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Gyroelongated Square Dipyramid
In geometry, the gyroelongated square bipyramid is a polyhedron with 16 triangular faces. it can be constructed from a square antiprism by attaching two equilateral square pyramids to each of its square faces. The same shape is also called hexakaidecadeltahedron, heccaidecadeltahedron, or tetrakis square antiprism; these last names mean a polyhedron with 16 triangular faces. It is an example of deltahedron, and of a Johnson solid. The dual polyhedron of the gyroelongated square bipyramid is a square truncated trapezohedron with eight pentagons and two squares as its faces. The gyroelongated square pyramid appears in chemistry as the basis for the bicapped square antiprismatic molecular geometry, and in mathematical optimization as a solution to the Thomson problem. Construction Like other gyroelongated bipyramids, the gyroelongated square bipyramid can be constructed by attaching two equilateral square pyramids onto the square faces of a square antiprism; this process is k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gyroelongated Bipyramid
In geometry, the gyroelongated bipyramids are an infinite set of polyhedra, constructed by elongating an bipyramid by inserting an antiprism between its congruent halves. Forms Three members of the set can be deltahedra, that is, constructed entirely of equilateral triangles: the gyroelongated square bipyramid, a Johnson solid; the icosahedron, a Platonic solid; and the '' gyroelongated triangular bipyramid'' if it is made with equilateral triangles, but because it has coplanar faces is not strictly convex. With pairs of triangles merged into rhombi, it can be seen as a trigonal trapezohedron. The other members can be constructed with isosceles triangles. Use Pentagonal Gyroelongated Bipyramid * The Pentagonal gyroelongated bipyramid is commonly used in 3D modeling. It appears in various 3D programs such as Autodesk Maya, Blender (software) or Cinema 4D. In most programs, it's referred to as an ico-spere, or Icosahedron. It is also commonly used as a dice in Dungeons and Dra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equilateral Triangles
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the special case of an isosceles triangle by modern definition, creating more special properties. The equilateral triangle can be found in various tilings, and in polyhedrons such as the deltahedron and antiprism. It appears in real life in popular culture, architecture, and the study of stereochemistry resembling the molecular known as the trigonal planar molecular geometry. Properties An equilateral triangle is a triangle that has three equal sides. It is a special case of an isosceles triangle in the modern definition, stating that an isosceles triangle is defined at least as having two equal sides. Based on the modern definition, this leads to an equilateral triangle in which one of the three sides may be considered its base. The fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canadian Journal Of Mathematics
The ''Canadian Journal of Mathematics'' () is a bimonthly mathematics journal published by the Canadian Mathematical Society. It was established in 1949 by H. S. M. Coxeter and G. de B. Robinson. The current editors-in-chief of the journal are Henry Kim and Robert McCann. The journal publishes articles in all areas of mathematics. See also * Canadian Mathematical Bulletin The ''Canadian Mathematical Bulletin'' () is a mathematics journal, established in 1958 and published quarterly by the Canadian Mathematical Society. The current editors-in-chief of the journal are Antonio Lei and Javad Mashreghi. The journal p ... References External links * Research Journals, Canadian Mathematical Society University of Toronto Press academic journals Mathematics journals Academic journals established in 1949 Bimonthly journals Multilingual journals Cambridge University Press academic journals Academic journals associated with learned and professional societies of Cana ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circumscribed Sphere
In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's Vertex (geometry), vertices. The word circumsphere is sometimes used to mean the same thing, by analogy with the term ''circumcircle''. As in the case of two-dimensional circumscribed circles (circumcircles), the radius of a sphere circumscribed around a polyhedron is called the circumradius of , and the center point of this sphere is called the circumcenter of . Existence and optimality When it exists, a circumscribed sphere need not be the Smallest-circle problem, smallest sphere containing the polyhedron; for instance, the tetrahedron formed by a vertex of a cube and its three neighbors has the same circumsphere as the cube itself, but can be contained within a smaller sphere having the three neighboring vertices on its equator. However, the smallest sphere containing a given polyhedron is always the circumsphere of the convex hull of a subset of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carbon Monoxide
