1.
Face (geometry)
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In solid geometry, a face is a flat surface that forms part of the boundary of a solid object, a three-dimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, in elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include side of a polyhedron, for example, any of the six squares that bound a cube is a face of the cube. Sometimes face is used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, some other polygons, which are not faces, are also important for polyhedra and tessellations. These include Petrie polygons, vertex figures and facets, 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 F is the number of faces. This equation is known as Eulers polyhedron formula, thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, in higher-dimensional geometry the faces of a polytope are features of all dimensions. A face of dimension k is called a k-face, for example, the polygonal faces of an ordinary polyhedron are 2-faces. In set theory, the set of faces of a polytope includes the polytope itself, for any n-polytope, −1 ≤ k ≤ n. For example, with meaning, the faces of a cube include the empty set, its vertices, edges and squares. Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P, from this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, abstract theory still requires that the set of faces include the polytope itself and the empty set. A cell is an element of a 4-dimensional polytope or 3-dimensional tessellation. Cells are facets for 4-polytopes and 3-honeycombs, examples, In higher-dimensional geometry, the facets of a n-polytope are the -faces of dimension one less than the polytope itself. A polytope is bounded by its facets, for example, The facets of a line segment are its 0-faces or vertices. The facets of a polygon are its 1-faces or edges, the facets of a polyhedron or plane tiling are its 2-faces. The facets of a 4D polytope or 3-honeycomb are its 3-faces, the facets of a 5D polytope or 4-honeycomb are its 4-faces
2.
Trapezoid
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The parallel sides are called the bases of the trapezoid and the other two sides are called the legs or the lateral sides. A scalene trapezoid is a trapezoid with no sides of equal measure, the first recorded use of the Greek word translated trapezoid was by Marinus Proclus in his Commentary on the first book of Euclids Elements. This article uses the term trapezoid in the sense that is current in the United States, in many other languages using a word derived from the Greek for this figure, the form closest to trapezium is used. A right trapezoid has two adjacent right angles, right trapezoids are used in the trapezoidal rule for estimating areas under a curve. An acute trapezoid has two adjacent acute angles on its longer base edge, while an obtuse trapezoid has one acute, an acute trapezoid is also an isosceles trapezoid, if its sides have the same length, and the base angles have the same measure. An obtuse trapezoid with two pairs of sides is a parallelogram. A parallelogram has central 2-fold rotational symmetry, a Saccheri quadrilateral is similar to a trapezoid in the hyperbolic plane, with two adjacent right angles, while it is a rectangle in the Euclidean plane. A Lambert quadrilateral in the plane has 3 right angles. A tangential trapezoid is a trapezoid that has an incircle, there is some disagreement whether parallelograms, which have two pairs of parallel sides, should be regarded as trapezoids. Some define a trapezoid as a quadrilateral having one pair of parallel sides. Others define a trapezoid as a quadrilateral with at least one pair of parallel sides, the latter definition is consistent with its uses in higher mathematics such as calculus. The former definition would make such concepts as the trapezoidal approximation to a definite integral ill-defined and this article uses the inclusive definition and considers parallelograms as special cases of a trapezoid. This is also advocated in the taxonomy of quadrilaterals, under the inclusive definition, all parallelograms are trapezoids. Rectangles have mirror symmetry on mid-edges, rhombuses have mirror symmetry on vertices, while squares have mirror symmetry on both mid-edges and vertices. Four lengths a, c, b, d can constitute the sides of a non-parallelogram trapezoid with a and b parallel only when | d − c | < | b − a | < d + c. The quadrilateral is a parallelogram when d − c = b − a =0, the angle between a side and a diagonal is equal to the angle between the opposite side and the same diagonal. The diagonals cut each other in mutually the same ratio, the diagonals cut the quadrilateral into four triangles of which one opposite pair are similar. The diagonals cut the quadrilateral into four triangles of which one pair have equal areas
3.
Polygon
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In elementary geometry, a polygon /ˈpɒlɪɡɒn/ is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit. These segments are called its edges or sides, and the points where two edges meet are the vertices or corners. The interior of the polygon is called its body. An n-gon is a polygon with n sides, for example, a polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The basic geometrical notion of a polygon has been adapted in various ways to suit particular purposes, mathematicians are often concerned only with the bounding closed polygonal chain and with simple polygons which do not self-intersect, and they often define a polygon accordingly. A polygonal boundary may be allowed to intersect itself, creating star polygons and these and other generalizations of polygons are described below. The word polygon derives from the Greek adjective πολύς much, many and it has been suggested that γόνυ knee may be the origin of “gon”. Polygons are primarily classified by the number of sides, Polygons may be characterized by their convexity or type of non-convexity, Convex, any line drawn through the polygon meets its boundary exactly twice. As a consequence, all its interior angles are less than 180°, equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. Non-convex, a line may be found which meets its boundary more than twice, equivalently, there exists a line segment between two boundary points that passes outside the polygon. Simple, the boundary of the polygon does not cross itself, there is at least one interior angle greater than 180°. Star-shaped, the interior is visible from at least one point. The polygon must be simple, and may be convex or concave, self-intersecting, the boundary of the polygon crosses itself. Branko Grünbaum calls these coptic, though this term does not seem to be widely used, star polygon, a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped, equiangular, all corner angles are equal. Cyclic, all lie on a single circle, called the circumcircle. Isogonal or vertex-transitive, all lie within the same symmetry orbit. The polygon is cyclic and equiangular
4.
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
5.
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
6.
List of spherical symmetry groups
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Spherical symmetry groups are also called point groups in three dimensions, however, this article is limited to the finite symmetries. There are five fundamental symmetry classes which have triangular fundamental domains, dihedral, cyclic, tetrahedral, octahedral and this article lists the groups by Schoenflies notation, Coxeter notation, orbifold notation, and order. John Conway uses a variation of the Schoenflies notation, based on the groups quaternion algebraic structure, the group order is defined as the subscript, unless the order is doubled for symbols with a plus or minus, ±, prefix, which implies a central inversion. The crystallography groups,32 in total, are a subset with element orders 2,3,4 and 6, there are four involutional groups, no symmetry, reflection symmetry, 2-fold rotational symmetry, and central point symmetry. There are four infinite cyclic symmetry families, with n=2 or higher, there are three infinite dihedral symmetry families, with n as 2 or higher. There are three types of symmetry, tetrahedral symmetry, octahedral symmetry, and icosahedral symmetry, named after the triangle-faced regular polyhedra with these symmetries. Crystallographic point group Triangle group List of planar symmetry groups Point groups in two dimensions Peter R. Cromwell, Polyhedra, Appendix I Sands, Donald E, mineola, New York, Dover Publications, Inc. p.165. Coxeter, edited by F. Arthur Sherk, Peter McMullen, thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M. Coxeter, Regular and Semi Regular Polytopes I, H. S. M, Coxeter, Regular and Semi-Regular Polytopes II, H. S. M. Coxeter, Regular and Semi-Regular Polytopes III, N. W, johnson, Geometries and Transformations, Chapter 11, Finite symmetry groups Finite spherical symmetry groups Weisstein, Eric W. Schoenflies symbol. Weisstein, Eric W. Crystallographic point groups, simplest Canonical Polyhedra of Each Symmetry Type, by David I
7.
Symmetry group
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In abstract algebra, the symmetry group of an object is the group of all transformations under which the object is invariant with composition as the group operation. For a space with a metric, it is a subgroup of the group of the space concerned. If not stated otherwise, this article considers symmetry groups in Euclidean geometry, the objects may be geometric figures, images, and patterns, such as a wallpaper pattern. The definition can be more precise by specifying what is meant by image or pattern. For symmetry of objects, one may also want to take their physical composition into account. The group of isometries of space induces an action on objects in it. The symmetry group is also called full symmetry group in order to emphasize that it includes the orientation-reversing isometries under which the figure is invariant. The subgroup of orientation-preserving isometries that leave the figure invariant is called its symmetry group. The proper symmetry group of an object is equal to its symmetry group if. The proper symmetry group is then a subgroup of the orthogonal group SO. A discrete symmetry group is a group such that for every point of the space the set of images of the point under the isometries in the symmetry group is a discrete set. There are also continuous symmetry groups, which contain rotations of arbitrarily small angles or translations of arbitrarily small distances, the group of all symmetries of a sphere O is an example of this, and in general such continuous symmetry groups are studied as Lie groups. With a categorization of subgroups of the Euclidean group corresponds a categorization of symmetry groups, for example, two 3D figures have mirror symmetry, but with respect to different mirror planes. Two 3D figures have 3-fold rotational symmetry, but with respect to different axes, two 2D patterns have translational symmetry, each in one direction, the two translation vectors have the same length but a different direction. When considering isometry groups, one may restrict oneself to those where for all points the set of images under the isometries is topologically closed. This includes all discrete isometry groups and also involved in continuous symmetries. A figure with this group is non-drawable and up to arbitrarily fine detail homogeneous. The group generated by all translations, this group cannot be the group of a pattern, it would be homogeneous
8.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
9.
Polyhedron
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In geometry, a polyhedron is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- + -hedron, a convex polyhedron is the convex hull of finitely many points, not all on the same plane. Cubes and pyramids are examples of convex polyhedra, a polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions. Convex polyhedra are well-defined, with several equivalent standard definitions, however, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of polyhedron have been given within particular contexts, some more rigorous than others, some of these definitions exclude shapes that have often been counted as polyhedra or include shapes that are often not considered as valid polyhedra. As Branko Grünbaum observed, The Original Sin in the theory of polyhedra goes back to Euclid, the writers failed to define what are the polyhedra. Nevertheless, there is agreement that a polyhedron is a solid or surface that can be described by its vertices, edges, faces. Natural refinements of this definition require the solid to be bounded, to have a connected interior, and possibly also to have a connected boundary. However, the polyhedra defined in this way do not include the self-crossing star polyhedra, their faces may not form simple polygons, definitions based on the idea of a bounding surface rather than a solid are also common. If a planar part of such a surface is not itself a convex polygon, ORourke requires it to be subdivided into smaller convex polygons, cromwell gives a similar definition but without the restriction of three edges per vertex. Again, this type of definition does not encompass the self-crossing polyhedra, however, there exist topological polyhedra that cannot be realized as acoptic polyhedra. One modern approach is based on the theory of abstract polyhedra and these can be defined as partially ordered sets whose elements are the vertices, edges, and faces of a polyhedron. A vertex or edge element is less than an edge or face element when the vertex or edge is part of the edge or face, additionally, one may include a special bottom element of this partial order and a top element representing the whole polyhedron. However, these requirements are relaxed, to instead require only that the sections between elements two levels apart from line segments. Geometric polyhedra, defined in other ways, can be described abstractly in this way, a realization of an abstract polyhedron is generally taken to be a mapping from the vertices of the abstract polyhedron to geometric points, such that the points of each face are coplanar. A geometric polyhedron can then be defined as a realization of an abstract polyhedron, realizations that forgo the requirement of planarity, that impose additional requirements of symmetry, or that map the vertices to higher dimensional spaces have also been considered. Unlike the solid-based and surface-based definitions, this perfectly well for star polyhedra. However, without restrictions, this definition allows degenerate or unfaithful polyhedra
10.
