Polyhedral Group
In geometry, the polyhedral group is any of the symmetry groups of the Platonic solids. Groups There are three polyhedral groups: *The tetrahedral group of order 12, rotational symmetry group of the regular tetrahedron. It is isomorphic to ''A''4. ** The conjugacy classes of ''T'' are: ***identity ***4 × rotation by 120°, order 3, cw ***4 × rotation by 120°, order 3, ccw ***3 × rotation by 180°, order 2 *The octahedral group of order 24, rotational symmetry group of the cube and the regular octahedron. It is isomorphic to ''S''4. **The conjugacy classes of ''O'' are: ***identity ***6 × rotation by ±90° around vertices, order 4 ***8 × rotation by ±120° around triangle centers, order 3 ***3 × rotation by 180° around vertices, order 2 ***6 × rotation by 180° around midpoints of edges, order 2 *The icosahedral group of order 60, rotational symmetry group of the regular dodecahedron and the regular icosahedron. It is isomorphic to ''A''5. **The conjugacy classes of ' ...
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Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ...
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Schoenflies Notation
The Schoenflies (or Schönflies) notation, named after the German mathematician Arthur Moritz Schoenflies, is a notation primarily used to specify point groups in three dimensions. Because a point group alone is completely adequate to describe the symmetry of a molecule, the notation is often sufficient and commonly used for spectroscopy. However, in crystallography, there is additional translational symmetry, and point groups are not enough to describe the full symmetry of crystals, so the full space group is usually used instead. The naming of full space groups usually follows another common convention, the Hermann–Mauguin notation, also known as the international notation. Although Schoenflies notation without superscripts is a pure point group notation, optionally, superscripts can be added to further specify individual space groups. However, for space groups, the connection to the underlying symmetry elements is much more clear in Hermann–Mauguin notation, so the latter n ...
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Sphere Symmetry Group Th
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre (geometry), centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the Greek mathematics, ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubble (physics), Bubbles such as soap bubbles take a spherical shape in equilibrium. spherical Earth, The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres rolling, roll smoothly in any direction, so mos ...
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Tetrakis Hexahedron Stereographic D2 Gyrations
Tetrakis may refer to: * Tetrakis (Paphlagonia), an ancient Greek city * Tetrakis cuboctahedron, convex polyhedron with 32 triangular faces * Tetrakis hexahedron, an Archimedean dual solid or a Catalan solid *Tetrakis square tiling, a tiling of the Euclidean plane See also *Tetracus ''Tetracus'' is an extinct genus of gymnures. Species are from the Oligocene of Belgium and France. Fossils can also be found in the Bouldnor Formation in the Hampshire Basin of southern England. Species: * †''Tetracus nanus'' (Aymard, 1846) ... * Tetrakis legomenon, a word that occurs only four times within a context * Tetricus (other) * Tetrix (other) * Truncated tetrakis cube, a convex polyhedron with 32 faces * {{disambiguation ...
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Tetrakis Hexahedron Stereographic D3 Gyrations
Tetrakis may refer to: * Tetrakis (Paphlagonia), an ancient Greek city * Tetrakis cuboctahedron, convex polyhedron with 32 triangular faces * Tetrakis hexahedron, an Archimedean dual solid or a Catalan solid *Tetrakis square tiling, a tiling of the Euclidean plane See also *Tetracus * Tetrakis legomenon, a word that occurs only four times within a context * Tetricus (other) * Tetrix (other) *Truncated tetrakis cube The truncated tetrakis cube, or more precisely an order-6 truncated tetrakis cube or hexatruncated tetrakis cube, is a convex polyhedron with 32 faces: 24 sets of 3 bilateral symmetry pentagons arranged in an octahedral arrangement, with 8 regular ...

