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Pentagonal Bipyramid
The pentagonal bipyramid (or pentagonal dipyramid) is a polyhedron with ten triangular faces. It is constructed by attaching two pentagonal pyramids to each of their bases. If the triangular faces are equilateral, the pentagonal bipyramid is an example of deltahedra, composite polyhedron, and Johnson solid. The pentagonal bipyramid may be represented as four-connected well-covered graph. This polyhedron may be used in the chemical compound as the description of an atom cluster known as pentagonal bipyramidal molecular geometry, as a solution in Thomson problem, as well as in decahedral nanoparticles. Special cases As a right bipyramid Like other bipyramids, the pentagonal bipyramid can be constructed by attaching the base of two pentagonal pyramids. These pyramids cover their pentagonal base, such that the resulting polyhedron has ten triangles as its faces, fifteen edges, and seven vertices. The pentagonal bipyramid is said to be right if the pyramids are symmetri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipyramid
In geometry, a bipyramid, dipyramid, or double pyramid is a polyhedron formed by fusing two Pyramid (geometry), pyramids together base (geometry), base-to-base. The polygonal base of each pyramid must therefore be the same, and unless otherwise specified the base Vertex (geometry), vertices are usually coplanar and a bipyramid is usually ''symmetric'', meaning the two pyramids are mirror images across their common base plane. When each apex (geometry), apex (, the off-base vertices) of the bipyramid is on a line perpendicular to the base and passing through its center, it is a ''right'' bipyramid; otherwise it is ''oblique''. When the base is a regular polygon, the bipyramid is also called ''regular''. Definition and properties A bipyramid is a polyhedron constructed by fusing two Pyramid (geometry), pyramids which share the same polygonal base (geometry), base; a pyramid is in turn constructed by connecting each vertex of its base to a single new vertex (geometry), vertex (th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagonal Bipyramidal Molecular Geometry
In chemistry, a pentagonal bipyramid is a molecular geometry with one atom at the centre with seven ligands at the corners of a pentagonal bipyramid. A perfect pentagonal bipyramid belongs to the molecular point group ''D5h''. The pentagonal bipyramid is a case where bond angles surrounding an atom are not identical (see also trigonal bipyramidal molecular geometry). This is one of the three common shapes for heptacoordinate transition metal complexes, along with the capped octahedron and the capped trigonal prism. Pentagonal bipyramids are claimed to be promising coordination geometries for lanthanide-based single-molecule magnets, since they present no extradiagonal crystal field terms, therefore minimising spin mixing, and all of their diagonal terms are in first approximation protected from low-energy vibrations, minimising vibronic coupling. Examples * Iodine heptafluoride (IF7) with 7 bonding groups * Rhenium heptafluoride (ReF7) * Peroxo chromium Chrom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equilateral Triangle
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. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
J13 Pentagonal Bipyramid
J13 may refer to: Roads * County Route J13 (California) * Johor State Route J13, in Malaysia Vehicles * GNR Class J13, a British steam locomotive class * , a ''Visby''-class destroyer of the Swedish Navy * Shenyang J-13, a cancelled Chinese light fighter aircraft Other uses * Bacterial pneumonia * Pentagonal bipyramid The pentagonal bipyramid (or pentagonal dipyramid) is a polyhedron with ten triangular faces. It is constructed by attaching two pentagonal pyramids to each of their bases. If the triangular faces are equilateral, the pentagonal bipyramid is an ..., a Johnson solid (J13) * J13, a Nissan J engine * July 13th, 2024 - the date US presidential candidate Donald Trump was shot in Butler, PA {{Letter-NumberCombDisambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Net (geometry)
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 of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Snub Disphenoid
In geometry, the snub disphenoid is a convex polyhedron with 12 equilateral triangles as its face (geometry), faces. It is an example of deltahedron and Johnson solid. It can be constructed in different approaches. This shape is also called Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron. The snub disphenoid can be visualized as an atom cluster surrounding a central atom, that is the dodecahedral molecular geometry. Its vertices may be placed in a sphere and can also be used as a minimum possible Lennard-Jones potential among all eight-sphere clusters. The dual polyhedron of the snub disphenoid is the elongated gyrobifastigium. Construction The snub disphenoid can be constructed in different ways. As suggested by the name, the snub disphenoid is constructed from a tetragonal disphenoid by cutting all the edges from its faces, and adding equilateral triangles (the light blue colors in the following image) that are twisted in a certain a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Octahedron
