in heaven, however, the ancient Vedic Rishis challenged this idea of afterlife as simplistic, because people do not live an equally moral or immoral life. Between generally virtuous lives, some are more virtuous, while evil too has degrees, the Vedic thinkers introduced the idea of an afterlife in heaven or hell in proportion to ones merit, and when this runs out, one returns and is reborn. The idea of rebirth following running out of merit appears in Buddhist texts as well, the Samsara, the life after death, and what impacts rebirth came to be seen as dependent on karma. The liberation from Saṃsāra developed as a goal and soteriological value in the Indian culture. This basic scheme underlies Hinduism, Jainism and Buddhism, where the aim is the timeless state of moksa, or, as the Buddhists first seem to have called it. Although the term occurs in the literatures of a number of ancient Indian traditions and it was later adopted by other Indian religions, but with different meanings and description, such as in the Hindu text Bhagavad Gita of the Mahabharata. Nirvana literally means blowing out or quenching and it is the most used as well as the earliest term to describe the soteriological goal in Buddhism, release from the cycle of rebirth. Nirvana is part of the Third Truth on cessation of dukkha in the Four Noble Truths doctrine of Buddhism and it is the goal of the Noble Eightfold Path
36.
Astrology
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Astrology is the study of the movements and relative positions of celestial objects as a means for divining information about human affairs and terrestrial events. Throughout most of its history astrology was considered a tradition and was common in academic circles, often in close relation with astronomy, alchemy, meteorology. It was present in political circles, and is mentioned in works of literature, from Dante Alighieri and Geoffrey Chaucer to William Shakespeare, Lope de Vega. Astrology thus lost its academic and theoretical standing, and common belief in it has largely declined, Astrology is now recognized to be pseudoscience. The word astrology comes from the early Latin word astrologia, which derives from the Greek ἀστρολογία—from ἄστρον astron, astrologia later passed into meaning star-divination with astronomia used for the scientific term. Many cultures have attached importance to astronomical events, and the Indians, Chinese, the majority of professional astrologers rely on such systems. Astrology has been dated to at least the 2nd millennium BCE, with roots in systems used to predict seasonal shifts. A form of astrology was practised in the first dynasty of Mesopotamia, Chinese astrology was elaborated in the Zhou dynasty. Hellenistic astrology after 332 BCE mixed Babylonian astrology with Egyptian Decanic astrology in Alexandria, Alexander the Greats conquest of Asia allowed astrology to spread to Ancient Greece and Rome. In Rome, astrology was associated with Chaldean wisdom, after the conquest of Alexandria in the 7th century, astrology was taken up by Islamic scholars, and Hellenistic texts were translated into Arabic and Persian. In the 12th century, Arabic texts were imported to Europe, major astronomers including Tycho Brahe, Johannes Kepler and Galileo practised as court astrologers. Astrological references appear in literature in the works of such as Dante Alighieri and Geoffrey Chaucer. Throughout most of its history, astrology was considered a scholarly tradition and it was accepted in political and academic contexts, and was connected with other studies, such as astronomy, alchemy, meteorology, and medicine. At the end of the 17th century, new concepts in astronomy. Astrology thus lost its academic and theoretical standing, and common belief in astrology has largely declined, Astrology, in its broadest sense, is the search for meaning in the sky. This was a first step towards recording the Moons influence upon tides and rivers, by the 3rd millennium BCE, civilisations had sophisticated awareness of celestial cycles, and may have oriented temples in alignment with heliacal risings of the stars. Scattered evidence suggests that the oldest known references are copies of texts made in the ancient world. The Venus tablet of Ammisaduqa thought to be compiled in Babylon around 1700 BCE, a scroll documenting an early use of electional astrology is doubtfully ascribed to the reign of the Sumerian ruler Gudea of Lagash
37.
