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Polyhedral Compounds
Polyhedral may refer to: *Dihedral (other), various meanings *Polyhedral compound * Polyhedral combinatorics * Polyhedral cone * Polyhedral cylinder * Polyhedral convex function * Polyhedral dice * Polyhedral dual * Polyhedral formula *Polyhedral graph *Polyhedral group *Polyhedral model *Polyhedral net *Polyhedral number *Polyhedral pyramid *Polyhedral prism *Polyhedral space *Polyhedral skeletal electron pair theory *Polyhedral symbol *Polyhedral symmetry *Polyhedral terrain See also * Polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ..., a geometric shape {{disamb Polyhedra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dihedral (other) , a geometric shape
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Dihedral or polyhedral may refer to: * Dihedral angle, the angle between two mathematical planes * Dihedral (aeronautics), the upward angle of a fixed-wing aircraft's wings where they meet at the fuselage, dihedral ''effect'' of an aircraft, longitudinal dihedral angle of a fixed-wing aircraft * Dihedral group, the group of symmetries of the ''n''-sided polygon in abstract algebra ** Also Dihedral symmetry in three dimensions * Dihedral kite, also known as a bowed kite * Dihedral doors, also known as butterfly doors * Dihedral prime, also known as a dihedral calculator prime * In rock climbing, an inside corner of rock See also * Anhedral (other) * Euhedral, a crystal structure * Polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedral Net
In geometry, a net of a polyhedron is an arrangement of non-overlapping edge-joined polygons in the plane which can be folded (along edges) to become the faces of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and solid geometry in general, as they allow for physical models of polyhedra to be constructed from material such as thin cardboard. An early instance of polyhedral nets appears in the works of Albrecht Dürer, whose 1525 book ''A Course in the Art of Measurement with Compass and Ruler'' (''Unterweysung der Messung mit dem Zyrkel und Rychtscheyd '') included nets for the Platonic solids and several of the Archimedean solids. These constructions were first called nets in 1543 by Augustin Hirschvogel. Existence and uniqueness Many different nets can exist for a given polyhedron, depending on the choices of which edges are joined and which are separated. The edges that are cut from a convex polyhedron to form a net must form a spanning tree o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedral Terrain
In computational geometry, a polyhedral terrain in three-dimensional Euclidean space is a polyhedral surface that intersects every line parallel to some particular line in a connected set (i.e., a point or a line segment) or the empty set. Without loss of generality, we may assume that the line in question is the ''z''-axis of the Cartesian coordinate system. Then a polyhedral terrain is the image of a piecewise-linear function in ''x'' and ''y'' variables. The polyhedral terrain is a generalization of the two-dimensional geometric object, the monotone polygonal chain. As the name may suggest, a major application area of polyhedral terrains include geographic information systems to model real-world terrain Terrain or relief (also topographical relief) involves the vertical and horizontal dimensions of land surface. The term bathymetry is used to describe underwater relief, while hypsometry studies terrain relative to sea level. The Latin wo ...s. Representation A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedral Symmetry
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedral Symbol
The polyhedral symbol is sometimes used in coordination chemistry A coordination complex consists of a central atom or ion, which is usually metallic and is called the ''coordination centre'', and a surrounding array of bound molecules or ions, that are in turn known as ''ligands'' or complexing agents. Many ... to indicate the approximate geometry of the coordinating atoms around the central atom. One or more italicised letters indicate the geometry, e.g. ''TP''-3 which is followed by a number that gives the coordination number of the central atom.NOMENCLATURE OF INORGANIC CHEMISTRY IUPAC Recommendations 2005 ed. N. G. Connelly et al. RSC Publishing https://iupac.org/wp-content/uploads/2016/07/Red_Book_2005.pdf The polyhedral symbol can be used in naming of compounds, in which case it is followed by the configuration index. Polyhedral symbols Configuration index The first step in determining the configuration index is to assign a priority number to each coordinating ligand ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedral Skeletal Electron Pair Theory
In chemistry the polyhedral skeletal electron pair theory (PSEPT) provides electron counting rules useful for predicting the structures of clusters such as borane and carborane clusters. The electron counting rules were originally formulated by Kenneth Wade, and were further developed by others including Michael Mingos; they are sometimes known as Wade's rules or the Wade–Mingos rules. The rules are based on a molecular orbital treatment of the bonding. These notes contained original material that served as the basis of the sections on the 4''n'', 5''n'', and 6''n'' rules. These rules have been extended and unified in the form of the Jemmis ''mno'' rules. Predicting structures of cluster compounds Different rules (4''n'', 5''n'', or 6''n'') are invoked depending on the number of electrons per vertex. The 4''n'' rules are reasonably accurate in predicting the structures of clusters having about 4 electrons per vertex, as is the case for many boranes and carboranes. For such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedral Space
