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Compound Of Small Stellated Dodecahedron And Great Dodecahedron
In geometry, the great dodecahedron is one of four Kepler–Poinsot polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagons), intersecting each other making a pentagrammic path, with five pentagons meeting at each vertex. Construction One way to construct a great dodecahedron is by faceting the regular icosahedron. In other words, it is constructed from the regular icosahedron by removing its polygonal faces without changing or creating new vertices. For each vertex of the icosahedron, the five neighboring vertices become those of a regular pentagon face of the great dodecahedron. The resulting shape has a pentagram as its vertex figure, so its Schläfli symbol is \ . The great dodecahedron may also be interpreted as the ''second stellation of dodecahedron''. The construction started from a regular dodecahedron by attaching 12 pentagonal pyramids onto each of its faces, known as the ''first stellation''. The second stellation appears when 30 wedges ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kepler–Poinsot Polyhedron
In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra. They may be obtained by stellating the regular convex dodecahedron and icosahedron, and differ from these in having regular pentagrammic faces or vertex figures. They can all be seen as three-dimensional analogues of the pentagram in one way or another. Characteristics Sizes The great icosahedron edge length is \phi^4 = \tfrac12\bigl(7+3\sqrt5\,\bigr) times the original icosahedron edge length. The small stellated dodecahedron, great dodecahedron, and great stellated dodecahedron edge lengths are respectively \phi^3 = 2+\sqrt5, \phi^2 = \tfrac12\bigl(3+\sqrt5\,\bigr), and \phi^5 = \tfrac12\bigl(11+5\sqrt5\,\bigr) times the original dodecahedron edge length. Non-convexity These figures have pentagrams (star pentagons) as faces or vertex figures. The small and great stellated dodecahedron have nonconvex regular pentagram faces. The great dodecahedron and great icosahedron have convex polyg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dodecadodecahedron
In geometry, the dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U36. It is the Rectification (geometry), rectification of the great dodecahedron (and that of its dual, the small stellated dodecahedron). It was discovered independently by , and . The edges of this model form 10 central hexagons, and these, projected onto a sphere, become 10 great circles. These 10, along with the great circles from projections of two other polyhedra, form the 31 great circles of the spherical icosahedron used in construction of geodesic domes. Wythoff constructions It has four Wythoff constructions between four Schwarz triangle families: 2 , 5 5/2, 2 , 5 5/3, 2 , 5/2 5/4, 2 , 5/3 5/4, but represent identical results. Similarly it can be given four extended Schläfli symbols: r, r, r, and r or as Coxeter-Dynkin diagrams: , , , and . Net A shape with the same exterior appearance as the dodecadodecahedron can be constructed by folding up these nets: 12 pentagrams and 20 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonconvex Uniform Polyhedron
In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, or both. The complete set of 57 nonprismatic uniform star polyhedra includes the 4 regular ones, called the Kepler–Poinsot polyhedron, Kepler–Poinsot polyhedra, 14 Quasiregular polyhedron#Nonconvex examples, quasiregular ones, and 39 semiregular ones. There are also two infinite sets of Uniform_polyhedron#.28p_2_2.29_Prismatic_.5Bp.2C2.5D.2C_I2.28p.29_family_.28Dph_dihedral_symmetry.29, ''uniform star prisms'' and ''uniform star antiprisms''. Just as (nondegenerate) star polygons (which have density (polytope), polygon density greater than 1) correspond to circular polygons with overlapping Tessellation, tiles, star polyhedra that do not pass through the center have polytope density greater than 1, and correspond to spherical polyhe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncation (geometry)
In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new Facet (geometry), facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids. Uniform truncation In general any polyhedron (or polytope) 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, a truncation operator applied to a regular polyhedron (or regular polytope) which creates a resulting uniform polyhedron (uniform polytope) with equal edge lengths. 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 \begin 5 \\ 3 \end, and Coxeter-Dynkin diagram or has a uniform truncation, the truncate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polytope Compound
In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram. The outer vertices of a compound can be connected to form a convex polyhedron called its convex hull. A compound is a faceting of its convex hull. Another convex polyhedron is formed by the small central space common to all members of the compound. This polyhedron can be used as the core for a set of stellations. Regular compounds A regular polyhedral compound can be defined as a compound which, like a regular polyhedron, is vertex-transitive, edge-transitive, and face-transitive. Unlike the case of polyhedra, this is not equivalent to the symmetry group acting transitively on its flags; the compound of two tetrahedra is the only regular compound with that property. There are five regular compounds of polyhedra: Best known is the regular compound of two tetrahedra, often called t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decagram (geometry)