Cone (geometry)
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A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex. A cone is formed by a set of segments, half-lines, or lines connecting a common point. If the enclosed points are included in the base, the cone is a solid object, otherwise it is a two-dimensional object in three-dimensional space. In the case of an object, the boundary formed by these lines or partial lines is called the lateral surface, if the lateral surface is unbounded. In the case of line segments, the cone does not extend beyond the base, while in the case of half-lines, in the case of lines, the cone extends infinitely far in both directions from the apex, in which case it is sometimes called a double cone. Either half of a cone on one side of the apex is called a nappe. The axis of a cone is the line, passing through the apex. If the base is right circular the intersection of a plane with this surface is a conic section, in general, however, the base may be any shape and the apex may lie anywhere. Contrasted with right cones are oblique cones, in which the axis passes through the centre of the base non-perpendicularly, a cone with a polygonal base is called a pyramid. Depending on the context, cone may also mean specifically a convex cone or a projective cone, cones can also be generalized to higher dimensions. The perimeter of the base of a cone is called the directrix, the base radius of a circular cone is the radius of its base, often this is simply called the radius of the cone. The aperture of a circular cone is the maximum angle between two generatrix lines, if the generatrix makes an angle θ to the axis, the aperture is 2θ. A cone with a region including its apex cut off by a plane is called a cone, if the truncation plane is parallel to the cones base. An elliptical cone is a cone with an elliptical base, a generalized cone is the surface created by the set of lines passing through a vertex and every point on a boundary. The slant height of a circular cone is the distance from any point on the circle to the apex of the cone. It is given by r 2 + h 2, where r is the radius of the cirf the cone and this application is primarily useful in determining the slant height of a cone when given other information regarding the radius or height. The volume V of any conic solid is one third of the product of the area of the base A B and the height h V =13 A B h. In modern mathematics, this formula can easily be computed using calculus – it is, up to scaling, the integral ∫ x 2 d x =13 x 3
11.
Pyramid (geometry)
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In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face and it is a conic solid with polygonal base. A pyramid with a base has n +1 vertices, n +1 faces. A right pyramid has its apex directly above the centroid of its base, nonright pyramids are called oblique pyramids. A regular pyramid has a polygon base and is usually implied to be a right pyramid. When unspecified, a pyramid is usually assumed to be a square pyramid. A triangle-based pyramid is often called a tetrahedron. Among oblique pyramids, like acute and obtuse triangles, a pyramid can be called if its apex is above the interior of the base and obtuse if its apex is above the exterior of the base. A right-angled pyramid has its apex above an edge or vertex of the base, in a tetrahedron these qualifiers change based on which face is considered the base. Pyramids are a subclass of the prismatoids, pyramids can be doubled into bipyramids by adding a second offset point on the other side of the base plane. A right pyramid with a base has isosceles triangle sides, with symmetry is Cnv or. It can be given an extended Schläfli symbol ∨, representing a point, a join operation creates a new edge between all pairs of vertices of the two joined figures. The trigonal or triangular pyramid with all equilateral triangles faces becomes the regular tetrahedron, a lower symmetry case of the triangular pyramid is C3v, which has an equilateral triangle base, and 3 identical isosceles triangle sides. The square and pentagonal pyramids can also be composed of convex polygons. Right pyramids with regular star polygon bases are called star pyramids, for example, the pentagrammic pyramid has a pentagram base and 5 intersecting triangle sides. A right pyramid can be named as ∨P, where is the point, ∨ is a join operator. It has C1v symmetry from two different base-apex orientations, and C2v in its full symmetry, a rectangular right pyramid, written as ∨, and a rhombic pyramid, as ∨, both have symmetry C2v. The volume of a pyramid is V =13 b h and this works for any polygon, regular or non-regular, and any location of the apex, provided that h is measured as the perpendicular distance from the plane containing the base
12.
Parallel planes
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In geometry, parallel lines are lines in a plane which do not meet, that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lines in space which do not meet must be in a common plane to be considered parallel. Parallel planes are planes in the same space that never meet. Parallel lines are the subject of Euclids parallel postulate, parallelism is primarily a property of affine geometries and Euclidean space is a special instance of this type of geometry. Some other spaces, such as space, have analogous properties that are sometimes referred to as parallelism. For example, A B ∥ C D indicates that line AB is parallel to line CD, in the Unicode character set, the parallel and not parallel signs have codepoints U+2225 and U+2226, respectively. In addition, U+22D5 represents the relation equal and parallel to, given parallel straight lines l and m in Euclidean space, the following properties are equivalent, Every point on line m is located at exactly the same distance from line l. Line m is in the plane as line l but does not intersect l. When lines m and l are both intersected by a straight line in the same plane, the corresponding angles of intersection with the transversal are congruent. Thus, the property is the one usually chosen as the defining property of parallel lines in Euclidean geometry. The other properties are consequences of Euclids Parallel Postulate. Another property that also involves measurement is that parallel to each other have the same gradient. The definition of parallel lines as a pair of lines in a plane which do not meet appears as Definition 23 in Book I of Euclids Elements. Alternative definitions were discussed by other Greeks, often as part of an attempt to prove the parallel postulate, proclus attributes a definition of parallel lines as equidistant lines to Posidonius and quotes Geminus in a similar vein. Simplicius also mentions Posidonius definition as well as its modification by the philosopher Aganis, at the end of the nineteenth century, in England, Euclids Elements was still the standard textbook in secondary schools. A major difference between these texts, both between themselves and between them and Euclid, is the treatment of parallel lines. These reform texts were not without their critics and one of them, Charles Dodgson, wrote a play, Euclid and His Modern Rivals, one of the early reform textbooks was James Maurice Wilsons Elementary Geometry of 1868
13.
Truncation (geometry)
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In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Keplers names for the Archimedean solids, in general any polyhedron can also be truncated with a degree of freedom as to how deep the cut is, as shown in Conway polyhedron notation truncation operation. A special kind of truncation, usually implied, is a uniform truncation, there are no degrees of freedom, and it represents a fixed geometric, just like the regular polyhedra. In general all single ringed uniform polytopes have a uniform truncation, for example, the icosidodecahedron, represented as Schläfli symbols r or, and Coxeter-Dynkin diagram or has a uniform truncation, the truncated icosidodecahedron, represented as tr or t. In the Coxeter-Dynkin diagram, the effect of a truncation is to ring all the adjacent to the ringed node. A truncated n-sided polygon will have 2n sides, a regular polygon uniformly truncated will become another regular polygon, t is. A complete truncation, r, is another regular polygon in its dual position, a regular polygon can also be represented by its Coxeter-Dynkin diagram, and its uniform truncation, and its complete truncation. Star polygons can also be truncated, a truncated pentagram will look like a pentagon, but is actually a double-covered decagon with two sets of overlapping vertices and edges. A truncated great heptagram gives a tetradecagram and this sequence shows an example of the truncation of a cube, using four steps of a continuous truncating process between a full cube and a rectified cube. The final polyhedron is a cuboctahedron, the middle image is the uniform truncated cube. It is represented by a Schläfli symbol t, a bitruncation is a deeper truncation, removing all the original edges, but leaving an interior part of the original faces. The truncated octahedron is a cube, 2t is an example. A complete bitruncation is called a birectification that reduces original faces to points, for polyhedra, this becomes the dual polyhedron. An octahedron is a birectification of the cube, = 2r is an example, another type of truncation is called cantellation, cuts edge and vertices, removing original edges and replacing them with rectangles. Higher dimensional polytopes have higher truncations, runcination cuts faces, edges, in 5-dimensions sterication cuts cells, faces, and edges. Edge-truncation is a beveling or chamfer for polyhedra, similar to cantellation but retains original vertices, in 4-polytopes edge-truncation replaces edges with elongated bipyramid cells. Alternation or partial truncation only removes some of the original vertices, a partial truncation or alternation - Half of the vertices and connecting edges are completely removed. The operation only applies to polytopes with even-sided faces, faces are reduced to half as many sides, and square faces degenerate into edges
14.
Right pyramid
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In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face and it is a conic solid with polygonal base. A pyramid with a base has n +1 vertices, n +1 faces. A right pyramid has its apex directly above the centroid of its base, nonright pyramids are called oblique pyramids. A regular pyramid has a polygon base and is usually implied to be a right pyramid. When unspecified, a pyramid is usually assumed to be a square pyramid. A triangle-based pyramid is often called a tetrahedron. Among oblique pyramids, like acute and obtuse triangles, a pyramid can be called if its apex is above the interior of the base and obtuse if its apex is above the exterior of the base. A right-angled pyramid has its apex above an edge or vertex of the base, in a tetrahedron these qualifiers change based on which face is considered the base. Pyramids are a subclass of the prismatoids, pyramids can be doubled into bipyramids by adding a second offset point on the other side of the base plane. A right pyramid with a base has isosceles triangle sides, with symmetry is Cnv or. It can be given an extended Schläfli symbol ∨, representing a point, a join operation creates a new edge between all pairs of vertices of the two joined figures. The trigonal or triangular pyramid with all equilateral triangles faces becomes the regular tetrahedron, a lower symmetry case of the triangular pyramid is C3v, which has an equilateral triangle base, and 3 identical isosceles triangle sides. The square and pentagonal pyramids can also be composed of convex polygons. Right pyramids with regular star polygon bases are called star pyramids, for example, the pentagrammic pyramid has a pentagram base and 5 intersecting triangle sides. A right pyramid can be named as ∨P, where is the point, ∨ is a join operator. It has C1v symmetry from two different base-apex orientations, and C2v in its full symmetry, a rectangular right pyramid, written as ∨, and a rhombic pyramid, as ∨, both have symmetry C2v. The volume of a pyramid is V =13 b h and this works for any polygon, regular or non-regular, and any location of the apex, provided that h is measured as the perpendicular distance from the plane containing the base
15.