Carbon monoxide (chemical formula CO) is a poisonous, flammable gas that is colorless, odorless, tasteless, and slightly less dense than air. Carbon monoxide consists of one carbon atom and one oxygen atom connected by a triple bond. It is the simplest oxocarbon, carbon oxide. In coordination complexes, the carbon monoxide ligand is called ''metal carbonyl, carbonyl''. It is a key ingredient in many processes in industrial chemistry. The most common source of carbon monoxide is the partial combustion of carbon-containing compounds. Numerous environmental and biological sources generate carbon monoxide. In industry, carbon monoxide is important in the production of many compounds, including drugs, fragrances, and fuels. Indoors CO is one of the most acutely toxic contaminants affecting indoor air quality. CO may be emitted from tobacco smoke and generated from malfunctioning fuel-burning stoves (wood, kerosene, natural gas, propane) and fuel-burning heating systems (wood, oil, n ... [...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 su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atom Cluster
Nanoclusters are atomically precise, crystalline materials most often existing on the 0-2 nanometer scale. They are often considered kinetically stable intermediates that form during the synthesis of comparatively larger materials such as semiconductor and metallic nanocrystals. The majority of research conducted to study nanoclusters has focused on characterizing their crystal structures and understanding their role in the nucleation and growth mechanisms of larger materials. Materials can be categorized into three different regimes, namely bulk, nanoparticles and nanoclusters. Bulk metals are electrical conductors and good optical reflectors and metal nanoparticles display intense colors due to surface plasmon resonance. However, when the size of metal nanoclusters is further reduced to form a nanocluster, the band structure becomes discontinuous and breaks down into discrete energy levels, somewhat similar to the energy levels of molecules. This gives nanoclusters similar q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chemical Compounds
A chemical compound is a chemical substance composed of many identical molecules (or molecular entities) containing atoms from more than one chemical element held together by chemical bonds. A molecule consisting of atoms of only one element is therefore not a compound. A compound can be transformed into a different substance by a chemical reaction, which may involve interactions with other substances. In this process, bonds between atoms may be broken or new bonds formed or both. There are four major types of compounds, distinguished by how the constituent atoms are bonded together. Molecular compounds are held together by covalent bonds; ionic compounds are held together by ionic bonds; intermetallic compounds are held together by metallic bonds; coordination complexes are held together by coordinate covalent bonds. Non-stoichiometric compounds form a disputed marginal case. A chemical formula specifies the number of atoms of each element in a compound molecule, us ... [...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] |
Gyroelongated Square Pyramid
In geometry, the gyroelongated square pyramid is the Johnson solid that can be constructed by attaching an equilateral square pyramid to a square antiprism. It occurs in chemistry; for example, the capped square antiprismatic molecular geometry. Construction The gyroelongated square pyramid is composite, since it can constructed by attaching one equilateral square pyramid to the square antiprism, a process known as the gyroelongation. This construction involves the covering of one of two square faces and replacing them with the four equilateral triangles, so that the resulting polyhedron has twelve equilateral triangles and one square. The convex polyhedron in which all of the faces are regular is the Johnson solid, and the gyroelongated square pyramid is one of them, enumerated as J_ , the tenth Johnson solid. Properties The surface area of a gyroelongated square pyramid with edge length a is: \left(1 + 3\sqrt\right)a^2 \approx 6.196a^2, the area of twelve equilateral ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dihedral Group
In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. The notation for the dihedral group differs in geometry and abstract algebra. In geometry, or refers to the symmetries of the n-gon, -gon, a group of order . In abstract algebra, refers to this same dihedral group. This article uses the geometric convention, . Definition The word "dihedral" comes from "di-" and "-hedron". The latter comes from the Greek word hédra, which means "face of a geometrical solid". Overall it thus refers to the two faces of a polygon. Elements A regular polygon with n sides has 2n different symmetries: n rotational symmetry, rotational symmetries and n reflection symmetry, reflection symmetries. Usually, we take n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