In geometry, a regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Regular octahedra occur in nature as crystal structures. An octahedron, more generally, can be any eight-sided polyhedron; many types of irregular octahedra also exist. A regular octahedron is convex, meaning that for any two points within it, the line segment connecting them lies entirely within it. It is one of the eight convex deltahedra because all of the faces are equilateral triangles. It is a composite polyhedron made by attaching two equilateral square pyramids. Its dual polyhedron is the cube, and they have the same three-dimensional symmetry groups, the octahedral symmetry \mathrm_\mathrm . A regular octahedron is a special case of an octahedron, any eight-sided polyhedron. It is the three-dimensional case of the more general concept of a cross-polytope. As a Platonic solid The regular octahedron is one of the Platonic solids, a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximal Independent Set
In graph theory, a maximal independent set (MIS) or maximal stable set is an Independent set (graph theory), independent set that is not a subset of any other independent set. In other words, there is no Vertex (graph theory), vertex outside the independent set that may join it because it is maximal with respect to the independent set property. For example, in the graph , a Path graph, path with three vertices , , and , and two edges and , the sets and are both maximal independent. The set is independent, but is not maximal independent, because it is a subset of the larger independent set In this same graph, the maximal cliques are the sets and A MIS is also a dominating set in the graph, and every dominating set that is independent must be maximal independent, so MISs are also called independent dominating sets. A graph may have many MISs of widely varying sizes; the largest, or possibly several equally large, MISs of a graph is called a maximum independent set, ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Polytope
In geometry, a simplicial polytope is a polytope whose facet_(mathematics), facets are all Simplex, simplices. For example, a ''simplicial polyhedron'' in three dimensions contains only Triangle, triangular faces (p.341) and corresponds via Steinitz's theorem to a maximal planar graph. They are Dual_polyhedron#Topological_duality, topologically dual to simple polytopes. Polytopes which are both simple and simplicial are either simplices or two-dimensional polygons. Examples Simplicial polyhedra include: * Bipyramids * Gyroelongated bipyramids *Deltahedron, Deltahedra (equilateral triangles) ** Platonic solid, Platonic *** tetrahedron, octahedron, icosahedron ** Johnson solids: ***triangular bipyramid, pentagonal bipyramid, snub disphenoid, triaugmented triangular prism, ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axis Of Symmetry
An axis (: axes) may refer to: Mathematics *A specific line (often a directed line) that plays an important role in some contexts. In particular: ** Coordinate axis of a coordinate system *** ''x''-axis, ''y''-axis, ''z''-axis, common names for the coordinate axes of a Cartesian coordinate system ** Axis of rotation ** Axis of symmetry ** Axis of a conic section Politics *Axis powers of World War II, 1936–1945. * Axis of evil (first used in 2002), U.S. President George W. Bush's description of Iran, Iraq, and North Korea *Axis of Resistance (first used in 2002), the Shia alliance of Iran, Syria, and Hezbollah * Axis of Upheaval (first used in 2024), foreign policy neologism of the Anti-western collaboration between Russia, China, Iran, and North Korea * Jakarta-Pyongyang-Peking Axis, diplomatic alignment and alliance between Indonesia, China, and North Korea during Sukarno's Presidency *Political spectrum, sometimes called an axis Science *Axis (anatomy), the second cerv ... [...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] |
Point Groups In Three Dimensions
In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group (mathematics), group of all isometry, isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrix, orthogonal matrices. O(3) itself is a subgroup of the Euclidean group E(3) of all isometries. Symmetry groups of geometric objects are isometry groups. Accordingly, analysis of isometry groups is analysis of possible symmetry, symmetries. All isometries of a Bounded set, bounded (finite) 3D object have one or more common fixed points. We follow the usual convention by choosing the Origin (mathematics), origin as one of them. The symmetry group of an object is sometimes also called its full symmetry group, as opposed to its proper symmetry group, the intersection of its full symmetry group with Euclidean group#Direct and indirect is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
