1.
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

2.
Dice
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Dice are small throwable objects with multiple resting positions, used for generating random numbers. Dice are suitable as gambling devices for games like craps and are used in non-gambling tabletop games. A traditional die is a cube, with each of its six faces showing a different number of dots from 1 to 6. When thrown or rolled, the die comes to rest showing on its surface a random integer from one to six. A variety of devices are also described as dice, such specialized dice may have polyhedral or irregular shapes. They may be used to produce other than one through six. Loaded and crooked dice are designed to favor some results over others for purposes of cheating or amusement. A dice tray, a used to contain thrown dice, is sometimes used for gambling or board games. Dice have been used since before recorded history, and it is uncertain where they originated, the oldest known dice were excavated as part of a backgammon-like game set at the Burnt City, an archeological site in south-eastern Iran, estimated to be from between 2800–2500 BCE. Other excavations from ancient tombs in the Indus Valley civilization indicate a South Asian origin, the Egyptian game of Senet was played with dice. Senet was played before 3000 BC and up to the 2nd century AD and it was likely a racing game, but there is no scholarly consensus on the rules of Senet. Dicing is mentioned as an Indian game in the Rigveda, Atharvaveda, there are several biblical references to casting lots, as in Psalm 22, indicating that dicing was commonplace when the psalm was composed. Knucklebones was a game of skill played by women and children, although gambling was illegal, many Romans were passionate gamblers who enjoyed dicing, which was known as aleam ludere. Dicing was even a popular pastime of emperors, letters by Augustus to Tacitus and his daughter recount his hobby of dicing. There were two sizes of Roman dice, tali were large dice inscribed with one, three, four, and six on four sides. Tesserae were smaller dice with sides numbered one to six. Twenty-sided dice date back to the 2nd century AD and from Ptolemaic Egypt as early as the 2nd century BC, dominoes and playing cards originated in China as developments from dice. The transition from dice to playing cards occurred in China around the Tang dynasty, in Japan, dice were used to play a popular game called sugoroku

3.
Dodecahedron
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In geometry, a dodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the dodecahedron, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form, all of these have icosahedral symmetry, order 120. The pyritohedron is a pentagonal dodecahedron, having the same topology as the regular one. The rhombic dodecahedron, seen as a case of the pyritohedron has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra are space-filling, there are a large number of other dodecahedra. The convex regular dodecahedron is one of the five regular Platonic solids, the dual polyhedron is the regular icosahedron, having five equilateral triangles around each vertex. Like the regular dodecahedron, it has twelve pentagonal faces. However, the pentagons are not constrained to be regular, and its 30 edges are divided into two sets – containing 24 and 6 edges of the same length. The only axes of symmetry are three mutually perpendicular twofold axes and four threefold axes. Note that the regular dodecahedron can occur as a shape for quasicrystals with icosahedral symmetry. Its name comes from one of the two common crystal habits shown by pyrite, the one being the cube. The coordinates of the eight vertices of the cube are, The coordinates of the 12 vertices of the cross-edges are. When h =1, the six cross-edges degenerate to points, when h =0, the cross-edges are absorbed in the facets of the cube, and the pyritohedron reduces to a cube. When h = √5 − 1/2, the inverse of the golden ratio, a reflected pyritohedron is made by swapping the nonzero coordinates above. The two pyritohedra can be superimposed to give the compound of two dodecahedra as seen in the image here, the regular dodecahedron represents a special intermediate case where all edges and angles are equal. A tetartoid is a dodecahedron with chiral tetrahedral symmetry, like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes, although regular dodecahedra do not exist in crystals, the tetartoid form does