Computer graphics
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Computer graphics are pictures and films created using computers. Usually, the term refers to computer-generated image data created with help from specialized hardware and software. It is a vast and recent area in computer science, the phrase was coined in 1960, by computer graphics researchers Verne Hudson and William Fetter of Boeing. It is often abbreviated as CG, though sometimes referred to as CGI. The overall methodology depends heavily on the sciences of geometry, optics. Computer graphics is responsible for displaying art and image data effectively and meaningfully to the user and it is also used for processing image data received from the physical world. Computer graphic development has had a significant impact on many types of media and has revolutionized animation, movies, advertising, video games, the term computer graphics has been used a broad sense to describe almost everything on computers that is not text or sound. Such imagery is found in and on television, newspapers, weather reports, a well-constructed graph can present complex statistics in a form that is easier to understand and interpret. In the media such graphs are used to illustrate papers, reports, thesis, many tools have been developed to visualize data. Computer generated imagery can be categorized into different types, two dimensional, three dimensional, and animated graphics. As technology has improved, 3D computer graphics have become more common, Computer graphics has emerged as a sub-field of computer science which studies methods for digitally synthesizing and manipulating visual content. Screens could display art since the Lumiere brothers use of mattes to create effects for the earliest films dating from 1895. New kinds of displays were needed to process the wealth of information resulting from such projects, early projects like the Whirlwind and SAGE Projects introduced the CRT as a viable display and interaction interface and introduced the light pen as an input device. Douglas T. Ross of the Whirlwind SAGE system performed an experiment in 1954 in which a small program he wrote captured the movement of his finger. Electronics pioneer Hewlett-Packard went public in 1957 after incorporating the decade prior, and established ties with Stanford University through its founders. This began the transformation of the southern San Francisco Bay Area into the worlds leading computer technology hub - now known as Silicon Valley. The field of computer graphics developed with the emergence of computer graphics hardware, further advances in computing led to greater advancements in interactive computer graphics. In 1959, the TX-2 computer was developed at MITs Lincoln Laboratory, the TX-2 integrated a number of new man-machine interfaces
16.
Viewing frustum
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In 3D computer graphics, the view frustum is the region of space in the modeled world that may appear on the screen, it is the field of view of the notional camera. Some authors use pyramid of vision as a synonym for view frustum itself, the exact shape of this region varies depending on what kind of camera lens is being simulated, but typically it is a frustum of a rectangular pyramid. The planes that cut the frustum perpendicular to the direction are called the near plane. Objects closer to the camera than the plane or beyond the far plane are not drawn. Sometimes, the far plane is placed far away from the camera so all objects within the frustum are drawn regardless of their distance from the camera. View frustum culling is the process of removing objects that lie completely outside the viewing frustum from the rendering process, rendering these objects would be a waste of time since they are not directly visible. To make culling fast, it is usually done using bounding volumes surrounding the rather than the objects themselves. VPN the view-plane normal – a normal to the view plane, VUV the view-up vector – the vector on the view plane that indicates the upward direction. VRP the viewing reference point – a point located on the plane. PRP the projection reference point – the point where the image is projected from, for parallel projection, the geometry is defined by a field of view angle, as well as an aspect ratio. Further, a set of z-planes define the near and far bounds of the frustum, together this information can be used to calculate a projection matrix for rendering transformations in a graphics pipeline
17.
Clipping (computer graphics)
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Clipping, in the context of computer graphics, is a method to selectively enable or disable rendering operations within a defined region of interest. Mathematically, clipping can be described using the terminology of constructive geometry, a rendering algorithm only draws pixels in the intersection between the clip region and the scene model. Lines and surfaces outside the volume are removed. Clip regions are commonly specified to improve render performance, a well-chosen clip allows the renderer to save time and energy by skipping calculations related to pixels that the user cannot see. Pixels that will be drawn are said to be within the clip region, pixels that will not be drawn are outside the clip region. More informally, pixels that will not be drawn are said to be clipped, in two-dimensional graphics, a clip region may be defined so that pixels are only drawn within the boundaries of a window or frame. Clip regions can also be used to selectively control pixel rendering for aesthetic or artistic purposes, a user application may render the image into a viewport. As the user zooms and scrolls to view a portion of the image. In addition, GUI widgets, overlays, and other windows or frames may obscure some pixels from the original image, in this sense, the clip region is the composite of the application-defined user clip and the device clip enforced by the systems software and hardware implementation. Application software can take advantage of this information to save computation time, energy. In three-dimensional graphics, the terminology of clipping can be used to describe many related features, typically, clipping refers to operations in the plane that work with rectangular shapes, and culling refers to more general methods to selectively process scene model elements. This terminology is not rigid, and exact usage varies among many sources, scene model elements include geometric primitives, points or vertices, line segments or edges, polygons or faces, and more abstract model objects such as curves, splines, surfaces, and even text. In complicated scene models, individual elements may be disabled for reasons including visibility within the viewport. Sophisticated algorithms exist to detect and perform such clipping. Many optimized clipping methods rely on hardware acceleration logic provided by a graphics processing unit. The concept of clipping can be extended to higher dimensionality using methods of algebraic geometry. Beyond projection of vertices & 2D clipping, near clipping is required to correctly rasterise 3D primitives, near clipping ensures that all the vertices used have valid 2D coordinates. Together with far-clipping it also helps prevent overflow of depth-buffer values, some early texture mapping hardware in video games suffered from complications associated with near clipping and UV coordinates
18.
Aerospace industry
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Aerospace engineering is the primary field of engineering concerned with the development of aircraft and spacecraft. It has two major and overlapping branches, aeronautical engineering and astronautical engineering, avionics engineering is similar, but deals with the electrical side of aerospace engineering. Aeronautical engineering was the term for the field. As flight technology advanced to include craft operating in outer space, Aerospace engineering, particularly the astronautics branch, is often colloquially referred to as rocket science. Flight vehicles are subjected to demanding conditions such as produced by changes in atmospheric pressure and temperature. The interaction between these technologies is known as aerospace engineering, because of the complexity and number of disciplines involved, aerospace engineering is carried out by teams of engineers, each having their own specialized area of expertise. Early knowledge of engineering was largely empirical with some concepts. Scientists understood some key elements of aerospace engineering, like fluid dynamics, many years later after the successful flights by the Wright brothers, the 1910s saw the development of aeronautical engineering through the design of World War I military aircraft. The first definition of aerospace engineering appeared in February 1958, the definition considered the Earths atmosphere and the outer space as a single realm, thereby encompassing both aircraft and spacecraft under a newly coined word aerospace. In response to the USSR launching the first satellite, Sputnik into space on October 4,1957, the National Aeronautics and Space Administration was founded in 1958 as a response to the Cold War. Some of the elements of aerospace engineering are, Radar cross-section – the study of vehicle signature apparent to Radar remote sensing, fluid mechanics – the study of fluid flow around objects. Specifically aerodynamics concerning the flow of air over bodies such as wings or through objects such as wind tunnels, astrodynamics – the study of orbital mechanics including prediction of orbital elements when given a select few variables. While few schools in the United States teach this at the undergraduate level, statics and Dynamics – the study of movement, forces, moments in mechanical systems. Mathematics – in particular, calculus, differential equations, and linear algebra, electrotechnology – the study of electronics within engineering. Propulsion – the energy to move a vehicle through the air is provided by internal combustion engines, jet engines and turbomachinery, a more recent addition to this module is electric propulsion and ion propulsion. Control engineering – the study of modeling of the dynamic behavior of systems and designing them, usually using feedback signals. This applies to the behavior of aircraft, spacecraft, propulsion systems. Aircraft structures – design of the configuration of the craft to withstand the forces encountered during flight
19.
Payload fairing
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A payload fairing is a nose cone used to protect a spacecraft against the impact of dynamic pressure and aerodynamic heating during launch through an atmosphere. More recently, a function is to maintain the cleanroom environment for precision instruments. Outside the atmosphere the fairing is jettisoned, exposing the payload, at this moment mechanical shocks and a spike in acceleration might be observed. The standard payload fairing is typically a combination, due to aerodynamic considerations. The type of fairing which separates into two halves upon jettisoning is called a clamshell fairing by way of analogy to the shell of a clam. In some cases the fairing may enclose both the payload and the stage of the rocket, such as on Atlas V. Since fairing is an part of a launch vehicle, being able to reuse them will lower the cost of launches. On March 30,2017, SpaceX successfully retrieved a fairing for the first time in history, failure of the fairing to separate in these cases may cause the craft to fail to reach orbit, due to the extra mass. The Augmented Target Docking Adapter, to be used for the Gemini 9A manned mission, was placed into orbit by an Atlas SLV-3 in June 1966. But when the Gemini crew rendezvoused with it, they discovered the fairing had failed to open and separate, two lanyards, which should have been removed before flight, were still in place. The cause was determined to be a crew error. In 1999, the launch of the IKONOS-1 Earth observation satellite failed after the payload fairing of the Athena II rocket did not open properly, the same happened to the Naro-1, South Koreas first carrier rocket, launched on 25 August 2009. During the launch half of the fairing failed to separate, and as a result. The satellite did not reach a stable orbit, on March 4,2011 NASAs Glory satellite launch failed to reach orbit after liftoff due to a fairing separation failure on the Orbital Sciences Taurus XL Launch Vehicle, ending up in the Indian Ocean. This failure represented the second failure of a fairing on an Orbital Sciences Taurus XL vehicle. NASA subsequently decided to switch the vehicle for the Orbiting Carbon Observatorys replacement, OCO-2
20.
