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Compound Of Two Great Icosahedra
The compound of two great icosahedra is a uniform polyhedron compound. It's composed of 2 great icosahedra, in the same arrangement as in the compound of 2 icosahedra. 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. The great icosahedron, as a uniform ''retrosnub tetrahedron'' , is similar to these snub-pair compounds: compound of two icosahedra, compound of two snub cubes and 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 he .... References *. External links * VRML model Polyhedral compounds {{polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
UC52-2 Great Icosahedra
SM ''UC-5'' was a German Type UC I minelayer submarine or U-boat in the German Imperial Navy (german: Kaiserliche Marine) during World War I. The U-boat had been ordered by November 1914 and was launched on 13 June 1915. She was commissioned into the German Imperial Navy on 19 June 1915 as SM ''UC-5''."SM" stands for "Seiner Majestät" (English "His Majesty's") and combined with the ''U'' for ''Unterseeboot'' would be translated as "His Majesty's Submarine". She served in World War I under the command of Herbert Pustkuchen (June - December 1915) and Ulrich Mohrbutter (December 1915 - April 1916). She ran aground and was abandoned but recovered by the Allies and displayed for propaganda purposes. Design A German Type UC I submarine, ''UC-5'' had a displacement of when at the surface and while submerged. She had a length overall of , a beam of , and a draught of . The submarine was powered by one Daimler-Motoren-Gesellschaft six-cylinder, four-stroke diesel engine producin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 (sometimes written ... [...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 decagon In geometry, a decagon (from the Greek δέκα ''déka'' and γωνία ''gonía,'' "ten angles") is a ten-sided polygon or 10-gon.. The total sum of the interior angles of a simple decagon is 1440°. A self-intersecting ''regular decagon'' i ...s, each alternating two different edge lengths. Together with its convex hull, it represents the snub dodecahedron-first projection of the nonuniform snub dodecahedral antiprism. 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: :(±1, ±''t'', ±) Truncated cuboctahedron This compound can be seen as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 (sometimes written ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Retrosnub Tetrahedron
In geometry, the great icosahedron is one of four Kepler–Poinsot polyhedra (nonconvex regular polyhedra), with Schläfli symbol and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence. The great icosahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the -dimensional simplex faces of the core -polytope (equilateral triangles for the great icosahedron, and line segments for the pentagram) until the figure regains regular faces. The grand 600-cell can be seen as its four-dimensional analogue using the same process. Images As a snub The ''great icosahedron'' can be constructed a uniform snub, with different colored faces and only tetrahedral symmetry: . This construction can be called a ''retrosnub tetrahedron'' or ''retrosnub tetratetrahedron'', similar to the snub tetrahedron symmetry of the icosahedron, as a p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. A regular octahedron is the dual polyhedron of a cube. It is a rectified tetrahedron. It is a square bipyramid in any of three orthogonal orientations. It is also a triangular antiprism in any of four orientations. An octahedron is the three-dimensional case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan () metric. Regular octahedron Dimensions If the edge length of a regular octahedron is ''a'', the radius of a circumscribed sphere (one that touches the octahedron at all vertices) is :r_u = \frac a \approx 0.707 \cdot a and the radius of an inscribed sphere (tangent to each of the octahedron's faces) is :r_i = \frac a \approx 0.408\cdot a while the midradius, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbit (group Theory)
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group ''acts'' on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. For example, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron. A group action on a vector space is called a representation of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of , the group of the invertible matrices of dimension over a field . The symmetric group acts on any set with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Polyhedron Compound
In geometry, a uniform polyhedron compound is a polyhedral compound whose constituents are identical (although possibly enantiomorphous) uniform polyhedra, in an arrangement that is also uniform, i.e. the symmetry group of the compound acts transitively on the compound's vertices. The uniform polyhedron compounds were first enumerated by John Skilling in 1976, with a proof that the enumeration is complete. The following table lists them according to his numbering. The prismatic compounds of prisms ( UC20 and UC21) exist only when , and when and are coprime. The prismatic compounds of antiprisms ( UC22, UC23, UC24 and UC25) exist only when , and when and are coprime. Furthermore, when , the antiprisms degenerate into tetrahedra with digon In geometry, a digon is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Polyhedron Compound
In geometry, a uniform polyhedron compound is a polyhedral compound whose constituents are identical (although possibly enantiomorphous) uniform polyhedra, in an arrangement that is also uniform, i.e. the symmetry group of the compound acts transitively on the compound's vertices. The uniform polyhedron compounds were first enumerated by John Skilling in 1976, with a proof that the enumeration is complete. The following table lists them according to his numbering. The prismatic compounds of prisms ( UC20 and UC21) exist only when , and when and are coprime. The prismatic compounds of antiprisms ( UC22, UC23, UC24 and UC25) exist only when , and when and are coprime. Furthermore, when , the antiprisms degenerate into tetrahedra with digon In geometry, a digon is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Of Two Great Icosahedra
The compound of two great icosahedra is a uniform polyhedron compound. It's composed of 2 great icosahedra, in the same arrangement as in the compound of 2 icosahedra. 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. The great icosahedron, as a uniform ''retrosnub tetrahedron'' , is similar to these snub-pair compounds: compound of two icosahedra, compound of two snub cubes and 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 he .... References *. External links * VRML model Polyhedral compounds {{polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
