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Herschel Graph
In graph theory, a branch of mathematics, the Herschel graph is a bipartite graph, bipartite undirected graph with 11 vertices and 18 edges. It is a polyhedral graph (the graph of a convex polyhedron), and is the smallest polyhedral graph that does not have a Hamiltonian cycle, a cycle passing through all its vertices. It is named after British astronomer Alexander Stewart Herschel, because of Herschel's studies of Hamiltonian cycles in polyhedral graphs (but not of this graph). Definition and properties The Herschel graph has three vertices of degree four (the three blue vertices aligned vertically in the center of the illustration) and eight vertices of degree three. Each two distinct degree-four vertices share two degree-three neighbors, forming a four-vertex cycle with these shared neighbors. There are three of these cycles, passing through six of the eight degree-three vertices (red in the illustration). Two more degree-three vertices (blue) do not participate in these four- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herschel Graph LS
Herschel or Herschell may refer to: People * Herschel (name), various people Places * Herschel, Eastern Cape, South Africa * Herschel, Saskatchewan * Herschel, Yukon * Herschel Bay, Canada * Herschel Heights, Alexander Island, Antarctica * Herschel Island, Canada * Herschel Island (Chile), an island of the Hermite Islands archipelago * Mount Herschel, Antarctica * Cape Sterneck, Antarctica Astronomy * Herschel (crater), various craters in the Solar System * 2000 Herschel, an asteroid * 35P/Herschel–Rigollet, a comet * Herschel Catalogue (other), various astronomical catalogues of nebulae * Herschel Medal, awarded by the UK Royal Astronomical Society * Herschel Museum of Astronomy, in Bath, United Kingdom * Herschel Space Observatory, operated by the European Space Agency * Herschel wedge, an optical prism used in solar observation * Herschel's Garnet Star, a red supergiant star * William Herschel Telescope, in the Canary Islands * Telescopium Herschelii, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skeleton (topology)
In mathematics, particularly in algebraic topology, the of a topological space presented as a simplicial complex (resp. CW complex) refers to the subspace that is the union of the simplices of (resp. cells of ) of dimensions In other words, given an inductive definition of a complex, the is obtained by stopping at the . These subspaces increase with . The is a discrete space, and the a topological graph. The skeletons of a space are used in obstruction theory, to construct spectral sequences by means of filtrations, and generally to make inductive arguments. They are particularly important when has infinite dimension, in the sense that the do not become constant as In geometry In geometry, a of P (functionally represented as skel''k''(''P'')) consists of all elements of dimension up to ''k''. For example: : skel0(cube) = 8 vertices : skel1(cube) = 8 vertices, 12 edges : skel2(cube) = 8 vertices, 12 edges, 6 square faces For simplicial sets The above ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Goldner–Harary Graph
In the mathematics, mathematical field of graph theory, the Goldner–Harary graph is a simple undirected graph with 11 vertices and 27 edges. It is named after Anita M. Goldner and Frank Harary, who proved in 1975 that it was the smallest Hamiltonian graph, non-Hamiltonian maximal planar graph. The same graph had already been given as an example of a non-Hamiltonian simplicial polyhedron by Branko Grünbaum in 1967. Properties The Goldner–Harary graph is a planar graph: it can be drawn in the plane with none of its edges crossing. When drawn on a plane, all its faces are triangular, making it a maximal planar graph. As with every maximal planar graph, it is also vertex connectivity, 3-vertex-connected: the removal of any two of its vertices leaves a connected Glossary of graph theory#Subgraphs, subgraph. The Goldner–Harary graph is also hamiltonian graph, non-Hamiltonian. The smallest possible number of vertices for a non-Hamiltonian polyhedral graph, polyhedral graph is 11. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herschel Hamiltonian Path
Herschel or Herschell may refer to: People * Herschel (name), various people Places * Herschel, Eastern Cape, South Africa * Herschel, Saskatchewan * Herschel, Yukon * Herschel Bay, Canada * Herschel Heights, Alexander Island, Antarctica * Herschel Island, Canada * Herschel Island (Chile), an island of the Hermite Islands archipelago * Mount Herschel, Antarctica * Cape Sterneck, Antarctica Astronomy * Herschel (crater), various craters in the Solar System * 2000 Herschel, an asteroid * 35P/Herschel–Rigollet, a comet * Herschel Catalogue (other), various astronomical catalogues of nebulae * Herschel Medal, awarded by the UK Royal Astronomical Society * Herschel Museum of Astronomy, in Bath, United Kingdom * Herschel Space Observatory, operated by the European Space Agency * Herschel wedge, an optical prism used in solar observation * Herschel's Garnet Star, a red supergiant star * William Herschel Telescope, in the Canary Islands * Telescopium Herschelii, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Polyhedron