4.
Square antiprism
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In geometry, the square antiprism is the second in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It is also known as an anticube, if all its faces are regular, it is a semiregular polyhedron or uniform polyhedron. One molecule with this geometry is the ion in the salt nitrosonium octafluoroxenate, however. Very few ions are cubical because such a shape would cause large repulsion between ligands, PaF3−8 is one of the few examples, in addition, the element sulfur forms octatomic S8 molecules as its most stable allotrope. The main building block of the One World Trade Center has the shape of a tall tapering square antiprism. It is not a true antiprism because of its taper, the top square has half the area of the bottom one, a twisted prism can be made with the same vertex arrangement. It can be seen as the form with 4 tetrahedrons excavated around the sides. However, after this it can no longer be triangulated into tetrahedra without adding new vertices and it has half of the symmetry of the uniform solution, Dn, +, order 8. The gyroelongated square pyramid is a Johnson solid constructed by augmenting one a square pyramid, similarly, the gyroelongated square bipyramid is a deltahedron constructed by replacing both squares of a square antiprism with a square pyramid. The snub disphenoid is another deltahedron, constructed by replacing the two squares of a square antiprism by pairs of equilateral triangles, the snub square antiprism can be seen as a square antiprism with a chain of equilateral triangles inserted around the middle. The sphenocorona and the sphenomegacorona are other Johnson solids that, like the square antiprism, the square antiprism is first in a series of snub polyhedra and tilings with vertex figure 3.3.4.3. n. Compound of three square antiprisms Weisstein, Eric W. Antiprism, square Antiprism interactive model Virtual Reality Polyhedra www. georgehart. com, The Encyclopedia of Polyhedra VRML model Conway Notation for Polyhedra Try, A4

5.
Octagonal prism
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In geometry, the octagonal prism is the sixth in an infinite set of prisms, formed by square sides and two regular octagon caps. If faces are all regular, it is a semiregular polyhedron, the octagonal prism can also be seen as a tiling on a sphere, In optics, octagonal prisms are used to generate flicker-free images in movie projectors. It is an element of three uniform honeycombs, It is also an element of two four-dimensional uniform 4-polytopes, Weisstein, Eric W. Octagonal prism, interactive model of an Octagonal Prism

6.
Role-playing game
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A role-playing game is a game in which players assume the roles of characters in a fictional setting. Players take responsibility for acting out these roles within a narrative, actions taken within many games succeed or fail according to a formal system of rules and guidelines. There are several forms of RPG, the original form, sometimes called the tabletop RPG, is conducted through discussion, whereas in live action role-playing games players physically perform their characters actions. In both of these forms, an arranger called a game master usually decides on the rules and setting to be used, acting as referee, while each of the other players plays the role of a single character. Several varieties of RPG also exist in media, such as multi-player text-based MUDs and their graphics-based successors. These games often share settings and rules with tabletop RPGs, despite this variety of forms, some game forms such as trading card games and wargames that are related to role-playing games may not be included. Role-playing activity may sometimes be present in games, but it is not the primary focus. The term is sometimes used to describe roleplay simulation games and exercises used in teaching, training. Both authors and major publishers of tabletop role-playing games consider them to be a form of interactive and collaborative storytelling, events, characters, and narrative structure give a sense of a narrative experience, and the game need not have a strongly-defined storyline. Interactivity is the difference between role-playing games and traditional fiction. Whereas a viewer of a show is a passive observer. Such role-playing games extend an older tradition of storytelling games where a party of friends collaborate to create a story. Participants in a game will generate specific characters and an ongoing plot. A consistent system of rules and a more or less realistic campaign setting in games aids suspension of disbelief, the level of realism in games ranges from just enough internal consistency to set up a believable story or credible challenge up to full-blown simulations of real-world processes. There is also a variety of systems of rules and game settings. Games that emphasize plot and character interaction over game mechanics and combat sometimes prefer the name storytelling game and these types of games tend to minimize or altogether eliminate the use of dice or other randomizing elements. Some games are played with characters created before the game by the GM and this type of game is typically played at gaming conventions, or in standalone games that do not form part of a campaign. Tabletop and pen-and-paper RPGs are conducted through discussion in a social gathering

