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Compound Of Two Icosahedra
This uniform polyhedron compound is a composition of 2 icosahedra. It has octahedral symmetry ''Oh''. As a holosnub, it is represented by Schläfli symbol β and Coxeter diagram . The triangles in this compound decompose into two orbits under action of the symmetry group: 16 of the triangles lie in coplanar pairs in octahedral planes, while the other 24 lie in unique planes. It shares the same vertex arrangement as a nonuniform truncated octahedron, having irregular hexagons alternating with long and short edges. The icosahedron, as a uniform ''snub tetrahedron'', is similar to these snub-pair compounds: compound of two snub cubes and compound of two snub dodecahedra. Together with its convex hull, it represents the icosahedron-first projection of the nonuniform snub tetrahedral antiprism. Cartesian coordinates Cartesian coordinates for the vertices of this compound are all the permutations of : (±1, 0, ±τ) where τ = (1+)/2 is the golden ratio In mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Polyhedron
In geometry, a uniform polyhedron has regular polygons as Face (geometry), faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruence (geometry), congruent. Uniform polyhedra may be Regular polyhedron, regular (if also Isohedral figure, face- and Isotoxal figure, edge-transitive), Quasiregular polyhedron, quasi-regular (if also edge-transitive but not face-transitive), or Semiregular polyhedron, semi-regular (if neither edge- nor face-transitive). The faces and vertices don't need to be Convex polyhedron, convex, so many of the uniform polyhedra are also Star polyhedron, star polyhedra. There are two infinite classes of uniform polyhedra, together with 75 other polyhedra. They are 2 infinite classes of Prism (geometry), prisms and antiprisms, the convex polyhedrons as in 5 Platonic solids and 13 Archimedean solids—2 Quasiregular polyhedron, quasiregular and 11 Semiregular polyhedron, semiregular&m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Of Two Snub Dodecahedra
This uniform polyhedron compound is a composition of the 2 enantiomers of the snub dodecahedron. The vertex arrangement of this compound is shared by a convex nonuniform truncated icosidodecahedron, with rectangular faces, alongside irregular hexagons and decagons, each alternating two different edge lengths. Together with its convex hull, it represents the snub dodecahedron-first projection of the nonuniform snub dodecahedral antiprism. See also *Compound of two icosahedra *Compound of two snub cubes This uniform polyhedron compound is a composition of the 2 enantiomers of the snub cube. As a holosnub, it is represented by Schläfli symbol βr and Coxeter diagram . The vertex arrangement of this compound is shared by a convex nonuniform t ... References *. Polyhedral compounds {{polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Of Two Snub Cubes
This uniform polyhedron compound is a composition of the 2 enantiomers of the snub cube. As a holosnub, it is represented by Schläfli symbol βr and Coxeter diagram . The vertex arrangement of this compound is shared by a convex nonuniform truncated cuboctahedron, having rectangular faces, alongside irregular hexagons and octagons, each alternating with two edge lengths. Together with its convex hull, it represents the snub cube-first projection of the nonuniform snub cubic antiprism. Cartesian coordinates Cartesian coordinates for the vertices are all the permutations of :(±1, ±''ξ'', ±1/''ξ'') where ''ξ'' is the real solution to :\xi^3+\xi^2+\xi=1, \, which can be written :\xi = \frac\left(\sqrt - \sqrt - 1\right) or approximately 0.543689. ξ is the reciprocal of the tribonacci constant. Equally, the tribonacci constant, ''t'', just like the snub cube, can compute the coordinates as the permutations of: :(±1, ±, ±''t'') Truncated cuboctahedron This comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Pyritohedron And Dual
Compound may refer to: Architecture and built environments * Compound (enclosure), a cluster of buildings having a shared purpose, usually inside a fence or wall ** Compound (fortification), a version of the above fortified with defensive structures * Compound (migrant labour), a hostel for migrant workers such as those historically connected with mines in South Africa * The Compound, an area of Palm Bay, Florida, US * Komboni or compound, a type of slum in Zambia Government and law * Composition (fine), a legal procedure in use after the English Civil War ** Committee for Compounding with Delinquents, an English Civil War institution that allowed Parliament to compound the estates of Royalists * Compounding treason, an offence under the common law of England * Compounding a felony, a previous offense under the common law of England Linguistics * Compound (linguistics), a word that consists of more than one radical element * Compound sentence (linguistics), a type of sentence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pyritohedron
