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Self-dual Polyhedra
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] |
Polyhedron Pair 6-8
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 the same plane. Cubes and pyramids are examples of convex polyhedra. A polyhedron is a 3-dimensional example of a polytope, a more general concept in any number of dimensions. Definition Convex polyhedra are well-defined, with several equivalent standard definitions. However, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of "polyhedron" have been given within particular contexts,. some more rigorous than others, and there is not universal agreement over which of these to choose. Some of these definitions exclude shapes that have often been counted as polyhedra (such as the self-crossing polyhedra) or include shapes that are often not considered as valid polyhe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dot Product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called the inner product (or rarely projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more). Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. In mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schlegel Diagram
In geometry, a Schlegel diagram is a projection of a polytope from \mathbb^d into \mathbb^ through a point just outside one of its facets. The resulting entity is a polytopal subdivision of the facet in \mathbb^ that, together with the original facet, is combinatorially equivalent to the original polytope. The diagram is named for Victor Schlegel, who in 1886 introduced this tool for studying combinatorial and topological properties of polytopes. In dimension 3, a Schlegel diagram is a projection of a polyhedron into a plane figure; in dimension 4, it is a projection of a 4-polytope to 3-space. As such, Schlegel diagrams are commonly used as a means of visualizing four-dimensional polytopes. Construction The most elementary Schlegel diagram, that of a polyhedron, was described by Duncan Sommerville as follows: :A very useful method of representing a convex polyhedron is by plane projection. If it is projected from any external point, since each ray cuts it twice, it will be r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-skeleton
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 def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a set of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Dual Polyhedron
A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, security guards, in some workplaces and schools and by inmates in prisons. In some countries, some other officials also wear uniforms in their duties; such is the case of the Commissioned Corps of the United States Public Health Service or the French prefects. For some organizations, such as police, it may be illegal for non members to wear the uniform. Etymology From the Latin ''unus'', one, and ''forma'', form. Corporate and work uniforms Workers sometimes wear uniforms or corporate clothing of one nature or another. Workers required to wear a uniform may include retail workers, bank and post-office workers, public-security and health-care workers, blue-collar employees, personal trainers in health clubs, instructors in summer ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines across the connected faces, joining adjacent points around the face. When done, these lines form a complete circuit, i.e. a polygon, around the vertex. This polygon is the vertex figure. More precise formal definitions can vary quite widely, according to circumstance. For example Coxeter (e.g. 1948, 1954) varies his definition as convenient for the current area of discussion. Most of the following definitions of a vertex figure apply equally well to infinite tessellation, tilings or, by extension, to Honeycomb (geometry), space-filling tessellation with polytope Cell (geometry), cells and other higher-dimensional polytopes. As a flat slice Make a slice through the corner of the polyhedron, cutting through all the edges ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Polyhedron
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also face- and edge-transitive), quasi-regular (if also edge-transitive but not face-transitive), or semi-regular (if neither edge- nor face-transitive). The faces and vertices need not be convex, so many of the uniform polyhedra are also star polyhedra. There are two infinite classes of uniform polyhedra, together with 75 other polyhedra: *Infinite classes: ** prisms, **antiprisms. * Convex exceptional: ** 5 Platonic solids: regular convex polyhedra, ** 13 Archimedean solids: 2 quasiregular and 11 semiregular convex polyhedra. * Star (nonconvex) exceptional: ** 4 Kepler–Poinsot polyhedra: regular nonconvex polyhedra, ** 53 uniform star polyhedra: 14 quasiregular and 39 semiregular. Hence 5 + 13 + 4 + 53 = 75. There are also many degen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Midsphere
In geometry, the midsphere or intersphere of a 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 ... is a sphere which is tangent to every Edge (geometry), 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 polyhedron, uniform polyhedra, including the regular polyhedron, regular, Quasiregular polyhedron, quasiregular and Semiregular polyhedron, semiregular polyhedra and their Dual polyhedron, duals all have midspheres. In the regular polyhedra, the inscribed sphere, midsphere, and circumscribe ... [...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] |
Dual Compound 6-8 Max
Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical number), a grammatical category used in some languages * Dual county, a Gaelic games county which in both Gaelic football and hurling * Dual diagnosis, a psychiatric diagnosis of co-occurrence of substance abuse and a mental problem * Dual fertilization, simultaneous application of a P-type and N-type fertilizer * Dual impedance, electrical circuits that are the dual of each other * Dual SIM cellphone supporting use of two SIMs * Aerochute International Dual a two-seat Australian powered parachute design Acronyms and other uses * Dual (brand), a manufacturer of Hifi equipment * DUAL (cognitive architecture), an artificial intelligence design model * DUAL algorithm, or diffusing update algorithm, used to update Internet protocol routing ta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called "points at infinity") to Euclidean points, and vice-versa. Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations (the affine transformations). The first issue for geometers is what kind of geometry is adequate for a novel situation. It is not possible to refer to angles in projective geometry as it is in Euclidean geometry, because angle is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