Multistage rocket
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A multistage rocket is a rocket that uses two or more stages, each of which contains its own engines and propellant. A tandem or serial stage is mounted on top of another stage, the result is effectively two or more rockets stacked on top of or attached next to each other. Taken together these are called a launch vehicle. Two-stage rockets are common, but rockets with as many as five separate stages have been successfully launched. By jettisoning stages when they run out of propellant, the mass of the rocket is decreased. This staging allows the thrust of the stages to more easily accelerate the rocket to its final speed. In serial or tandem staging schemes, the first stage is at the bottom and is usually the largest, in parallel staging schemes solid or liquid rocket boosters are used to assist with lift-off. These are sometimes referred to as stage 0, in the typical case, the first-stage and booster engines fire to propel the entire rocket upwards. When the boosters run out of fuel, they are detached from the rest of the rocket, the first stage then burns to completion and falls off. This leaves a smaller rocket, with the stage on the bottom. Known in rocketry circles as staging, this process is repeated until the final stages motor burns to completion, by dropping the stages which are no longer useful to the mission, the rocket lightens itself. The thrust of future stages is able to provide more acceleration than if the stage were still attached, or a single. When a stage drops off, the rest of the rocket is still traveling near the speed that the whole assembly reached at burn-out time and this means that it needs less total fuel to reach a given velocity and/or altitude. A further advantage is that each stage can use a different type of rocket motor each tuned for its operating conditions. Thus the lower-stage motors are designed for use at atmospheric pressure, on the downside, staging requires the vehicle to lift motors which are not yet being used, as well as making the entire rocket more complex and harder to build. In addition, each staging event is a significant point of failure during a launch, with the possibility of failure, ignition failure. Nevertheless, the savings are so great that every rocket ever used to deliver a payload into orbit has had staging of some sort. In terms of staging, the rocket stages usually have a lower specific impulse rating, trading efficiency for superior thrust in order to quickly push the rocket into higher altitudes.3 to 2.0
21.
Saturn V
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The Saturn V was an American human-rated expendable rocket used by NASA between 1967 and 1973. The Saturn V was launched 13 times from the Kennedy Space Center in Florida with no loss of crew or payload, to date, the Saturn V remains the only launch vehicle to launch missions to carry humans beyond low Earth orbit. A total of 15 flight-capable vehicles were built, but only 13 were flown, an additional three vehicles were built for ground testing purposes. A total of 24 astronauts were launched to the Moon, three of them twice, in the four years spanning December 1968 through December 1972 and it was known that Americas rival, the Soviet Union, would also try to secure some of the Germans. Von Braun was put into the design division of the Army due to his prior direct involvement in the creation of the V-2 rocket. Between 1945 and 1958, his work was restricted to conveying the ideas, finally, they turned to von Braun and his team, who during these years created and experimented with the Jupiter series of rockets. The Juno I was the rocket launched the first American satellite in January 1958. The Jupiter series was one step in von Brauns journey to the Saturn V. The Saturn Vs design stemmed from the designs of the Jupiter series rockets, as the success of the Jupiter series became evident, the Saturn series emerged. Between 1960 and 1962, the Marshall Space Flight Center designed a series of Saturn rockets that could be used for various Earth orbit or lunar missions. NASA planned to use the C-3 as part of the Earth Orbit Rendezvous concept, the C-4 would need only two launches to carry out an EOR lunar mission. On January 10,1962, NASA announced plans to build the C-5. The three-stage rocket would consist of, the S-IC first stage, with five F-1 engines, the S-II second stage, with five J-2 engines, the C-5 was designed for a 90, 000-pound payload capacity to the Moon. The C-5 would undergo component testing even before the first model was constructed, by testing all components at once, far fewer test flights would be required before a manned launch. The C-5 was confirmed as NASAs choice for the Apollo program in early 1963, the C-1 became the Saturn I, and C-1B became Saturn IB. Von Braun headed a team at the Marshall Space Flight Center in building a vehicle capable of launching a spacecraft on a trajectory to the Moon. Before they moved under NASAs jurisdiction, von Brauns team had begun work on improving the thrust, creating a less complex operating system. It was during these revisions that the decision to reject the single engine of the V-2s design came about, the Saturn I and IB reflected these changes, but were not large enough to send a manned spacecraft to the Moon
22.
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
23.
Prism (geometry)
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In geometry, a prism is a polyhedron comprising an n-sided polygonal base, a second base which is a translated copy of the first, and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the bases are translations of the bases, prisms are named for their bases, so a prism with a pentagonal base is called a pentagonal prism. The prisms are a subclass of the prismatoids, a right prism is a prism in which the joining edges and faces are perpendicular to the base faces. This applies if the faces are rectangular. If the joining edges and faces are not perpendicular to the base faces, for example a parallelepiped is an oblique prism of which the base is a parallelogram, or equivalently a polyhedron with six faces which are all parallelograms. A truncated prism is a prism with nonparallel top and bottom faces, some texts may apply the term rectangular prism or square prism to both a right rectangular-sided prism and a right square-sided prism. A right p-gonal prism with rectangular sides has a Schläfli symbol ×, a right rectangular prism is also called a cuboid, or informally a rectangular box. A right square prism is simply a box, and may also be called a square cuboid. A right rectangular prism has Schläfli symbol ××, an n-prism, having regular polygon ends and rectangular sides, approaches a cylindrical solid as n approaches infinity. The term uniform prism or semiregular prism can be used for a prism with square sides. A uniform p-gonal prism has a Schläfli symbol t, right prisms with regular bases and equal edge lengths form one of the two infinite series of semiregular polyhedra, the other series being the antiprisms. The dual of a prism is a bipyramid. The volume of a prism is the product of the area of the base, the volume is therefore, V = B ⋅ h where B is the base area and h is the height. The volume of a prism whose base is a regular n-sided polygon with side s is therefore. The surface area of a prism is 2 · B + P · h, where B is the area of the base, h the height. The surface area of a prism whose base is a regular n-sided polygon with side length s and height h is therefore. The rotation group is Dn of order 2n, except in the case of a cube, which has the symmetry group O of order 24. The symmetry group Dnh contains inversion iff n is even, a prismatic polytope is a higher-dimensional generalization of a prism
24.
Octahedron
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In geometry, an octahedron is a polyhedron with eight faces, twelve edges, and six vertices. A regular octahedron is a Platonic solid composed of eight equilateral triangles, a regular octahedron is the dual polyhedron of a cube. It is a square bipyramid in any of three orthogonal orientations and it is also a triangular antiprism in any of four orientations. An octahedron is the case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan metric, the second and third correspond to the B2 and A2 Coxeter planes. The octahedron 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. An octahedron with edge length √2 can be placed with its center at the origin and its vertices on the coordinate axes, the Cartesian coordinates of the vertices are then. In an x–y–z Cartesian coordinate system, the octahedron with center coordinates, additionally the inertia tensor of the stretched octahedron is I =. These reduce to the equations for the regular octahedron when x m = y m = z m = a 22, the interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. Correspondingly, an octahedron is the result of cutting off from a regular tetrahedron. One can also divide the edges of an octahedron in the ratio of the mean to define the vertices of an icosahedron. There are five octahedra that define any given icosahedron in this fashion, octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space, called the octet truss by Buckminster Fuller. This is the only such tiling save the regular tessellation of cubes, another is a tessellation of octahedra and cuboctahedra. The octahedron is unique among the Platonic solids in having a number of faces meeting at each vertex. Consequently, it is the member of that group to possess mirror planes that do not pass through any of the faces. Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid, truncation of two opposite vertices results in a square bifrustum. The octahedron is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices and it is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size
25.
Apex (geometry)
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In geometry, an apex is the vertex which is in some sense the highest of the figure to which it belongs. The term is used to refer to the vertex opposite from some base. In an isosceles triangle, the apex is the vertex where the two sides of equal length meet, opposite the third side. In a pyramid or cone, the apex is the vertex at the top, in a pyramid, the vertex is the point that is part of all the lateral faces, or where all the lateral edges meet
26.
Prismatoid
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In geometry, a prismatoid is a polyhedron whose vertices all lie in two parallel planes. Its lateral faces can be trapezoids or triangles, if both planes have the same number of vertices, and the lateral faces are either parallelograms or trapezoids, it is called a prismoid. For example, in four dimensions, two polyhedra can be placed in two parallel 3-spaces, and connected with polyhedral sides
27.
Bifrustum
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An n-agonal bifrustum is a polyhedron composed of three parallel planes of n-agons, with the middle plane largest and usually the top and bottom congruent. It can be constructed as two congruent frusta combined across a plane of symmetry, and also as a bipyramid with the two polar vertices truncated and they are duals to the family of elongated bipyramids. Three bifrustums are duals to three Johnson solids, J14-16, in general, a n-agonal bifrustum has 2n trapezoids,2 n-agons, and is dual to the elongated dipyramids
28.
Egyptian mathematics
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Ancient Egyptian mathematics is the mathematics that was developed and used in Ancient Egypt c.3000 to c.300 BC. Written evidence of the use of mathematics dates back to at least 3000 BC with the ivory labels found in Tomb U-j at Abydos and these labels appear to have been used as tags for grave goods and some are inscribed with numbers. Further evidence of the use of the base 10 number system can be found on the Narmer Macehead which depicts offerings of 400,000 oxen,1,422,000 goats and 120,000 prisoners. The evidence of the use of mathematics in the Old Kingdom is scarce, the lines in the diagram are spaced at a distance of one cubit and show the use of that unit of measurement. The earliest true mathematical documents date to the 12th dynasty, the Rhind Mathematical Papyrus which dates to the Second Intermediate Period is said to be based on an older mathematical text from the 12th dynasty. The Moscow Mathematical Papyrus and Rhind Mathematical Papyrus are so-called mathematical problem texts and they consist of a collection of problems with solutions. These texts may have been written by a teacher or a student engaged in solving typical mathematics problems, an interesting feature of Ancient Egyptian mathematics is the use of unit fractions. Scribes used tables to help work with these fractions. The Egyptian Mathematical Leather Roll for instance is a table of unit fractions which are expressed as sums of unit fractions. The Rhind Mathematical Papyrus and some of the other texts contain 2 n tables and these tables allowed the scribes to rewrite any fraction of the form 1 n as a sum of unit fractions. During the New Kingdom mathematical problems are mentioned in the literary Papyrus Anastasi I, in the workers village of Deir el-Medina several ostraca have been found that record volumes of dirt removed while quarrying the tombs. Our understanding of ancient Egyptian mathematics is impeded by the paucity of available sources. The Reisner Papyrus dates to the early Twelfth dynasty of Egypt and was found in Nag el-Deir, the Rhind Mathematical Papyrus dates from the Second Intermediate Period, but its author, Ahmes, identifies it as a copy of a now lost Middle Kingdom papyrus. The RMP is the largest mathematical text, from the New Kingdom we have a handful of mathematical texts and inscription related to computations, The Papyrus Anastasi I is a literary text from the New Kingdom. It is written as a written by a scribe named Hori. A segment of the letter describes several mathematical problems, ostracon Senmut 153 is a text written in hieratic. Ostracon Turin 57170 is a written in hieratic. Ostraca from Deir el-Medina contain computations, ostracon IFAO1206 for instance shows the calculations of volumes, presumably related to the quarrying of a tomb
29.
