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Compound Of Twenty Octahedra
The compound of twenty octahedra is a uniform polyhedron compound. It's composed of a symmetric arrangement of 20 octahedra (considered as triangular antiprisms). It is a special case of the compound of 20 octahedra with rotational freedom, in which pairs of octahedral vertices coincide. Related polyhedra This compound shares its edge arrangement with the great dirhombicosidodecahedron, the great disnub dirhombidodecahedron, and the compound of twenty tetrahemihexahedra. It may be constructed as the exclusive or of the two enantiomorphs of the great snub dodecicosidodecahedron. See also *Compound of three octahedra * Compound of four octahedra *Compound of five octahedra The compound of five octahedra is one of the five regular polyhedron compounds. This polyhedron can be seen as either a polyhedral stellation or a compound. This compound was first described by Edmund Hess in 1876. It is unique among the regula ... * Compound of ten octahedra References *. Pol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exclusive Or
Exclusive or or exclusive disjunction is a logical operation that is true if and only if its arguments differ (one is true, the other is false). It is symbolized by the prefix operator J and by the infix operators XOR ( or ), EOR, EXOR, , , , , , and . The negation of XOR is the logical biconditional, which yields true if and only if the two inputs are the same. It gains the name "exclusive or" because the meaning of "or" is ambiguous when both operands are true; the exclusive or operator ''excludes'' that case. This is sometimes thought of as "one or the other but not both". This could be written as "A or B, but not, A and B". Since it is associative, it may be considered to be an ''n''-ary operator which is true if and only if an odd number of arguments are true. That is, ''a'' XOR ''b'' XOR ... may be treated as XOR(''a'',''b'',...). Truth table The truth table of A XOR B shows that it outputs true whenever the inputs differ: Equivalences, elimination, and introdu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Of Five Octahedra
The compound of five octahedra is one of the five regular polyhedron compounds. This polyhedron can be seen as either a polyhedral stellation or a compound. This compound was first described by Edmund Hess in 1876. It is unique among the regular compounds for not having a regular convex hull. As a stellation It is the second stellation of the icosahedron, and given as Wenninger model index 23. It can be constructed by a rhombic triacontahedron with rhombic-based pyramids added to all the faces, as shown by the five colored model image. (This construction does not generate the ''regular'' compound of five octahedra, but shares the same topology and can be smoothly deformed into the regular compound.) It has a density of greater than 1. As a compound It can also be seen as a polyhedral compound of five octahedra arranged in icosahedral symmetry (Ih). The spherical and stereographic projections of this compound look the same as those of the disdyakis triacontahedron. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Of Four Octahedra
The compound of four octahedra is a uniform polyhedron compound. It's composed of a symmetric arrangement of 4 octahedron, octahedra, considered as triangular antiprisms. It can be constructed by superimposing four identical octahedra, and then rotating each by 60 degrees about a separate axis (that passes through the centres of two opposite octahedral faces). Its dual is the compound of four cubes. Cartesian coordinates Cartesian coordinates for the vertices of this compound are all the permutations of : (±2, ±1, ±2) See also *Compound of three octahedra *Compound of five octahedra *Compound of ten octahedra *Compound of twenty octahedra *Compound of four cubes References *. Polyhedral compounds {{polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Of Three Octahedra
In mathematics, the compound of three octahedra or octahedron 3-compound is a polyhedral compound formed from three regular octahedra, all sharing a common center but rotated with respect to each other. Although appearing earlier in the mathematical literature, it was rediscovered and popularized by M. C. Escher, who used it in the central image of his 1948 woodcut ''Stars''. Construction A regular octahedron can be circumscribed around a cube in such a way that the eight edges of two opposite squares of the cube lie on the eight faces of the octahedron. The three octahedra formed in this way from the three pairs of opposite cube squares form the compound of three octahedra.. The eight cube vertices are the same as the eight points in the compound where three edges cross each other. Each of the octahedron edges that participates in these triple crossings is divided by the crossing point in the ratio 1: . The remaining octahedron edges cross each other in pairs, within the interior ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Of Twenty Tetrahemihexahedra