Polyhedral space is a certain metric space. A ( Euclidean) polyhedral space is a (usually finite) simplicial complex in which every simplex has a flat metric. (Other spaces of interest are spherical and hyperbolic polyhedral spaces, where every simplex has a metric of constant positive or negative curvature). In the sequel all polyhedral spaces are taken to be Euclidean polyhedral spaces. Examples All 1-dimensional polyhedral spaces are just metric graphs. A good source of 2-dimensional examples constitute triangulations of 2-dimensional surfaces. The surface of a convex polyhedron in R^3 is a 2-dimensional polyhedral space. Any PL-manifold (which is essentially the same as a simplicial manifold, just with some technical assumptions for convenience) is an example of a polyhedral space. In fact, one can consider pseudomanifolds, although it makes more sense to restrict the attention to normal manifolds. Metric singularities In the study of polyhedral spaces (particularly of those ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedral Prism
In geometry, a prism is a polyhedron comprising an polygon base, a second base which is a translated copy (rigidly moved without rotation) of the first, and other faces, necessarily all parallelograms, joining corresponding sides of the two bases. All cross-sections parallel to the bases are translations of the bases. Prisms are named after their bases, e.g. a prism with a pentagonal base is called a pentagonal prism. Prisms are a subclass of prismatoids. Like many basic geometric terms, the word ''prism'' () was first used in Euclid's Elements. Euclid defined the term in Book XI as “a solid figure contained by two opposite, equal and parallel planes, while the rest are parallelograms”. However, this definition has been criticized for not being specific enough in relation to the nature of the bases, which caused confusion among later geometry writers. Oblique prism An oblique prism is a prism in which the joining edges and faces are ''not perpendicular'' to the base f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedral Pyramid
In geometry, a pyramid () is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a ''lateral face''. It is a conic solid with polygonal base. A pyramid with an base has vertices, faces, and edges. All pyramids are self-dual. A right pyramid has its apex directly above the centroid of its base. Nonright pyramids are called oblique pyramids. A regular pyramid has a regular polygon base and is usually implied to be a ''right pyramid''. When unspecified, a pyramid is usually assumed to be a ''regular'' square pyramid, like the physical pyramid structures. A triangle-based pyramid is more often called a tetrahedron. Among oblique pyramids, like acute and obtuse triangles, a pyramid can be called ''acute'' if its apex is above the interior of the base and ''obtuse'' if its apex is above the exterior of the base. A right-angled pyramid has its apex above an edge or vertex of the base. In a tetrahed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedral Number
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean * polygonal number * a number represented as a discrete -dimensional regular geometric pattern of -dimensional balls such as a polygonal number (for ) or a polyhedral number (for ). * a member of the subset of the sets above containing only triangular numbers, pyramidal numbers, and their analogs in other dimensions. Terminology Some kinds of figurate number were discussed in the 16th and 17th centuries under the name "figural number". In historical works about Greek mathematics the preferred term used to be ''figured number''. In a use going back to Jacob Bernoulli's Ars Conjectandi, the term ''figurate number'' is used for triangular numbers made up of successive integers, tetrahedral numbers made up of successive triangular numbers, etc. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedral Model
The polyhedral model (also called the polytope method) is a mathematical framework for programs that perform large numbers of operations -- too large to be explicitly enumerated -- thereby requiring a ''compact'' representation. Nested loop programs are the typical, but not the only example, and the most common use of the model is for loop nest optimization in program optimization. The polyhedral method treats each loop iteration within nested loops as lattice points inside mathematical objects called polyhedra, performs affine transformations or more general non-affine transformations such as tiling on the polytopes, and then converts the transformed polytopes into equivalent, but optimized (depending on targeted optimization goal), loop nests through polyhedra scanning. Simple example Consider the following example written in C: const int n = 100; int i, j, a n]; for (i = 1; i < n; i++) The essential problem with this code is that each iteration of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedral 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 facetting 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 calle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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