7.
Tetrahedron
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In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra, the tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a polygon base. In the case of a tetrahedron the base is a triangle, like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. For any tetrahedron there exists a sphere on which all four vertices lie, a regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have known since antiquity. In a regular tetrahedron, not only are all its faces the same size and shape, regular tetrahedra alone do not tessellate, but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron, the compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. This form has Coxeter diagram and Schläfli symbol h, the tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, and the together form the stellated octahedron. In other words, if C is the centroid of the base and this follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other. The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron, the symmetries of a regular tetrahedron correspond to half of those of a cube, those that map the tetrahedra to themselves, and not to each other. The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion, the regular tetrahedron has 24 isometries, forming the symmetry group Td, isomorphic to the symmetric group, S4. The first corresponds to the A2 Coxeter plane, the two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these intersects the tetrahedron the resulting cross section is a rectangle. When the intersecting plane is one of the edges the rectangle is long. When halfway between the two edges the intersection is a square, the aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly, if the tetrahedron is bisected on this plane, both halves become wedges

8.
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

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.
Topology
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In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, important topological properties include connectedness and compactness. Topology developed as a field of study out of geometry and set theory, through analysis of such as space, dimension. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs, Leonhard Eulers Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the fields first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, by the middle of the 20th century, topology had become a major branch of mathematics. It defines the basic notions used in all branches of topology. Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to geometry and together they make up the geometric theory of differentiable manifolds. Geometric topology primarily studies manifolds and their embeddings in other manifolds, a particularly active area is low-dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots, Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by Leonhard Euler and his 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750 Euler wrote to a friend that he had realised the importance of the edges of a polyhedron and this led to his polyhedron formula, V − E + F =2. Some authorities regard this analysis as the first theorem, signalling the birth of topology, further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti. Listing introduced the term Topologie in Vorstudien zur Topologie, written in his native German, in 1847, the term topologist in the sense of a specialist in topology was used in 1905 in the magazine Spectator. Their work was corrected, consolidated and greatly extended by Henri Poincaré, in 1895 he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a case of a general topological space. In 1914, Felix Hausdorff coined the term topological space and gave the definition for what is now called a Hausdorff space, currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski

11.
Bipyramid
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An n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base. An n-gonal bipyramid has 2n triangle faces, 3n edges, and 2 + n vertices, the referenced n-gon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the two pyramid halves. A right bipyramid has two points above and below the centroid of its base, nonright bipyramids are called oblique bipyramids. A regular bipyramid has a regular polygon internal face and is implied to be a right bipyramid. A right bipyramid can be represented as + P for internal polygon P, a concave bipyramid has a concave interior polygon. The face-transitive regular bipyramids are the dual polyhedra of the uniform prisms, a bipyramid can be projected on a sphere or globe as n equally spaced lines of longitude going from pole to pole, and bisected by a line around the equator. Bipyramid faces, projected as spherical triangles, represent the fundamental domains in the dihedral symmetry Dnh, the volume of a bipyramid is V =2/3Bh where B is the area of the base and h the height from the base to the apex. This works for any location of the apex, provided that h is measured as the distance from the plane which contains the base. The volume of a bipyramid whose base is a regular n-sided polygon with side length s and whose height is h is therefore, only three kinds of bipyramids can have all edges of the same length, the triangular, tetragonal, and pentagonal bipyramids. The rotation group is Dn of order 2n, except in the case of an octahedron, which has the larger symmetry group O of order 24. The digonal faces of a spherical 2n-bipyramid represents the fundamental domains of symmetry in three dimensions, Dnh, order 4n. The reflection domains can be shown as alternately colored triangles as mirror images, a scalenohedron is topologically identical to a 2n-bipyramid, but contains congruent scalene triangles. In one type the 2n vertices around the center alternate in rings above, in the other type, the 2n vertices are on the same plane, but alternate in two radii. The first has 2-fold rotation axes mid-edge around the sides, reflection planes through the vertices, in crystallography, 8-sided and 12-sided scalenohedra exist. All of these forms are isohedra, the second has symmetry Dn, order 2n. The smallest scalenohedron has 8 faces and is identical to the regular octahedron. The second type is a rhombic bipyramid, the first type has 6 vertices can be represented as, where z is a parameter between 0 and 1, creating a regular octahedron at z =0, and becoming a disphenoid with merged coplanar faces at z =1. For z >1, it becomes concave, self-intersecting bipyramids exist with a star polygon central figure, defined by triangular faces connecting each polygon edge to these two points