Anointing
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Anointing is the ritual act of pouring aromatic oil over a persons head or entire body. By extension, the term is applied to related acts of sprinkling, dousing, or smearing a person or object with any perfumed oil, milk, butter. Scented oils are used as perfumes and sharing them is an act of hospitality, in present usage, anointing is typically used for ceremonial blessings such as the coronation of European monarchs. This continues an earlier Hebrew practice most famously observed in the anointings of Aaron as high priest, the concept is important to the figures of the Messiah and the Christ who appear prominently in Jewish and Christian theology and eschatology. Anointing—particularly the anointing of the sick—may also be known as unction, the present verb derives from the now obsolete adjective anoint, equivalent to anointed. The adjective is first attested in 1303, derived from Old French enoint, the past participle of enoindre, from Latin inungere and it is thus cognate with unction. The oil used in a ceremonial anointment may be called chrism, several related words such as chrismation and chrismarium derive from the same root. Anointing served and serves three purposes, it is regarded as a means of health and comfort, as a token of honor. Used in conjunction with bathing, anointment with oil closes pores and it was regarded as counteracting the influence of the sun, reducing sweating. Aromatic oils naturally masked body and other odors, and other forms of fat could be combined with perfumes. Applications of oils and fats are used as traditional medicines. The Bible records olive oil being applied to the sick and poured into wounds and it is still used in traditional Indian medicine to remove illness, bad luck, and demonic possession. For sanitary and religious reasons, the bodies of the dead are sometimes anointed, in medieval and early modern Christianity, the practice was particularly associated with protection against vampires and ghouls who might otherwise take possession of the corpse. Anointing guests with oil as a mark of hospitality and token of honor is recorded in Egypt, Greece and it was a common custom among the ancient Hebrews and continued among the Arabs into the 20th century. For about 3,000 years, Persian Zoroastrians honor their guests with rose extract while holding a mirror in front of their guests face, the guests hold their palms out, collect the rose water, and then spread the perfumed liquid upon their faces and sometimes heads. The words of rooj kori aka might be said as well, east African Arabs traditionally anointed themselves with lions fat to gain courage and provoke fear in other animals. Australian Aborigines would rub themselves with a human victims caul fat to gain his powers, in religions like Christianity where animal sacrifice is no longer practiced, it is common to consecrate the oil in a special ceremony. The most famous example of this is on the throne of Tutankhamun, anointment of the corpse with sweet-smelling oils was an important part of mummification
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Papyrus
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The word papyrus /pəˈpaɪrəs/ refers to a thick precursor to modern paper made from the pith of the papyrus plant, Cyperus papyrus. Papyrus can also refer to a document written on sheets of papyrus joined together side by side and rolled up into a scroll, the plural for such documents is papyri. Papyrus is first known to have used in ancient Egypt. It was also used throughout the Mediterranean region and in Kingdom of Kush, the Ancient Egyptians used papyrus as a writing material, as well as employing it commonly in the construction of other artifacts such as reed boats, mats, rope, sandals, and baskets. Papyrus was first manufactured in Egypt as far back as the fourth millennium BCE, the earliest archaeological evidence of papyrus was excavated in 2012 and 2013 at Wadi al-Jarf, an ancient Egyptian harbor located on the Red Sea coast. The papyrus rolls describe the last years of building the Great Pyramid of Giza, in the first centuries BCE and CE, papyrus scrolls gained a rival as a writing surface in the form of parchment, which was prepared from animal skins. Sheets of parchment were folded to form quires from which book-form codices were fashioned, early Christian writers soon adopted the codex form, and in the Græco-Roman world, it became common to cut sheets from papyrus rolls to form codices. Codices were an improvement on the scroll, as the papyrus was not pliable enough to fold without cracking. Papyrus had the advantage of being cheap and easy to produce. Unless the papyrus was of quality, the writing surface was irregular. Its last appearance in the Merovingian chancery is with a document of 692, the latest certain dates for the use of papyrus are 1057 for a papal decree, under Pope Victor II, and 1087 for an Arabic document. Its use in Egypt continued until it was replaced by more inexpensive paper introduced by Arabs who originally learned of it from the Chinese, by the 12th century, parchment and paper were in use in the Byzantine Empire, but papyrus was still an option. Papyrus was made in several qualities and prices, pliny the Elder and Isidore of Seville described six variations of papyrus which were sold in the Roman market of the day. These were graded by quality based on how fine, firm, white, grades ranged from the superfine Augustan, which was produced in sheets of 13 digits wide, to the least expensive and most coarse, measuring six digits wide. Materials deemed unusable for writing or less than six digits were considered commercial quality and were pasted edge to edge to be used only for wrapping, until the middle of the 19th century, only some isolated documents written on papyrus were known. They did not contain literary works, the first modern discovery of papyri rolls was made at Herculaneum in 1752. Until then, the papyri known had been a few surviving from medieval times. The English word papyrus derives, via Latin, from Greek πάπυρος, Greek has a second word for it, βύβλος
39.