In geometry, a decagram is a 10-point star polygon. There is one regular decagram, containing the vertices of a regular decagon, but connected by every third point. Its Schläfli symbol is . The name ''decagram'' combines a numeral prefix, ''wikt:deca-, deca-'', with the Greek language, Greek suffix ''wikt:-gram, -gram''. The ''-gram'' suffix derives from ''γραμμῆς'' (''grammēs'') meaning a line. Regular decagram For a regular decagram with unit edge lengths, the proportions of the crossing points on each edge are as shown below. Applications Decagrams have been used as one of the decorative motifs in girih tiles. : Isotoxal variations An isotoxal polygon has two vertices and one edge. There are isotoxal decagram forms, which alternates vertices at two radii. Each form has a freedom of one angle. The first is a variation of a double-wound of a pentagon , and last is a variation of a double-wound of a pentagram . The middle is a variation of a regular decagram, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Polyhedron
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all can also be constructed as 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 many classes of polyhedra defined by their symmetries, the duals belong to a corresponding symmetry class. For example, the regular polyhedrathe (convex) Platonic solids and (star) Kepler–Poinsot polyhedraform dual pairs, where the regular tetrahedron is self-dual. The dual of an isogonal polyhedron (one in which any two vertices are equivalent under symmetries of the polyhedron) is an isohedral polyhedron (one in which any two faces are equivalent .., and vice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedron Compound
In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram. The outer vertices of a compound can be connected to form a convex polyhedron called its convex hull. A compound is a faceting of its convex hull. Another convex polyhedron is formed by the small central space common to all members of the compound. This polyhedron can be used as the core for a set of stellations. Regular compounds A regular polyhedral compound can be defined as a compound which, like a regular polyhedron, is vertex-transitive, edge-transitive, and face-transitive. Unlike the case of polyhedra, this is not equivalent to the symmetry group acting transitively on its flags; the compound of two tetrahedra is the only regular compound with that property. There are five regular compounds of polyhedra: Best known is the regular compound of two tetrahedra, often called t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twisty Puzzle
Twist may refer to: In arts and entertainment Film, television, and stage * ''Twist'' (2003 film), a 2003 independent film loosely based on Charles Dickens's novel ''Oliver Twist'' * ''Twist'' (2021 film), a 2021 modern rendition of ''Oliver Twist'' starring Rafferty Law * ''The Twist'' (1976 film), a 1976 film co-written and directed by Claude Chabrol * ''The Twist'' (1992 film), a 1992 documentary film directed by Ron Mann * ''Twist'' (stage play), a 1995 stage thriller by Miles Tredinnick * Twist, the main character on television series '' The Fresh Beat Band'' and its spin-off ''Fresh Beat Band of Spies'' * Oliver Twist (other), name of several film, television, and musical adaptations based on Charles Dickens's novel ''Oliver Twist'' * "Twist" (''Only Murders in the Building''), a 2021 episode of the TV series ''Only Murders in the Building'' * Jack Twist, a character in the 2005 film ''Brokeback Mountain'' * Twist Morgan, a character in the television seri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander's Star
Alexander's Star is a puzzle similar to the Rubik's Cube, in the shape of a great dodecahedron. History Alexander's Star was invented by Adam Alexander, an American mathematician, in 1982. It was patented on 26 March 1985, with US patent number 4,506,891, and sold by the Ideal Toy Company. It came in two varieties: painted surfaces or stickers. Since the design of the puzzle practically forces the stickers to peel with continual use, the painted variety is likely a later edition. Description The puzzle has 30 moving pieces, which rotate in star-shaped groups of five around its outermost vertices. The purpose of the puzzle is to rearrange the moving pieces so that each star is surrounded by five faces of the same color, and opposite stars are surrounded by the same color. This is equivalent to solving just the edges of a six-color Megaminx. The puzzle is solved when each pair of parallel planes is made up of only one colour. To see a plane, however, one has to look "past ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wenzel Jamnitzer
Wenzel Jamnitzer (sometimes Jamitzer, or Wenzel ''Gemniczer'') (1507/1508 – 19 December 1585) was a Northern Mannerist goldsmith, artist, and printmaker in etching, who worked in Nuremberg. He was the best known German goldsmith of his era, and court goldsmith to a succession of Holy Roman Emperors. A native of Vienna, Jamnitzer was a member of a Moravian German family which, for more than 160 years, had produced works under the names ''Jamnitzer, Jemniczer, Gemniczer, and Jamitzer''. Wenzel, with his brother Albrecht, was trained by his father Hans the Elder. Later, Wenzel's son Hans Jamnitzer (1539–1603) and grandson Christof Jamnitzer (1563–1618) continued his business. Jamnitzer worked as a court goldsmith for all the German emperors of his era, including Charles V, Ferdinand I, Maximilian II, and Rudolf II. Also, he probably invented an embossing machine. In 1534, Jamnitzer settled in Nuremberg. He made vases and jewelry boxes with great skill, in a sty ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