In geometry, the midsphere or intersphere of a convex polyhedron is a sphere which is tangent to every edge of the polyhedron. Not every polyhedron has a midsphere, but the uniform polyhedra, including the regular, quasiregular and semiregular polyhedra and their duals ( Catalan solids) all have midspheres. The radius of the midsphere is called the midradius. A polyhedron that has a midsphere is said to be midscribed about this sphere. When a polyhedron has a midsphere, one can form two perpendicular circle packings on the midsphere, one corresponding to the adjacencies between vertices of the polyhedron, and the other corresponding in the same way to its polar polyhedron, which has the same midsphere. The length of each polyhedron edge is the sum of the distances from its two endpoints to their corresponding circles in this circle packing. Every convex polyhedron has a combinatorially equivalent polyhedron, the canonical polyhedron, that does have a midsphere, centered at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Gathering
The Gathering may refer to: Film and television * ''The Gathering'' (1977 film), an American television film directed by Randal Kleiser * The Gathering (1998 film), an American thriller film directed by Danny Carrales * ''The Gathering'' (2003 film), a British thriller/horror film directed by Brian Gilbert * ''The Gathering'' (miniseries), a 2007 American thriller starring Peter Fonda * ''The Gathering'' (audio drama), a 2006 audio drama based on the television programme ''Doctor Who'' * The Gathering, a contest among immortals in the Highlander franchise * '' Babylon 5: The Gathering'', the 1993 pilot movie for ''Babylon 5'' TV episodes * "The Gathering" (''Gargoyles'') * "The Gathering" (''Ghost Whisperer'') * "The Gathering" (''Highlander: The Series''), pilot * "The Gathering" (''Outlander'') * "The Gathering" (''Star Wars: The Clone Wars'') * "The Gathering" (''Torchwood'') Literature * ''The Gathering'' (Armstrong novel), a 2011 novel by Kelley Armstrong * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isosceles Triangle
In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter version thus including the equilateral triangle as a special case. Examples of isosceles triangles include the isosceles right triangle, the Golden triangle (mathematics), golden triangle, and the faces of bipyramids and certain Catalan solids. The mathematical study of isosceles triangles dates back to ancient Egyptian mathematics and Babylonian mathematics. Isosceles triangles have been used as decoration from even earlier times, and appear frequently in architecture and design, for instance in the pediments and gables of buildings. The two equal sides are called the ''legs'' and the third side is called the base (geometry), ''base'' of the triangle. The other dimensions of the triangle, such ... [...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] |
Triangular Prism
In geometry, a triangular prism or trigonal prism is a Prism (geometry), prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a ''right triangular prism''. A right triangular prism may be both Semiregular polyhedron, semiregular and Uniform polyhedron, uniform. The triangular prism can be used in constructing another polyhedron. Examples are some of the Johnson solids, the truncated right triangular prism, and Schönhardt polyhedron. Properties A triangular prism has 6 vertices, 9 edges, and 5 faces. Every prism has 2 congruent faces known as its ''bases'', and the bases of a triangular prism are triangles. The triangle has 3 vertices, each of which pairs with another triangle's vertex, making up another 3 edges. These edges form 3 parallelograms as other faces. If the prism's edges are perpendicular to the base, the lateral faces are rectangles, and the prism is called a ''right triangular prism''. This ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Hull
In geometry, the convex hull, 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 set, 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 projective duality, dual problem of intersecting Half-space (geome ... [...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] |
Kite (geometry)
In Euclidean geometry, a kite is a quadrilateral with reflection symmetry across a diagonal. Because of this symmetry, a kite has two equal angles and two pairs of adjacent equal-length sides. Kites are also known as deltoids, but the word ''deltoid'' may also refer to a deltoid curve, an unrelated geometric object sometimes studied in connection with quadrilaterals.See H. S. M. Coxeter's review of in : "It is unfortunate that the author uses, instead of 'kite', the name 'deltoid', which belongs more properly to a curve, the three-cusped hypocycloid." A kite may also be called a dart, particularly if it is not convex. Every kite is an orthodiagonal quadrilateral (its diagonals are at right angles) and, when convex, a tangential quadrilateral (its sides are tangent to an inscribed circle). The convex kites are exactly the quadrilaterals that are both orthodiagonal and tangential. They include as special cases the right kites, with two opposite right angles; the rhombus, rhombi, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