In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, 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. Some dodecahedra have the same combinatorial structure as the regular dodecahedron (in terms of the graph formed by its vertices and edges), but their pentagonal faces are not regular: The pyritohedron, a common crystal form in pyrite, has pyritohedral symmetry, while the tetartoid has tetrahedral symmetry. The rhombic dodecahedron can be seen as a limiting case of the pyritohedron, and it has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra, are space-filling. There are numerous other dodecahedra. While the regular dodecahedron sha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golden Ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \frac = \frac = \varphi, where the Greek letter Phi (letter), phi ( or ) denotes the golden ratio. The constant satisfies the quadratic equation and is an irrational number with a value of The golden ratio was called the extreme and mean ratio by Euclid, and the divine proportion by Luca Pacioli; it also goes by other names. Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the Straightedge and compass construction, construction of the dodecahedron and icosahedron. A golden rectangle—that is, a rectangle with an aspect ratio of —may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has bee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Coordinates
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular oriented lines, called '' coordinate lines'', ''coordinate axes'' or just ''axes'' (plural of ''axis'') of the system. The point where the axes meet is called the '' origin'' and has as coordinates. The axes directions represent an orthogonal basis. The combination of origin and basis forms a coordinate frame called the Cartesian frame. Similarly, the position of any point in three-dimensional space can be specified by three ''Cartesian coordinates'', which are the signed distances from the point to three mutually perpendicular planes. More generally, Cartesian coordinates specify the point in an -dimensional Euclidean space for any dimension . These coordinates are the signed distances from the point to mutually perpendicular fixed h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated Octahedral Prism
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedron, uniform polyhedra, and faces are regular polygons. There are 47 non-Prism (geometry), prismatic Convex polytope, convex uniform 4-polytopes. There are two infinite sets of convex prismatic forms, along with 17 cases arising as prisms of the convex uniform polyhedra. There are also an unknown number of non-convex star forms. History of discovery * Convex Regular polytopes: ** 1852: Ludwig Schläfli proved in his manuscript ''Theorie der vielfachen Kontinuität'' that there are exactly 6 regular polytopes in 4 dimensions and only 3 in 5 or more dimensions. * Schläfli-Hess polychoron, Regular star 4-polytopes (star polyhedron cells and/or vertex figures) ** 1852: Ludwig Schläfli also found 4 of the 10 regular star 4-polytopes, discounting 6 with cells or vertex figures small stellated dodecahedron, and great dodecahedron, . ** ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Of Two Snub Dodecahedra
This uniform polyhedron compound is a composition of the 2 enantiomers of the snub dodecahedron. The vertex arrangement of this compound is shared by a convex nonuniform truncated icosidodecahedron, with rectangular faces, alongside irregular hexagons and decagons, each alternating two different edge lengths. Together with its convex hull, it represents the snub dodecahedron-first projection of the nonuniform snub dodecahedral antiprism. See also *Compound of two icosahedra *Compound of two snub cubes This uniform polyhedron compound is a composition of the 2 enantiomers of the snub cube. As a holosnub, it is represented by Schläfli symbol βr and Coxeter diagram . The vertex arrangement of this compound is shared by a convex nonuniform t ... References *. Polyhedral compounds {{polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Of Two Snub Cubes
This uniform polyhedron compound is a composition of the 2 enantiomers of the snub cube. As a holosnub, it is represented by Schläfli symbol βr and Coxeter diagram . The vertex arrangement of this compound is shared by a convex nonuniform truncated cuboctahedron, having rectangular faces, alongside irregular hexagons and octagons, each alternating with two edge lengths. Together with its convex hull, it represents the snub cube-first projection of the nonuniform snub cubic antiprism. Cartesian coordinates Cartesian coordinates for the vertices are all the permutations of :(±1, ±''ξ'', ±1/''ξ'') where ''ξ'' is the real solution to :\xi^3+\xi^2+\xi=1, \, which can be written :\xi = \frac\left(\sqrt - \sqrt - 1\right) or approximately 0.543689. ξ is the reciprocal of the tribonacci constant. Equally, the tribonacci constant, ''t'', just like the snub cube, can compute the coordinates as the permutations of: :(±1, ±, ±''t'') Truncated cuboctahedron This comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Snub Tetrahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical than others. The best known is the (convex, non- stellated) regular icosahedron—one of the Platonic solids—whose faces are 20 equilateral triangles. Regular icosahedra There are two objects, one convex and one nonconvex, that can both be called regular icosahedra. Each has 30 edges and 20 equilateral triangle faces with five meeting at each of its twelve vertices. Both have icosahedral symmetry. The term "regular icosahedron" generally refers to the convex variety, while the nonconvex form is called a ''great icosahedron''. Convex regular icosahedron The convex regular icosahedron is usually referred to simply as the ''regular icosahedron'', one of the five regular Platonic solids, and is represented by its Schläfli symbol , containi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