Moscow Mathematical Papyrus
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Golenishchev bought the papyrus in 1892 or 1893 in Thebes. It later entered the collection of the Pushkin State Museum of Fine Arts in Moscow, approximately 5½ m long and varying between 3.8 and 7.6 cm wide, its format was divided into 25 problems with solutions by the Soviet Orientalist Vasily Vasilievich Struve in 1930. It is a well-known mathematical papyrus along with the Rhind Mathematical Papyrus, the Moscow Mathematical Papyrus is older than the Rhind Mathematical Papyrus, while the latter is the larger of the two. The problems in the Moscow Papyrus follow no particular order, the papyrus is well known for some of its geometry problems. Problems 10 and 14 compute a surface area and the volume of a frustum respectively, the remaining problems are more common in nature. Problems 2 and 3 are ships part problems, one of the problems calculates the length of a ships rudder and the other computes the length of a ships mast given that it is 1/3 + 1/5 of the length of a cedar log originally 30 cubits long. Aha problems involve finding unknown quantities if the sum of the quantity, the Rhind Mathematical Papyrus also contains four of these type of problems. Problems 1,19, and 25 of the Moscow Papyrus are Aha problems, for instance problem 19 asks one to calculate a quantity taken 1 and ½ times and added to 4 to make 10. In other words, in mathematical notation one is asked to solve 3 /2 × x +4 =10 Most of the problems are pefsu problems,10 of the 25 problems. A pefsu measures the strength of the beer made from a heqat of grain pefsu = number loaves of bread number of heqats of grain A higher pefsu number means weaker bread or beer, the pefsu number is mentioned in many offering lists. Then reckon what you need for a des-jug of beer like the beer called 1/2 1/4 malt-date beer The result is 1/2 of the heqat measure needed for des-jug of beer made from Upper-Egyptian grain. Calculate 1/2 of 5 heqat, the result will be 2 1/2 Take this 2 1/2 four times The result is 10, then you say to him, Behold. The beer quantity is found to be correct, problems 11 and 23 are Baku problems. These calculate the output of workers, problem 11 asks if someone brings in 100 logs measuring 5 by 5, then how many logs measuring 4 by 4 does this correspond to. Problem 23 finds the output of a given that he has to cut. Seven of the problems are geometry problems and range from computing areas of triangles, to finding the surface area of a hemisphere. The 10th problem of the Moscow Mathematical Papyrus asks for a calculation of the area of a hemisphere or possibly the area of a semi-cylinder. Below we assume that the problem refers to the area of a hemisphere, the text of problem 10 runs like this, Example of calculating a basket
30.
13th dynasty
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The Thirteenth Dynasty of ancient Egypt is often combined with Dynasties XI, XII and XIV under the group title Middle Kingdom. Some writers separate it from these dynasties and join it to Dynasties XIV through XVII as part of the Second Intermediate Period, Dynasty XIII lasted from approximately 1803 BC until approximately 1649 BC, i. e. for 154 years. The 13th dynasty was a continuation of the preceding 12th dynasty. As direct heirs to the kings of the 12th dynasty, pharaohs of the 13th dynasty reigned from Memphis over Middle and Upper Egypt, all the way to the second cataract to the south. The power of the 13th dynasty waned progressively over its 150 years of existence and it came to an end with the conquest of Memphis by the Hyksos rulers of the 15th dynasty. In later texts, this dynasty is described as an era of chaos. Unfortunately, the chronology of this dynasty is difficult to determine as there are few monuments dating from the period. Many of the names are only known from odd fragmentary inscriptions or from scarabs. The names and order in the table are based on Dodson and Hilton, following these kings, the remaining rulers of the 13th Dynasty are only attested by finds from Upper Egypt. This may indicate the abandonment of the old capital Itjtawy in favor of Thebes, daphna Ben Tor believes that this event was triggered by the invasion of the eastern Delta and the Memphite region by Canaanite rulers. For some authors, this marks the end of the Middle Kingdom and this analysis is rejected by Ryholt and Baker however, who note that the stele of Seheqenre Sankhptahi, reigning toward the end of the dynasty, strongly suggests that he reigned over Memphis. Unfortunately, the stele is of unknown provenance and this is now the dominant hypothesis in Egyptology and Sobekhotep Sekhemre Khutawy is referred to as Sobekhotep I in this article. Ryholt thus credits Sekhemre Khutawy Sobkhotep I with a reign of 3 to 4 years c.1800 BC, Dodson and Hilton similarly believe that Sekhemre Khutawy Sobekhotep predated Khaankhre Sobekhotep. After allowing discipline at the forts to deteriorate, the government eventually withdrew its garrisons and, not long afterward. In the north, Lower Egypt was overrun by the Hyksos, an independent line of kings created Dynasty XIV that arose in the western Delta during later Dynasty XIII. Their regime, called Dynasty XV, was claimed to have replaced Dynasties XIII, however, recent archaeological finds at Edfu could indicate that the Hyksos 15th dynasty was already in existence at least by the mid-13th dynasty reign of king Sobekhotep IV. In a recently published paper in Egypt and the Levant, Nadine Moeller, Gregory Marouard, the preserved contexts of these seals shows that Sobekhotep IV and Khyan were most likely contemporaries of one another. Therefore, Manethos statement that the Hyksos 15th dynasty violently replaced the 13th dynasty could be a piece of later Egyptian propaganda, thus the seals of Sobekhotep IV might not indicate that he was a contemporary of Khyan
31.
Volume
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Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance or shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic metre, three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, volumes of a complicated shape can be calculated by integral calculus if a formula exists for the shapes boundary. Where a variance in shape and volume occurs, such as those that exist between different human beings, these can be calculated using techniques such as the Body Volume Index. One-dimensional figures and two-dimensional shapes are assigned zero volume in the three-dimensional space, the volume of a solid can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas, the combined volume of two substances is usually greater than the volume of one of the substances. However, sometimes one substance dissolves in the other and the volume is not additive. In differential geometry, volume is expressed by means of the volume form, in thermodynamics, volume is a fundamental parameter, and is a conjugate variable to pressure. Any unit of length gives a unit of volume, the volume of a cube whose sides have the given length. For example, a cubic centimetre is the volume of a cube whose sides are one centimetre in length, in the International System of Units, the standard unit of volume is the cubic metre. The metric system also includes the litre as a unit of volume, thus 1 litre =3 =1000 cubic centimetres =0.001 cubic metres, so 1 cubic metre =1000 litres. Small amounts of liquid are often measured in millilitres, where 1 millilitre =0.001 litres =1 cubic centimetre. Capacity is defined by the Oxford English Dictionary as the applied to the content of a vessel, and to liquids, grain, or the like. Capacity is not identical in meaning to volume, though closely related, Units of capacity are the SI litre and its derived units, and Imperial units such as gill, pint, gallon, and others. Units of volume are the cubes of units of length, in SI the units of volume and capacity are closely related, one litre is exactly 1 cubic decimetre, the capacity of a cube with a 10 cm side. In other systems the conversion is not trivial, the capacity of a fuel tank is rarely stated in cubic feet, for example. The density of an object is defined as the ratio of the mass to the volume, the inverse of density is specific volume which is defined as volume divided by mass. Specific volume is an important in thermodynamics where the volume of a working fluid is often an important parameter of a system being studied
32.
Factorization
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In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 ×5, in all cases, a product of simpler objects is obtained. The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viètes formulas relate the coefficients of a polynomial to its roots, the opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms. Integer factorization for large integers appears to be a difficult problem, there is no known method to carry it out quickly. Its complexity is the basis of the security of some public key cryptography algorithms. A matrix can also be factorized into a product of matrices of special types, One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types, QR decomposition, LQ, QL, RQ and this situation is generalized by factorization systems. By the fundamental theorem of arithmetic, every integer greater than 1 has a unique prime factorization. Given an algorithm for integer factorization, one can factor any integer down to its constituent primes by repeated application of this algorithm, for very large numbers, no efficient classical algorithm is known. Modern techniques for factoring polynomials are fast and efficient, but use sophisticated mathematical ideas and these techniques are used in the construction of computer routines for carrying out polynomial factorization in Computer algebra systems. This article is concerned with classical techniques. While the general notion of factoring just means writing an expression as a product of simpler expressions, when factoring polynomials this means that the factors are to be polynomials of smaller degree. Thus, while x 2 − y = is a factorization of the expression, another issue concerns the coefficients of the factors. It is not always possible to do this, and a polynomial that can not be factored in this way is said to be irreducible over this type of coefficient, thus, x2 -2 is irreducible over the integers and x2 +4 is irreducible over the reals. In the first example, the integers 1 and -2 can also be thought of as real numbers, and if they are, then x 2 −2 = shows that this polynomial factors over the reals. Similarly, since the integers 1 and 4 can be thought of as real and hence complex numbers, x2 +4 splits over the complex numbers, i. e. x 2 +4 =. The fundamental theorem of algebra can be stated as, Every polynomial of n with complex number coefficients splits completely into n linear factors
33.