Tetrakis Hexahedron Stereographic D4 Gyrations
Tetrakis may refer to: * Tetrakis (Paphlagonia), an ancient Greek city * Tetrakis cuboctahedron, convex polyhedron with 32 triangular faces * Tetrakis hexahedron, an Archimedean dual solid or a Catalan solid *Tetrakis square tiling, a tiling of the Euclidean plane See also *Tetracus * Tetrakis legomenon, a word that occurs only four times within a context * Tetricus (other) * Tetrix (other) *Truncated tetrakis cube The truncated tetrakis cube, or more precisely an order-6 truncated tetrakis cube or hexatruncated tetrakis cube, is a convex polyhedron with 32 faces: 24 sets of 3 bilateral symmetry pentagons arranged in an octahedral arrangement, with 8 regular ...

Sphere Symmetry Group T
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres rolling, roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings. Basic terminology As mentioned earlier is the s ...
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Rhomb
In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a "diamond", after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (which some authors call a calisson after the French sweet – also see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle. Every rhombus is simple (non-self-intersecting), and is a special case of a parallelogram and a kite. A rhombus with right angles is a square. Etymology The word "rhombus" comes from grc, ῥόμβος, rhombos, meaning something that spins, which derives from the verb , romanized: , meaning "to turn round and round." The word was used both by Eucli ...
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Purple Fire
Purple is any of a variety of colors with hue between red and blue. In the RGB color model used in computer and television screens, purples are produced by mixing red and blue light. In the RYB color model historically used by painters, purples are created with a combination of red and blue pigments. In the CMYK color model used in printing, purples are made by combining magenta pigment with either cyan pigment, black pigment, or both. Purple has long been associated with royalty, originally because Tyrian purple dye, made from the mucus secretion of a species of snail, was extremely expensive in antiquity. Purple was the color worn by Roman magistrates; it became the imperial color worn by the rulers of the Byzantine Empire and the Holy Roman Empire, and later by Roman Catholic bishops. Similarly in Japan, the color is traditionally associated with the emperor and aristocracy. According to contemporary surveys in Europe and the United States, purple is the color most often ...
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Armed Forces Red Triangle
Armed (May, 1941–1964) was an American Thoroughbred gelding race horse who was the American Horse of the Year in 1947 and Champion Older Male Horse in both 1946 and 1947. He was inducted into the National Museum of Racing and Hall of Fame in 1963. Background Armed was sired by the great stakes winner Bull Lea, the sire of Citation. His dam was Armful, whose sire was Belmont Stakes winner Chance Shot and whose grandsire was the great Fair Play. Besides being small for his age and very headstrong, Armed had the habits of biting and kicking hay out of his handler's pitchfork. Since he was also practically untrainable, his trainer, Ben A. Jones, sent him back to Calumet Farm to be gelded and turned out to grow up. He returned to the track late in his two-year-old season and resumed training. Racing career His first start was as a three-year-old the following February, and he won at Hialeah Park by eight lengths. He won again less than a week later but then won only ...
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Tetrahedral Symmetry
150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation. The group of all (not necessarily orientation preserving) symmetries is isomorphic to the group S4, the symmetric group of permutations of four objects, since there is exactly one such symmetry for each permutation of the vertices of the tetrahedron. The set of orientation-preserving symmetries forms a group referred to as the alternating subgroup A4 of S4. Details Chiral and full (or achiral tetrahedral symmetry and pyritohedral symmetry) are discrete point symmetries (or equivalently, symmetries on the sphere). They are among the crystallographic point groups of the cubic crystal system. Seen in stereographic projection the edges of the tetrakis hexahedron form 6 circles (or centrally radial lines) in the plane. Ea ...
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Degree (graph Theory)
In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex's degree, for the two ends of the edge. The degree of a vertex v is denoted \deg(v) or \deg v. The maximum degree of a graph G, denoted by \Delta(G), and the minimum degree of a graph, denoted by \delta(G), are the maximum and minimum of its vertices' degrees. In the multigraph shown on the right, the maximum degree is 5 and the minimum degree is 0. In a regular graph, every vertex has the same degree, and so we can speak of ''the'' degree of the graph. A complete graph (denoted K_n, where n is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum possible degree, n-1. In a signed graph, the number of positive edges connected to the vertex v is called positive deg(v) and the number of connected negative edges is entitled negative deg(v). Handshaking lemma ...
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