This uniform polyhedron compound is a symmetric arrangement of 20 tetrahemihexahedra. It is chiral with icosahedral symmetry (I). John Skilling notes, in his enumeration of uniform compounds of uniform polyhedra, that this compound of 20 tetrahemihexahedra is unique in that it cannot be obtained by "adding symmetry to a group in which the basic polyhedron is uniform". Each tetrahemihexahedron in this compound is embedded with symmetry group C3, which does not act transitively over the tetrahemihexahedron's six vertices. However, the compound as a whole can achieve uniformity because two tetrahemihexahedra coincide at each vertex. Related polyhedra This compound shares its edge arrangement with the great dirhombicosidodecahedron, the great disnub dirhombidodecahedron, and the compound of 20 octahedra. The edges and 20 of the triangular faces occur in one enantiomer of the great snub dodecicosidodecahedron In geometry, the great snub dodecicosidodecahedron (or great snub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Disnub Dirhombidodecahedron
In geometry, the great disnub dirhombidodecahedron, also called ''Skilling's figure'', is a degenerate uniform star polyhedron. It was proven in 1970 that there are only 75 uniform polyhedra other than the infinite families of prisms and antiprisms. John Skilling discovered another degenerate example, the great disnub dirhombidodecahedron, by relaxing the condition that edges must be single. More precisely, he allowed any even amount of faces to meet at each edge, as long as the set of faces couldn't be separated into two connected sets (Skilling, 1975). Due to its geometric realization having some double edges where 4 faces meet, it is considered a degenerate uniform polyhedron but not strictly a uniform polyhedron. The number of edges is ambiguous, because the underlying abstract polyhedron has 360 edges, but 120 pairs of these have the same image in the geometric realization, so that the geometric realization has 120 single edges and 120 double edges where 4 faces meet, for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Dirhombicosidodecahedron
In geometry, the great dirhombicosidodecahedron (or great snub disicosidisdodecahedron) is a nonconvex uniform polyhedron, indexed last as . It has 124 faces (40 triangles, 60 squares, and 24 pentagrams), 240 edges, and 60 vertices. This is the only non-degenerate uniform polyhedron with more than six faces meeting at a vertex. Each vertex has 4 squares which pass through the vertex central axis (and thus through the centre of the figure), alternating with two triangles and two pentagrams. Another unusual feature is that the faces all occur in coplanar pairs. This is also the only uniform polyhedron that cannot be made by the Wythoff construction from a spherical triangle. It has a special Wythoff symbol relating it to a spherical quadrilateral. This symbol suggests that it is a sort of snub polyhedron, except that instead of the non-snub faces being surrounded by snub triangles as in most snub polyhedra, they are surrounded by snub squares. It has been nicknamed "Miller ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Snub Dodecicosidodecahedron
In geometry, the great snub dodecicosidodecahedron (or great snub dodekicosidodecahedron) is a nonconvex uniform polyhedron, indexed as U64. It has 104 faces (80 triangles and 24 pentagrams), 180 edges, and 60 vertices. It has Coxeter diagram, . It has the unusual feature that its 24 pentagram faces occur in 12 coplanar pairs. Related polyhedra It shares its vertices and edges, as well as 20 of its triangular faces and all its pentagrammic faces, with the great dirhombicosidodecahedron, (although the latter has 60 edges not contained in the great snub dodecicosidodecahedron). It shares its other 60 triangular faces (and its pentagrammic faces again) with the great disnub dirhombidodecahedron. The edges and triangular faces also occur in the compound of twenty octahedra. In addition, 20 of the triangular faces occur in one enantiomer of the compound of twenty tetrahemihexahedra, and the other 60 triangular faces occur in the other enantiomer. Gallery See also * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset. Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets of points. The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its dual problem of intersecting half-spaces, are fundamental problems of co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
