1.
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
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.
Gen Con
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Gen Con is the largest tabletop-game convention in North America by both attendance and number of events. It features traditional pen-and-paper, board, and card games, including role-playing games, miniatures wargames, live action role-playing games, collectible card games, Gen Con also features computer games. Attendees engage in a variety of tournament and interactive game sessions, in 2015, Gen Con had 61,423 unique attendees, making it one of the largest conventions in North America. Established in 1968 as a convention by Gary Gygax, who later co-created Dungeons & Dragons, Gen Con was first held in Lake Geneva. Other Gen Con conventions have been held sporadically in various locations around the United States, in 1976, Gen Con became the property of TSR, Inc. the gaming company co-founded by Gygax. TSR were acquired by Wizards of the Coast in 1997, which was acquired by Hasbro. Hasbro then sold Gen Con to the former CEO of Wizards of the Coast, Peter Adkison, Gen Con spent a short time under Chapter 11 bankruptcy protection, due to a lawsuit brought against them by Lucasfilm in 2008. The organization emerged from bankruptcy protection a year later, while holding its regularly scheduled events. Some IFW gamers in the Chicago area could not make the journey to Malvern, so they had a gathering that same weekend at the Lake Geneva. Later this gathering would come to be referred to as Gen Con 0, in 1968, Gygax rented Lake Genevas Horticultural Hall to hold a follow-up IFW convention, the Lake Geneva Wargames Convention, later known as the Gen Con gaming convention. The IFW, which Gygax co-founded, put up $35 of the $50 Horticultural Hall fee to sponsor this first Gen Con, at the second Gen Con in August 1969, Gygax met Rob Kuntz and Dave Arneson. During these early conventions, the events centered around board games, Gen Cons name is a derivation of Geneva Convention, due to the conventions origins in Lake Geneva. It is also a play on words, as the Geneva Conventions are a set of important international treaties regarding war, starting in 1971, Gen Con was co-sponsored by the Lake Geneva Tactical Studies Association. Beginning in 1975, Gen Con was managed and hosted by TSR, in 1978 the convention moved to the University of Wisconsin–Parkside campus in Kenosha, where it remained through 1984. A Gen Con West was held in California for only three years, 1976–1978. From 1978 to 1984, Gen Con South was held in Jacksonville, Florida, and Gen Con East was held in 1981 and 1982, first in Cherry Hill, New Jersey, and then in Chester, Pennsylvania. In 1985, Gen Con moved to the Milwaukee Exposition & Convention Center & Arena in Milwaukee, after the move, attendance steadily rose from 5,000 to a peak of 30,000 in 1995, making Gen Con the premier event in the role-playing game industry. In 1992, Gen Con broke all attendance records for any U. S. gaming convention
4.
Platonic solid
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In three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent regular polygonal faces with the number of faces meeting at each vertex. Five solids meet those criteria, Geometers have studied the mathematical beauty and they are named for the ancient Greek philosopher Plato who theorized in his dialogue, the Timaeus, that the classical elements were made of these regular solids. The Platonic solids have been known since antiquity, dice go back to the dawn of civilization with shapes that predated formal charting of Platonic solids. The ancient Greeks studied the Platonic solids extensively, some sources credit Pythagoras with their discovery. In any case, Theaetetus gave a description of all five. The Platonic solids are prominent in the philosophy of Plato, their namesake, Plato wrote about them in the dialogue Timaeus c.360 B. C. in which he associated each of the four classical elements with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, there was intuitive justification for these associations, the heat of fire feels sharp and stabbing. Air is made of the octahedron, its components are so smooth that one can barely feel it. Water, the icosahedron, flows out of hand when picked up. By contrast, a highly nonspherical solid, the hexahedron represents earth and these clumsy little solids cause dirt to crumble and break when picked up in stark difference to the smooth flow of water. Moreover, the cubes being the regular solid that tessellates Euclidean space was believed to cause the solidity of the Earth. Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarks. the god used for arranging the constellations on the whole heaven. Aristotle added an element, aithēr and postulated that the heavens were made of this element. Euclid completely mathematically described the Platonic solids in the Elements, the last book of which is devoted to their properties, propositions 13–17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order. For each solid Euclid finds the ratio of the diameter of the sphere to the edge length. In Proposition 18 he argues there are no further convex regular polyhedra. Andreas Speiser has advocated the view that the construction of the 5 regular solids is the goal of the deductive system canonized in the Elements
5.
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
6.
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
7.
Cube
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In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube is the only regular hexahedron and is one of the five Platonic solids and it has 6 faces,12 edges, and 8 vertices. The cube is also a square parallelepiped, an equilateral cuboid and it is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. The cube is dual to the octahedron and it has cubical or octahedral symmetry. The cube has four special orthogonal projections, centered, on a vertex, edges, face, the first and third correspond to the A2 and B2 Coxeter planes. The cube can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. In analytic geometry, a surface with center and edge length of 2a is the locus of all points such that max = a. For a cube of length a, As the volume of a cube is the third power of its sides a × a × a, third powers are called cubes, by analogy with squares. A cube has the largest volume among cuboids with a surface area. Also, a cube has the largest volume among cuboids with the same linear size. They were unable to solve this problem, and in 1837 Pierre Wantzel proved it to be impossible because the root of 2 is not a constructible number. The cube has three uniform colorings, named by the colors of the faces around each vertex,111,112,123. The cube has three classes of symmetry, which can be represented by coloring the faces. The highest octahedral symmetry Oh has all the faces the same color, the dihedral symmetry D4h comes from the cube being a prism, with all four sides being the same color. The lowest symmetry D2h is also a symmetry, with sides alternating colors. Each symmetry form has a different Wythoff symbol, a cube has eleven nets, that is, there are eleven ways to flatten a hollow cube by cutting seven edges. To color the cube so that no two adjacent faces have the color, one would need at least three colors
8.