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
40.
God
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In monotheism, God is conceived of as the Supreme Being and principal object of faith. The concept of God as described by most theologians includes the attributes of omniscience, omnipotence, omnipresence, divine simplicity, many theologians also describe God as being omnibenevolent and all loving. Furthermore, some religions attribute only a purely grammatical gender to God, incorporeity and corporeity of God are related to conceptions of transcendence and immanence of God, with positions of synthesis such as the immanent transcendence of Chinese theology. God has been conceived as personal or impersonal. In theism, God is the creator and sustainer of the universe, while in deism, God is the creator, in pantheism, God is the universe itself. In atheism, God is not believed to exist, while God is deemed unknown or unknowable within the context of agnosticism, God has also been conceived as the source of all moral obligation, and the greatest conceivable existent. Many notable philosophers have developed arguments for and against the existence of God, there are many names for God, and different names are attached to different cultural ideas about Gods identity and attributes. In the ancient Egyptian era of Atenism, possibly the earliest recorded monotheistic religion, this deity was called Aten, premised on being the one true Supreme Being and creator of the universe. In the Hebrew Bible and Judaism, He Who Is, I Am that I Am, in the Christian doctrine of the Trinity, God, consubstantial in three persons, is called the Father, the Son, and the Holy Spirit. In Judaism, it is common to refer to God by the titular names Elohim or Adonai, in Islam, the name Allah is used, while Muslims also have a multitude of titular names for God. In Hinduism, Brahman is often considered a concept of God. In Chinese religion, God is conceived as the progenitor of the universe, intrinsic to it, other religions have names for God, for instance, Baha in the Baháí Faith, Waheguru in Sikhism, and Ahura Mazda in Zoroastrianism. The earliest written form of the Germanic word God comes from the 6th-century Christian Codex Argenteus, the English word itself is derived from the Proto-Germanic * ǥuđan. The reconstructed Proto-Indo-European form * ǵhu-tó-m was likely based on the root * ǵhau-, in the English language, the capitalized form of God continues to represent a distinction between monotheistic God and gods in polytheism. The same holds for Hebrew El, but in Judaism, God is also given a proper name, in many translations of the Bible, when the word LORD is in all capitals, it signifies that the word represents the tetragrammaton. Allāh is the Arabic term with no plural used by Muslims and Arabic speaking Christians and Jews meaning The God, Ahura Mazda is the name for God used in Zoroastrianism. Mazda, or rather the Avestan stem-form Mazdā-, nominative Mazdå and it is generally taken to be the proper name of the spirit, and like its Sanskrit cognate medhā, means intelligence or wisdom. Both the Avestan and Sanskrit words reflect Proto-Indo-Iranian *mazdhā-, from Proto-Indo-European mn̩sdʰeh1, literally meaning placing ones mind, Waheguru is a term most often used in Sikhism to refer to God
41.