Heron of Alexandria
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Hero of Alexandria was a Greek mathematician and engineer who was active in his native city of Alexandria, Roman Egypt. He is considered the greatest experimenter of antiquity and his work is representative of the Hellenistic scientific tradition, Hero published a well recognized description of a steam-powered device called an aeolipile. Among his most famous inventions was a windwheel, constituting the earliest instance of wind harnessing on land and he is said to have been a follower of the atomists. Some of his ideas were derived from the works of Ctesibius, much of Heros original writings and designs have been lost, but some of his works were preserved in Arabic manuscripts. Hero described the construction of the aeolipile which was a reaction engine. It was created almost two millennia before the industrial revolution, another engine used air from a closed chamber heated by an altar fire to displace water from a sealed vessel, the water was collected and its weight, pulling on a rope, opened temple doors. Some historians have conflated the two inventions to assert that the aeolipile was capable of useful work. The first vending machine was one of his constructions, when a coin was introduced via a slot on the top of the machine. This was included in his list of inventions in his book Mechanics and Optics, when the coin was deposited, it fell upon a pan attached to a lever. The lever opened up a valve which let some water flow out, the pan continued to tilt with the weight of the coin until it fell off, at which point a counter-weight would snap the lever back up and turn off the valve. A windwheel operating an organ, marking the first instance of wind powering a machine in history, the sound of thunder was produced by the mechanically-timed dropping of metal balls onto a hidden drum. The force pump was used in the Roman world. A syringe-like device was described by Hero to control the delivery of air or liquids. In optics, Hero formulated the principle of the shortest path of light, If a ray of light propagates from point A to point B within the same medium, a standalone fountain that operates under self-contained hydrostatic energy A programmable cart that was powered by a falling weight. The program consisted of strings wrapped around the drive axle, Hero described a method for iteratively computing the square root of a number. Today, however, his name is most closely associated with Heros formula for finding the area of a triangle from its side lengths, the most comprehensive edition of Heros works was published in five volumes in Leipzig by the publishing house Teubner in 1903. The Mechanical Technology of Greek and Roman Antiquity, A Study of the Literary Sources, madison, WI, University of Wisconsin Press. Greek and Roman Artillery, Technical Treatises, Schellenberg, H. M. Anmerkungen zu Hero von Alexandria und seinem Werk über den Geschützbau, in, Schellenberg, H. M. / Hirschmann, V. E. / Krieckhaus, A
34.
Imaginary number
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An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property i2 = −1. The square of an imaginary number bi is −b2, for example, 5i is an imaginary number, and its square is −25. Zero is considered to be real and imaginary. Originally coined in the 17th century as a term and regarded as fictitious or useless. Some authors use the term pure imaginary number to denote what is called here an imaginary number, the concept had appeared in print earlier, for instance in work by Gerolamo Cardano. At the time imaginary numbers, as well as numbers, were poorly understood and regarded by some as fictitious or useless. The use of numbers was not widely accepted until the work of Leonhard Euler. The geometric significance of numbers as points in a plane was first described by Caspar Wessel. This idea first surfaced with the articles by James Cockle beginning in 1848, geometrically, imaginary numbers are found on the vertical axis of the complex number plane, allowing them to be presented perpendicular to the real axis. One way of viewing imaginary numbers is to consider a number line, positively increasing in magnitude to the right. This vertical axis is called the imaginary axis and is denoted iℝ, I. In this representation, multiplication by –1 corresponds to a rotation of 180 degrees about the origin, note that a 90-degree rotation in the negative direction also satisfies this interpretation. This reflects the fact that −i also solves the equation x2 = −1, in general, multiplying by a complex number is the same as rotating around the origin by the complex numbers argument, followed by a scaling by its magnitude. Care must be used when working with numbers expressed as the principal values of the square roots of negative numbers. For example,6 =36 = ≠ −4 −9 = =6 i 2 = −6, Imaginary unit de Moivres formula NaN Octonion Quaternion Nahin, Paul. An Imaginary Tale, the Story of the Square Root of −1, explains many applications of imaginary expressions. How can one show that imaginary numbers really do exist, – an article that discusses the existence of imaginary numbers. In our time, Imaginary numbers Discussion of imaginary numbers on BBC Radio 4, 5Numbers programme 4 BBC Radio 4 programme Why Use Imaginary Numbers
35.
Pi
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The number π is a mathematical constant, the ratio of a circles circumference to its diameter, commonly approximated as 3.14159. It has been represented by the Greek letter π since the mid-18th century, being an irrational number, π cannot be expressed exactly as a fraction. Still, fractions such as 22/7 and other numbers are commonly used to approximate π. The digits appear to be randomly distributed, in particular, the digit sequence of π is conjectured to satisfy a specific kind of statistical randomness, but to date no proof of this has been discovered. Also, π is a number, i. e. a number that is not the root of any non-zero polynomial having rational coefficients. This transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass, ancient civilizations required fairly accurate computed values for π for practical reasons. It was calculated to seven digits, using techniques, in Chinese mathematics. The extensive calculations involved have also used to test supercomputers. Because its definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres. Because of its role as an eigenvalue, π appears in areas of mathematics. It is also found in cosmology, thermodynamics, mechanics, attempts to memorize the value of π with increasing precision have led to records of over 70,000 digits. In English, π is pronounced as pie, in mathematical use, the lowercase letter π is distinguished from its capitalized and enlarged counterpart ∏, which denotes a product of a sequence, analogous to how ∑ denotes summation. The choice of the symbol π is discussed in the section Adoption of the symbol π, π is commonly defined as the ratio of a circles circumference C to its diameter d, π = C d The ratio C/d is constant, regardless of the circles size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of π implicitly makes use of geometry, although the notion of a circle can be extended to any curved geometry. Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be defined independently of geometry using limits. An integral such as this was adopted as the definition of π by Karl Weierstrass, definitions of π such as these that rely on a notion of circumference, and hence implicitly on concepts of the integral calculus, are no longer common in the literature. One such definition, due to Richard Baltzer, and popularized by Edmund Landau, is the following, the cosine can be defined independently of geometry as a power series, or as the solution of a differential equation
36.
Radius (geometry)
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In classical geometry, a radius of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is the length of any of them. The name comes from Latin radius, meaning ray but also the spoke of a chariot wheel, the plural of radius can be either radii or the conventional English plural radiuses. The typical abbreviation and mathematic variable name for radius is r, by extension, the diameter d is defined as twice the radius, d ≐2 r ⇒ r = d 2. If an object does not have a center, the term may refer to its circumradius, in either case, the radius may be more than half the diameter, which is usually defined as the maximum distance between any two points of the figure. The inradius of a figure is usually the radius of the largest circle or sphere contained in it. The inner radius of a ring, tube or other object is the radius of its cavity. For regular polygons, the radius is the same as its circumradius, the inradius of a regular polygon is also called apothem. In graph theory, the radius of a graph is the minimum over all u of the maximum distance from u to any other vertex of the graph. The radius of the circle with perimeter C is r = C2 π, alternatively, this can be expressed as r = C τ, with τ being equal to 2 π exactly, although this has yet to gain mainstream usage. For many geometrical figures, the radius has a relationship with other measures of the figure. The radius of a circle with area A is r = A π and this formula uses the sine rule. The polar coordinate system is a coordinate system in which each point on a plane is determined by a distance from a fixed point. The fixed point is called the pole, and the ray from the pole in the direction is the polar axis. The distance from the pole is called the radial coordinate or radius, in the cylindrical coordinate system, there is a chosen reference axis and a chosen reference plane perpendicular to that axis. The origin of the system is the point where all three coordinates can be given as zero and this is the intersection between the reference plane and the axis. The distance from the axis may be called the distance or radius. The radius and the azimuth are together called the coordinates, as they correspond to a two-dimensional polar coordinate system in the plane through the point. The third coordinate may be called the height or altitude, longitudinal position, in a spherical coordinate system, the radius describes the distance of a point from a fixed origin
37.
Rolo
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First manufactured in the UK by Mackintoshs in 1937, they are made by Nestlé, except in the United States where production has been licensed by The Hershey Company. The Rolo product was developed in England by Mackintoshs, simply a combination of Mackintoshs Toffee, Rolo was first sold in 1937. They were also produced in Norwich until 1994, when all UK production moved to Fawdon in Tyneside, there have now been Rolo biscuits, ice-cream, muffins, birthday cake, desserts, cake bars, doughnuts, mini Rolos, big Rolos, yogurts and Easter eggs made. In May 2011, McDonalds combined chocolate pieces and caramel sauce with their soft-serve McFlurry product to simulate the Rolo flavour profile in a cross-branded product, initially the New England Confectionery Company acquired a license to produce Rolos in the United States. However, they have produced in the U. S. by The Hershey Company since 1969. Initially, the U. S. wrappers from Hershey indicated that the confectionery had been produced in England, Rolo was advertised for many years with the slogan Do you love anyone enough to give them your last Rolo. In 1996 the Rolo ad Elephant won the Grand Prix in the section Film Lions at the Cannes Lions International Advertising Festival and this ad was produced by Ammirati Puris Lintas, which now belongs to Lowe Worldwide. In this ad an elephant gets fooled by a boy and decades later takes revenge. List of chocolate bar brands Historical Rolo Wrappers 1970s Rolo Commercial
38.