Conway polyhedron notation
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In geometry, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations. Conway and Hart extended the idea of using operators, like truncation defined by Kepler, the basic descriptive operators can generate all the Archimedean solids and Catalan solids from regular seeds. For example tC represents a cube, and taC, parsed as t, is a truncated cuboctahedron. The simplest operator dual swaps vertex and face elements, like a cube is an octahedron. Applied in a series, these allow many higher order polyhedra to be generated. A resulting polyhedron will have a fixed topology, while exact geometry is not constrained, the seed polyhedra are the Platonic solids, represented by the first letter of their name, the prisms for n-gonal forms, antiprisms, cupolae and pyramids. Any polyhedron can serve as a seed, as long as the operations can be executed on it, for example regular-faced Johnson solids can be referenced as Jn, for n=1.92. In general, it is difficult to predict the appearance of the composite of two or more operations from a given seed polyhedron. For instance ambo applied twice becomes the same as the operation, aa=e, while a truncation after ambo produces bevel. There has been no general theory describing what polyhedra can be generated in by any set of operators, instead all results have been discovered empirically. Elements are given from the seed to the new forms, assuming seed is a polyhedron, An example image is given for each operation. The basic operations are sufficient to generate the reflective uniform polyhedra, some basic operations can be made as composites of others. Special forms The kis operator has a variation, kn, which only adds pyramids to n-sided faces, the truncate operator has a variation, tn, which only truncates order-n vertices. The operators are applied like functions from right to left, for example, a cuboctahedron is an ambo cube, i. e. t = aC, and a truncated cuboctahedron is t = t = taC. Chirality operator r – reflect – makes the image of the seed. Alternately an overline can be used for picking the other chiral form, the operations are visualized here on cube seed examples, drawn on the surface of the cube, with blue faces that cross original edges, and pink faces that center at original vertices. The first row generates the Archimedean solids and the row the Catalan solids. Comparing each new polyhedron with the cube, each operation can be visually understood, the truncated icosahedron, tI or zD, which is Goldberg polyhedron G, creates more polyhedra which are neither vertex nor face-transitive
9.
Dual polyhedron
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Such dual figures remain combinatorial or abstract polyhedra, but not all are also geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron, duality preserves the symmetries of a polyhedron. Therefore, for classes of polyhedra defined by their symmetries. Thus, the regular polyhedra – the Platonic solids and Kepler-Poinsot polyhedra – form dual pairs, the dual of an isogonal polyhedron, having equivalent vertices, is one which is isohedral, having equivalent faces. The dual of a polyhedron is also isotoxal. Duality is closely related to reciprocity or polarity, a transformation that. There are many kinds of duality, the kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality. The duality of polyhedra is often defined in terms of polar reciprocation about a concentric sphere. In coordinates, for reciprocation about the sphere x 2 + y 2 + z 2 = r 2, the vertex is associated with the plane x 0 x + y 0 y + z 0 z = r 2. The vertices of the dual are the reciprocal to the face planes of the original. Also, any two adjacent vertices define an edge, and these will reciprocate to two adjacent faces which intersect to define an edge of the dual and this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, and r 1 and r 2 respectively the distances from its centre to the pole and its polar, then, r 1. R2 = r 02 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, the choice of center for the sphere is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will intersect at a single point. Failing that, a sphere, inscribed sphere, or midsphere is commonly used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required plane at infinity. Some theorists prefer to stick to Euclidean space and say there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, the concept of duality here is closely related to the duality in projective geometry, where lines and edges are interchanged
10.
Trigonal trapezohedron
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In geometry, a trigonal trapezohedron or trigonal deltohedron is a three-dimensional figure formed by six congruent rhombi. Six identical rhombic faces can construct two configurations of trigonal trapezohedra, the acute or prolate form has three acute angles corners of the rhombic faces meeting at two polar axis vertices. The obtuse or oblate or flat form has three obtuse angle corners of the rhombic faces meeting at the two polar axis vertices, the trigonal trapezohedra is a special case of a rhombohedron. A general rhombohedron allows up to three types of rhombic faces, a trigonal trapezohedron is a special kind of parallelepiped, and are the only parallelepipeds with six congruent faces. Since all of the edges must have the length, every trigonal trapezohedron is also a rhombohedron. It is the simplest of the trapezohedra, a sequence of polyhedra which are dual to the antiprisms. The dual of a trigonal trapezohedron is a triangular antiprism, a trigonal trapezohedron with square faces is a cube. A lower symmetry variation of the trigonal trapezohedron has only rotational symmetry, D3, a golden rhombohedron is one of two special case of the trigonal trapezohedron with golden rhombus faces. The acute or prolate form has three acute angles corners of the rhombic faces meeting at two polar axis vertices, the obtuse or oblate or flat form has three obtuse angle corners of the rhombic faces meeting at the two polar axis vertices. Cartesian coordinates for a golden rhombohedron with one pole at the origin are, and vector additions thereof, the rhombic hexecontahedron can be constructed by 20 acute golden rhombohedra meeting at a point. A regular octahedron augumented by 2 regular tetrahedra creates a trigonal trapezohedron, truncated triangular trapezohedron Weisstein, Eric W. Trapezohedron