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Herschel Graph
In graph theory, a branch of mathematics, the Herschel graph is a 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 and eight vertices of degree three. Each of the three pairs of degree-four vertices form a cycle of four vertices, in which they alternate with two of the degree-three vertices. Each of the two remaining degree-three vertices is adjacent to a degree-three vertex from each of these three cycles. The Herschel graph is a polyhedral graph; this means that it is a planar graph, one that can be drawn in the plane with none of its edges crossing, and ... [...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 * 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, a constellation * Uranus, for a time known as Herschel Other uses * Allan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enneahedron
In geometry, an enneahedron (or nonahedron) is a polyhedron with nine faces. There are 2606 types of convex enneahedron, each having a different pattern of vertex, edge, and face connections. None of them are regular. Examples The most familiar enneahedra are the octagonal pyramid and the heptagonal prism. The heptagonal prism is a uniform polyhedron, with two regular heptagon faces and seven square faces. The octagonal pyramid has eight isosceles triangular faces around a regular octagonal base. Two more enneahedra are also found among the Johnson solids: the elongated square pyramid and the elongated triangular bipyramid. The three-dimensional associahedron, with six pentagonal faces and three quadrilateral faces, is an enneahedron. Five Johnson solids have enneahedral duals: the triangular cupola, gyroelongated square pyramid, self-dual elongated square pyramid, triaugmented triangular prism (whose dual is the associahedron), and tridiminished icosahedron. Another enneahedron i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tait's Conjecture
In mathematics, Tait's conjecture states that "Every 3-connected planar cubic graph has a Hamiltonian cycle (along the edges) through all its vertices". It was proposed by and disproved by , who constructed a counterexample with 25 faces, 69 edges and 46 vertices. Several smaller counterexamples, with 21 faces, 57 edges and 38 vertices, were later proved minimal by . The condition that the graph be 3-regular is necessary due to polyhedra such as the rhombic dodecahedron, which forms a bipartite graph with six degree-four vertices on one side and eight degree-three vertices on the other side; because any Hamiltonian cycle would have to alternate between the two sides of the bipartition, but they have unequal numbers of vertices, the rhombic dodecahedron is not Hamiltonian. The conjecture was significant, because if true, it would have implied the four color theorem: as Tait described, the four-color problem is equivalent to the problem of finding 3-edge-colorings of bridgeless ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Goldner–Harary Graph
In the mathematical field of graph theory, the Goldner–Harary graph is a simple undirected graph with 11 vertices and 27 edges. It is named after A. Goldner and Frank Harary, who proved in 1975 that it was the smallest 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 3-vertex-connected: the removal of any two of its vertices leaves a connected subgraph. The Goldner–Harary graph is also non-Hamiltonian. The smallest possible number of vertices for a non-Hamiltonian polyhedral graph is 11. Therefore, the Goldner–Harary graph is a minimal example of graphs of this type. However, the Herschel graph, another non-Ha ... [...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 * 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, a constellation * Uranus, for a time known as Herschel Other uses * Allan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Polyhedron
In geometry, the midsphere or intersphere of a polyhedron is a sphere which is tangent to every edge of the polyhedron. That is to say, it touches any given edge at exactly one point. Not every polyhedron has a midsphere, but for every convex polyhedron there is a combinatorially equivalent polyhedron, the canonical polyhedron, that does have a midsphere. The radius of the midsphere is called the midradius. Examples The uniform polyhedra, including the regular, quasiregular and semiregular polyhedra and their duals all have midspheres. In the regular polyhedra, the inscribed sphere, midsphere, and circumscribed sphere all exist and are concentric, and the midsphere touches each edge at its midpoint. Not every irregular tetrahedron has a midsphere. The tetrahedra that have a midsphere have been called "Crelle's tetrahedra"; they form a four-dimensional subfamily of the six-dimensional space of all tetrahedra (as parameterized by their six edge lengths). Tangent circles If is ... [...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'' (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 * ''The Gathering'' (Carmody novel), a 1993 novel by Isobelle Carmody * ''The Gath ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isosceles Triangle
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. 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, 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 of the triangle. The other dimensions of the triangle, such as its height, area, and perimeter, can be calculated by simple formulas from the lengths of the legs an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equilateral Triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle. Principal properties Denoting the common length of the sides of the equilateral triangle as a, we can determine using the Pythagorean theorem that: *The area is A=\frac a^2, *The perimeter is p=3a\,\! *The radius of the circumscribed circle is R = \frac *The radius of the inscribed circle is r=\frac a or r=\frac *The geometric center of the triangle is the center of the circumscribed and inscribed circles *The altitude (height) from any side is h=\frac a Denoting the radius of the circumscribed circle as ''R'', we can determine using trigonometry that: *The area of the triangle is \mathrm=\fracR^2 Many of these quantities have simple r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Prism
In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is ''oblique''. A uniform triangular prism is a right triangular prism with equilateral bases, and square sides. Equivalently, it is a polyhedron of which two faces are parallel, while the surface normals of the other three are in the same plane (which is not necessarily parallel to the base planes). These three faces are parallelograms. All cross-sections parallel to the base faces are the same triangle. As a semiregular (or uniform) polyhedron A right triangular prism is semiregular or, more generally, a uniform polyhedron if the base faces are equilateral triangles, and the other three faces are squares. It can be seen as a truncated trigonal hosohedron, represented by Schläfli symbol t. Alternately it can be seen as the Cartesian product of a triangle ... [...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 com ... [...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 vers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