United States one-dollar bill
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The United States one-dollar bill is a denomination of United States currency. The first U. S. President, George Washington, painted by Gilbert Stuart, is featured on the obverse. The one-dollar bill has the oldest design of all U. S. currency currently being produced, the design seen today debuted in 1963 when it was first issued as a Federal Reserve Note. The inclusion of the motto, In God We Trust, on all currency was required by law in 1955, an individual dollar bill is also less formally known as a one, a single, a buck, a bone, and a bill. The Federal Reserve says the life of a $1 bill in circulation is 5.8 years before it is replaced because of wear. Approximately 42% of all U. S. currency produced in 2009 were one-dollar bills,1862, The first one-dollar bill was issued as a Legal Tender Note with a portrait of Salmon P. Chase, the Secretary of the Treasury under President Abraham Lincoln. 1869, The $1 United States Note was redesigned with a portrait of George Washington in the center, the obverse of the note also featured green and blue tinting. Although this note is technically a United States Note, TREASURY NOTE appeared on it instead of UNITED STATES NOTE,1874, The Series of 1869 United States Note was revised. Changes on the obverse included removing the green and blue tinting, adding a red floral design around the word WASHINGTON D. C. and this note was also issued as Series of 1875 and 1878. 1880, The red floral design around the words ONE DOLLAR, later versions also had blue serial numbers and a small seal moved to the left side of the note. 1886, The first woman to appear on U. S. currency, the reverse of the note featured an ornate design that occupied the entire note, excluding the borders. 1890, One-dollar Treasury or Coin Notes were issued for government purchases of silver bullion from the mining industry. The reverse featured the large word ONE in the center surrounded by a design that occupied almost the entire note. 1891, The reverse of the Series of 1890 Treasury Note was redesigned because the treasury felt that it was too busy, more open space was incorporated into the new design. 1896, The famous Educational Series Silver Certificate was issued, the entire obverse was covered with artwork of allegorical figures representing history instructing youth in front of Washington D. C. The reverse featured portraits of George and Martha Washington surrounded by a design that occupied almost the entire note. 1899, The $1 Silver Certificate was again redesigned, the obverse featured a vignette of the United States Capitol behind a bald eagle perched on an American flag. Below that were small portraits of Abraham Lincoln to the left,1917, The obverse of the $1 United States Note was changed slightly with the removal of ornamental frames that surrounded the serial numbers
39.
Great Seal of the United States
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The Great Seal of the United States is used to authenticate certain documents issued by the U. S. federal government. The phrase is used both for the seal itself, and more generally for the design impressed upon it. The Great Seal was first used publicly in 1782, the obverse of the great seal is used as the national coat of arms of the United States. It is officially used on such as United States passports, military insignia, embassy placards. As a coat of arms, the design has official colors, since 1935, both sides of the Great Seal have appeared on the reverse of the one-dollar bill. The Seal of the President of the United States is directly based on the Great Seal, the design on the obverse of the seal is the coat of arms of the United States. The shield, though sometimes drawn incorrectly, has two differences from the American flag. First, it has no stars on the blue chief, second, unlike the American flag, the outermost stripes are white, not red, so as not to violate the heraldic rule of tincture. The supporter of the shield is an eagle with its wings outstretched. Although not specified by law, the branch is usually depicted with 13 leaves and 13 olives. The eagle has its head turned towards the branch, on its right side. In its beak, the eagle clutches a scroll with the motto E pluribus unum, over its head there appears a glory with 13 mullets on a blue field. In the current dies of the seal, the 13 stars above the eagle are arranged in rows of 1-4-3-4-1. The 1782 resolution of Congress adopting the arms, still in force, legally blazoned the shield as Paleways of 13 pieces, argent and gules, a more technically proper blazon would have been argent, six pallets gules. But the phrase used was chosen to preserve the reference to the 13 original states, the 1782 resolution adopting the seal blazons the image on the reverse as A pyramid unfinished. In the zenith an eye in a triangle, surrounded by a glory, the pyramid is conventionally shown as consisting of 13 layers to refer to the 13 original states. The adopting resolution provides that it is inscribed on its base with the date MDCCLXXVI in Roman numerals, where the top of the pyramid should be, the Eye of Providence watches over it. Two mottos appear, Annuit cœptis signifies that Providence has approved of undertakings, Novus ordo seclorum, freely taken from Virgil, is Latin for a new order of the ages
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Eye of Providence
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The Eye of Providence is a symbol showing an eye often surrounded by rays of light or a glory and usually enclosed by a triangle. It represents the eye of God watching over mankind, in the modern era, a notable depiction of the eye is the reverse of the Great Seal of the United States, which appears on the United States one-dollar bill. Seventeenth-century depictions of the Eye of Providence sometimes show it surrounded by clouds or sunbursts, in 1782, the Eye of Providence was adopted as part of the symbolism on the reverse side of the Great Seal of the United States. It was first suggested as an element of the Great Seal by the first of three committees in 1776 and is thought to be the suggestion of the artistic consultant. In his original proposal to the committee, Du Simitiere placed the Eye over shields symbolizing each of the thirteen states of the Union. On the version of the seal that was approved, the Eye is positioned above an unfinished pyramid of thirteen steps. The symbolism is explained by the motto that appears above the Eye, Annuit Cœptis, meaning approves undertakings. Perhaps due to its use in the design of the Great Seal, the Eye has made its way into other American seals and logos, for example, today, the Eye of Providence is often associated with Freemasonry. The Eye first appeared as part of the iconography of the Freemasons in 1797. Here, it represents the eye of God and is a reminder that mans thoughts. Typically, the Masonic Eye of Providence has a semi-circular glory below the eye, sometimes the Eye is enclosed by a triangle. However, common Masonic use of the Eye dates to 14 years after the creation of the Great Seal, furthermore, among the members of the various design committees for the Great Seal, only Benjamin Franklin was a Mason. Indeed, many Masonic organizations have explicitly denied any connection to the creation of the Seal, a common occurrence is in the context of a reference to the Illuminati. The logo for the WWE tag team, The Ascension, media related to Eye of Providence at Wikimedia Commons
41.
Ziggurat
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A ziggurat was a massive structure built in ancient Mesopotamia and the western Iranian plateau. It had the form of a step pyramid of successively receding stories or levels. Notable ziggurats include the Great Ziggurat of Ur near Nasiriyah, the Ziggurat of Aqar Quf near Baghdad, Ziggurats were built by the ancient Sumerians, Babylonians, Elamites, Akkadians, and Assyrians for local religions. Each ziggurat was part of a complex which included other buildings. The precursors of the ziggurat were raised platforms that date from the Ubaid period during the fourth millennium BCE, the earliest ziggurats began as a platform, the ziggurat was a pyramidal structure with a flat top. Sun-baked bricks made up the core of the ziggurat with facings of fired bricks on the outside, each step was slightly smaller than the step below it. The facings were often glazed in different colors and may have had astrological significance, kings sometimes had their names engraved on these glazed bricks. The number of floors ranged from two to seven, according to archaeologist Harriet Crawford, It is usually assumed that the ziggurats supported a shrine, though the only evidence for this comes from Herodotus, and physical evidence is nonexistent. It has also suggested by a number of scholars that this shrine was the scene of the sacred marriage. Herodotus describes the furnishing of the shrine on top of the ziggurat at Babylon, the god Marduk was also said to come and sleep in his shrine. The likelihood of such a shrine ever being found is sadly remote, erosion has usually reduced the surviving ziggurats to a fraction of their original height, but textual evidence may yet provide more facts about the purpose of these shrines. Access to the shrine would have been by a series of ramps on one side of the ziggurat or by a ramp from base to summit. The Mesopotamian ziggurats were not places for worship or ceremonies. They were believed to be dwelling places for the gods and each city had its own patron god, only priests were permitted on the ziggurat or in the rooms at its base, and it was their responsibility to care for the gods and attend to their needs. The priests were very powerful members of Sumerian society, one of the best-preserved ziggurats is Chogha Zanbil in western Iran. The Sialk ziggurat, in Kashan, Iran, is the oldest known ziggurat, Ziggurat designs ranged from simple bases upon which a temple sat, to marvels of mathematics and construction which spanned several terraced stories and were topped with a temple. An example of a simple ziggurat is the White Temple of Uruk, the ziggurat itself is the base on which the White Temple is set. Its purpose is to get the temple closer to the heavens, the Mesopotamians believed that these pyramid temples connected heaven and earth
42.
Step pyramid
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A step pyramid or stepped pyramid is an architectural structure that uses flat platforms, or steps, receding from the ground up, to achieve a completed shape similar to a geometric pyramid. Step pyramids are structures which characterized several cultures throughout history, in locations throughout the world. These pyramids typically are large and made of layers of stone. The term refers to pyramids of similar design that emerged separately from one another, ziggurats were huge religious monuments built in the ancient Mesopotamian valley and western Iranian plateau, having the form of a terraced step pyramid of successively receding stories or levels. There are 32 ziggurats known at, and near, Mesopotamia, twenty-eight of them are in Iraq, and four of them are in Iran. Ziggurats were built by the Sumerians, Babylonians, Elamites and Assyrians as monuments to local religions, the earliest ziggurats probably date from the latter part of the Early Dynastic Period of Sumer. Built in receding tiers upon a rectangular, oval, or square platform, sun-baked bricks made up the core of the ziggurat with facings of fired bricks on the outside. The facings were often glazed in different colors and may have had astrological significance, kings sometimes had their names engraved on these glazed bricks. The number of tiers ranged from two to seven, with a shrine or temple at the summit, access to the shrine was provided by a series of ramps on one side of the ziggurat or by a spiral ramp from base to summit. It was also called Hill of Heaven or Mountain of the gods, the earliest Egyptian pyramids were step pyramids. During the Third Dynasty of Egypt, the architect Imhotep designed Egypts first step pyramid as a tomb for the pharaoh and this structure, the Pyramid of Djoser, was composed of a series of six successively smaller mastabas, one on top of another. Later pharaohs, including Sekhemkhet and Khaba, built structures, known as the Buried Pyramid. In the Fourth Dynasty of Egypt, the Egyptians began to build true pyramids with smooth sides, the earliest of these pyramids, located at Meidum, began as a step pyramid built for Sneferu. Sneferu later made other pyramids, the Bent Pyramid and Red Pyramid at Dahshur, with this innovation, the age of Egyptian stepped pyramids came to an end. One of the structures of Igboland was the Nsude Pyramids, at the Nigerian town of Nsude. Ten pyramidal structures were built of clay/mud, the first base section was 60 ft. in circumference and 3 ft. in height. The next stack was 45 ft. in circumference, circular stacks continued, till it reached the top. The structures were temples for the god Ala/Uto, who was believed to reside at the top, a stick was placed at the top to represent the gods residence
43.