Hinduism
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Hinduism is a religion, or a way of life, found most notably in India and Nepal. Hinduism has been called the oldest religion in the world, and some practitioners and scholars refer to it as Sanātana Dharma, scholars regard Hinduism as a fusion or synthesis of various Indian cultures and traditions, with diverse roots and no founder. This Hindu synthesis started to develop between 500 BCE and 300 CE following the Vedic period, although Hinduism contains a broad range of philosophies, it is linked by shared concepts, recognisable rituals, cosmology, shared textual resources, and pilgrimage to sacred sites. Hindu texts are classified into Shruti and Smriti and these texts discuss theology, philosophy, mythology, Vedic yajna, Yoga, agamic rituals, and temple building, among other topics. Major scriptures include the Vedas and Upanishads, the Bhagavad Gita, prominent themes in Hindu beliefs include the four Puruṣārthas, the proper goals or aims of human life, namely Dharma, Artha, Kama and Moksha, karma, samsara, and the various Yogas. Hindu practices include such as puja and recitations, meditation, family-oriented rites of passage, annual festivals. Some Hindus leave their world and material possessions, then engage in lifelong Sannyasa to achieve Moksha. Hinduism prescribes the eternal duties, such as honesty, refraining from injuring living beings, patience, forbearance, self-restraint, Hinduism is the worlds third largest religion, with over one billion followers or 15% of the global population, known as Hindus. The majority of Hindus reside in India, Nepal, Mauritius, the Caribbean, the word Hindu is derived from the Indo-Aryan/Sanskrit word Sindhu, the Indo-Aryan name for the Indus River in the northwestern part of the Indian subcontinent. The term Hindu in these ancient records is a geographical term, the Arabic term al-Hind referred to the people who live across the River Indus. This Arabic term was taken from the pre-Islamic Persian term Hindū. By the 13th century, Hindustan emerged as an alternative name of India. It was only towards the end of the 18th century that European merchants and colonists began to refer to the followers of Indian religions collectively as Hindus. The term Hinduism, then spelled Hindooism, was introduced into the English language in the 18th-century to denote the religious, philosophical, because of the wide range of traditions and ideas covered by the term Hinduism, arriving at a comprehensive definition is difficult. The religion defies our desire to define and categorize it, Hinduism has been variously defined as a religion, a religious tradition, a set of religious beliefs, and a way of life. From a Western lexical standpoint, Hinduism like other faiths is appropriately referred to as a religion, in India the term dharma is preferred, which is broader than the western term religion. Hindu traditionalists prefer to call it Sanatana Dharma, the study of India and its cultures and religions, and the definition of Hinduism, has been shaped by the interests of colonialism and by Western notions of religion. Since the 1990s, those influences and its outcomes have been the topic of debate among scholars of Hinduism, Hinduism as it is commonly known can be subdivided into a number of major currents
42.
Buddhism
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Buddhism is a religion and dharma that encompasses a variety of traditions, beliefs and spiritual practices largely based on teachings attributed to the Buddha. Buddhism originated in India sometime between the 6th and 4th centuries BCE, from where it spread through much of Asia, two major extant branches of Buddhism are generally recognized by scholars, Theravada and Mahayana. Buddhism is the worlds fourth-largest religion, with over 500 million followers or 7% of the global population, Buddhist schools vary on the exact nature of the path to liberation, the importance and canonicity of various teachings and scriptures, and especially their respective practices. In Theravada the ultimate goal is the attainment of the state of Nirvana, achieved by practicing the Noble Eightfold Path, thus escaping what is seen as a cycle of suffering. Theravada has a following in Sri Lanka and Southeast Asia. Mahayana, which includes the traditions of Pure Land, Zen, Nichiren Buddhism, Shingon, rather than Nirvana, Mahayana instead aspires to Buddhahood via the bodhisattva path, a state wherein one remains in the cycle of rebirth to help other beings reach awakening. Vajrayana, a body of teachings attributed to Indian siddhas, may be viewed as a branch or merely a part of Mahayana. Tibetan Buddhism, which preserves the Vajrayana teachings of eighth century India, is practiced in regions surrounding the Himalayas, Tibetan Buddhism aspires to Buddhahood or rainbow body. Buddhism is an Indian religion attributed to the teachings of Buddha, the details of Buddhas life are mentioned in many early Buddhist texts but are inconsistent, his social background and life details are difficult to prove, the precise dates uncertain. Some hagiographic legends state that his father was a king named Suddhodana, his mother queen Maya, and he was born in Lumbini gardens. Some of the stories about Buddha, his life, his teachings, Buddha was moved by the innate suffering of humanity. He meditated on this alone for a period of time, in various ways including asceticism, on the nature of suffering. He famously sat in meditation under a Ficus religiosa tree now called the Bodhi Tree in the town of Bodh Gaya in Gangetic plains region of South Asia. He reached enlightenment, discovering what Buddhists call the Middle Way, as an enlightened being, he attracted followers and founded a Sangha. Now, as the Buddha, he spent the rest of his teaching the Dharma he had discovered. Dukkha is a concept of Buddhism and part of its Four Noble Truths doctrine. It can be translated as incapable of satisfying, the unsatisfactory nature, the Four Truths express the basic orientation of Buddhism, we crave and cling to impermanent states and things, which is dukkha, incapable of satisfying and painful. This keeps us caught in saṃsāra, the cycle of repeated rebirth, dukkha