Indigenous peoples of the Americas
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The indigenous peoples of the Americas are the pre-Columbian peoples of the Americas and their descendants. The term Amerindian is used in Quebec, the Guianas, Indigenous peoples of the United States are commonly known as Native Americans or American Indians, and Alaska Natives. Application of the term Indian originated with Christopher Columbus, who, in his search for Asia, eventually, the Americas came to be known as the West Indies, a name still used to refer to the islands of the Caribbean Sea. This led to the blanket term Indies and Indians for the indigenous inhabitants, although some indigenous peoples of the Americas were traditionally hunter-gatherers—and many, especially in the Amazon basin, still are—many groups practiced aquaculture and agriculture. The impact of their agricultural endowment to the world is a testament to their time, although some societies depended heavily on agriculture, others practiced a mix of farming, hunting, and gathering. In some regions the indigenous peoples created monumental architecture, large-scale organized cities, chiefdoms, states, and empires. Many parts of the Americas are still populated by peoples, some countries have sizable populations, especially Belize, Bolivia, Chile, Ecuador, Greenland, Guatemala, Mexico. At least a different indigenous languages are spoken in the Americas. Some, such as the Quechuan languages, Aymara, Guaraní, Mayan languages, many also maintain aspects of indigenous cultural practices to varying degrees, including religion, social organization, and subsistence practices. Like most cultures, over time, cultures specific to many indigenous peoples have evolved to incorporate traditional aspects, some indigenous peoples still live in relative isolation from Western culture and a few are still counted as uncontacted peoples. The specifics of Paleo-Indian migration to and throughout the Americas, including the dates and routes traveled, are the subject of ongoing research. According to archaeological and genetic evidence, North and South America were the last continents in the world with human habitation. During the Wisconsin glaciation, 50–17,000 years ago, falling sea levels allowed people to move across the bridge of Beringia that joined Siberia to northwest North America. Alaska was a glacial refugium because it had low snowfall, allowing a small population to exist, the Laurentide Ice Sheet covered most of North America, blocking nomadic inhabitants and confining them to Alaska for thousands of years. Indigenous genetic studies suggest that the first inhabitants of the Americas share a single population, one that developed in isolation. The isolation of these peoples in Beringia might have lasted 10–20,000 years, around 16,500 years ago, the glaciers began melting, allowing people to move south and east into Canada and beyond. These people are believed to have followed herds of now-extinct Pleistocene megafauna along ice-free corridors that stretched between the Laurentide and Cordilleran Ice Sheets. Another route proposed involves migration - either on foot or using primitive boats - along the Pacific Northwest coast to the south, archeological evidence of the latter would have been covered by the sea level rise of more than 120 meters since the last ice age
44.
Chinese pyramids
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About 38 of them are located around 25 kilometres -35 kilometres north-west of Xian, on the Qin Chuan Plains in Shaanxi Province. The most famous is the Mausoleum of the First Qin Emperor, although known in the West for at least a century, their existence has been made controversial by sensationalist publicity and the problems of Chinese archaeology in early 20th century. The earliest tombs in China are found just north of Beijing in the Inner Mongolia Autonomous Region and they belong to the Neolithic Hongshan culture. The site of Niuheliang in Liaoning contains a pyramidal structure, in 1667 the Jesuit Father Athanasius Kircher wrote about Chinese pyramids in his book China monumentis Illustrata. The existence of pyramids in China remained little known in the Western world until the 1910s and they were documented in large numbers around Xian, first in 1912 by the Western traders Fred Meyer Schroder and Oscar Mamen, and also in 1913 by the expedition of Victor Segalen. He wrote about the First Emperors tomb, and about the other tombs in the region in his Mission Archeologique en Chine. The introduction of pyramids in China to popular attention came soon after World War II, many early stories were focused on the existence of a Great White Pyramid. This is the tomb of Emperor Wu of Han located in Xingping, U. S. Army Air Corps pilot James Gaussman is said to have seen a white jewel-topped pyramid during a flight between India and China during World War II. Colonel Maurice Sheahan, Far Eastern director of the Trans World Airline, a photo of Sheahans pyramid appeared in The New York Sunday News on March 30,1947. This photograph later became attributed to James Gaussman, chris Maier showed that the pyramid in the photo is the Maoling Mausoleum of Emperor Wu just outside Xian. Pseudohistorians, through promoting their theories, have increased awareness of these pyramids. Hartwig Hausdorf speculated it was built by aliens, and Philip Coppens repeated this theory, despite claims to the contrary, the existence of these pyramid-shaped tomb mounds was known by scientists in the West before the publicity caused by the story in 1947. The location, reported 40 miles southwest of Sian, is in an area of archaeological importance. Some of the pyramids of Xian are currently tourist attractions, such as for example the Han Yang Ling Mausoleum of the Western Han Dynasty, the original height was 76 metres, the present height is 47 metres, and the dimensions are 357 by 354 metres. It was built during the short-lived imperial Qin Dynasty, Tomb of Emperor Qin Ershi in Xian. Maoling Mausoleum group, Tomb of Emperor Wu of Han 34. 338085°N108. 569684°E /34.338085,108.569684, the size is 222 metres x 217 metres. Some are among the biggest Chinese mausoleums, such as Qianling, joint tomb of Emperor Gaozong of Tang and it is a natural hill shaped by man. Tomb of Emperor Xiaojing of Tang 34. 63276°N112. 81109°E /34.63276,112 and it belongs to the Capital Cities and Tombs of the Ancient Koguryo Kingdom on the World heritage list
45.
John Hancock Center
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The John Hancock Center is a 100-story,1, 128-foot supertall skyscraper at 875 North Michigan Avenue, Chicago, Illinois, United States. It was constructed under the supervision of Skidmore, Owings and Merrill, with chief designer Bruce Graham, when the building topped out on May 6,1968, it was the second tallest building in the world and the tallest outside of New York City. When measured to the top of its masts, it stands at 1,500 feet. The building was named for John Hancock Mutual Life Insurance Company, from the 95th floor restaurant, diners can look out at Chicago and Lake Michigan. The Observatory, which competes with the Willis Towers Skydeck, has a 360° view of the city, up to four states, the Observatory has Chicagos only open-air SkyWalk and also features a free multimedia tour in six languages. The 44th-floor sky lobby features Americas highest indoor swimming pool, construction of the tower was interrupted in 1967 due to a flaw in an innovative engineering method used to pour concrete in stages that was discovered when the building was 20 stories high. The engineers were getting the same settlements for the 20 stories that had been built as what they had expected for the entire 99 stories. This forced the owner to stop development until the problem could be resolved. This situation is similar to the one faced during the construction of 111 West Wacker, wolmans bankruptcy resulted in John Hancock taking over the project, which retained the original design, architect, engineer, and main contractor. The buildings first resident was Ray Heckla, the building engineer. Heckla moved his family in April 1969, before the building was completed, wearing a wetsuit and using a climbing device that enabled him to ascend the I-beams on the buildings side, Goodwin battled repeated attempts by the Chicago Fire Department to knock him off. Fire Commissioner William Blair ordered Chicago firemen to stop Goodwin by directing a fully engaged fire hose at him, fearing for Goodwins life, Mayor Jane Byrne intervened and allowed him to continue to the top. The John Hancock Center was featured in the 1988 movie Poltergeist III, on December 18,1997, comedian Chris Farley was found dead in his apartment on the 60th floor of the John Hancock Center. On March 9,2002, part of a scaffold fell 43 stories after being torn loose by wind gusts around 60 mph crushing several cars, killing three people in two of them. The remaining part of the stage swung back-and-forth in the gusts repeatedly slamming against the building, damaging cladding panels, breaking windows, shorenstein had bought the building in 1998 for $220 million. Golub defaulted on its debt and the building was acquired in 2012 by Deutsche Bank AG who subsequently carved up the building, the observation deck was sold to Paris-based Montparnasse 56 Group for $35 million and $45 in July 2012. That same month, Prudential Real Estate Investors acquired the retail, in November 2012, Boston-based American Tower Corp affiliate paid $70 million for the antennas. In June 2013, a venture of Chicago-based real estate investment firm Hearn Co, the Chicago firm did not disclose a price, but sources said it was about $145 million
46.
Chicago
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Chicago, officially the City of Chicago, is the third-most populous city in the United States. With over 2.7 million residents, it is the most populous city in the state of Illinois, and it is the county seat of Cook County. In 2012, Chicago was listed as a global city by the Globalization and World Cities Research Network. Chicago has the third-largest gross metropolitan product in the United States—about $640 billion according to 2015 estimates, the city has one of the worlds largest and most diversified economies with no single industry employing more than 14% of the workforce. In 2016, Chicago hosted over 54 million domestic and international visitors, landmarks in the city include Millennium Park, Navy Pier, the Magnificent Mile, Art Institute of Chicago, Museum Campus, the Willis Tower, Museum of Science and Industry, and Lincoln Park Zoo. Chicagos culture includes the arts, novels, film, theater, especially improvisational comedy. Chicago also has sports teams in each of the major professional leagues. The city has many nicknames, the best-known being the Windy City, the name Chicago is derived from a French rendering of the Native American word shikaakwa, known to botanists as Allium tricoccum, from the Miami-Illinois language. The first known reference to the site of the current city of Chicago as Checagou was by Robert de LaSalle around 1679 in a memoir, henri Joutel, in his journal of 1688, noted that the wild garlic, called chicagoua, grew abundantly in the area. In the mid-18th century, the area was inhabited by a Native American tribe known as the Potawatomi, the first known non-indigenous permanent settler in Chicago was Jean Baptiste Point du Sable. Du Sable was of African and French descent and arrived in the 1780s and he is commonly known as the Founder of Chicago. In 1803, the United States Army built Fort Dearborn, which was destroyed in 1812 in the Battle of Fort Dearborn, the Ottawa, Ojibwe, and Potawatomi tribes had ceded additional land to the United States in the 1816 Treaty of St. Louis. The Potawatomi were forcibly removed from their land after the Treaty of Chicago in 1833, on August 12,1833, the Town of Chicago was organized with a population of about 200. Within seven years it grew to more than 4,000 people, on June 15,1835, the first public land sales began with Edmund Dick Taylor as U. S. The City of Chicago was incorporated on Saturday, March 4,1837, as the site of the Chicago Portage, the city became an important transportation hub between the eastern and western United States. Chicagos first railway, Galena and Chicago Union Railroad, and the Illinois, the canal allowed steamboats and sailing ships on the Great Lakes to connect to the Mississippi River. A flourishing economy brought residents from rural communities and immigrants from abroad, manufacturing and retail and finance sectors became dominant, influencing the American economy. The Chicago Board of Trade listed the first ever standardized exchange traded forward contracts and these issues also helped propel another Illinoisan, Abraham Lincoln, to the national stage