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Gyroelongated Triangular Cupola
In geometry, the gyroelongated triangular cupola is one of the Johnson solids (''J''22). It can be constructed by attaching a hexagonal antiprism to the base of a triangular cupola (''J''3). This is called "gyroelongation", which means that an antiprism is joined to the base of a solid, or between the bases of more than one solid. The gyroelongated triangular cupola can also be seen as a gyroelongated triangular bicupola (''J''44) with one triangular cupola removed. Like all cupolae, the base (geometry), base polygon has twice as many sides as the top (in this case, the bottom polygon is a hexagon because the top is a triangle). Formulae The following formulae for volume and surface area can be used if all face (geometry), faces are regular polygon, regular, with edge length ''a'':Stephen Wolfram,Gyroelongated triangular cupola from Wolfram Alpha. Retrieved July 22, 2010. :V=\left(\frac\sqrt\right)a^3\approx3.51605...a^3 :A=\left(3+\frac\right)a^2\approx12.5263...a^2 Dual ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johnson Solid
In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two Solid geometry, solids with such a property: the first solids are the Pyramid (geometry), pyramids, Cupola (geometry), cupolas, and a Rotunda (geometry), rotunda; some of the solids may be constructed by attaching with those previous solids, whereas others may not. Definition and background A Johnson solid is a convex polyhedron whose faces are all regular polygons. The convex polyhedron means as bounded intersections of finitely many Half-space (geometry), half-spaces, or as the convex hull of finitely many points. Although there is no restriction that any given regular polygon cannot be a face of a Johnson solid, some authors required that Johnson solids are not Uniform polyhedron, uniform. This means that a Johnson solid is not a Platonic solid, Arc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base (geometry)
In geometry, a base is a side of a polygon or a face of a polyhedron, particularly one oriented perpendicular to the direction in which height is measured, or on what is considered to be the "bottom" of the figure. This term is commonly applied in plane geometry to triangles, parallelograms, trapezoids, and in solid geometry to cylinders, cones, pyramids, parallelepipeds, prisms, and frustums. The side or point opposite the base is often called the '' apex'' or ''summit'' of the shape. Of a triangle In a triangle, any arbitrary side can be considered the ''base''. The two endpoints of the base are called ''base vertices'' and the corresponding angles are called ''base angles''. The third vertex opposite the base is called the ''apex''. The extended base of a triangle (a particular case of an extended side) is the line that contains the base. When the triangle is obtuse and the base is chosen to be one of the sides adjacent to the obtuse angle, then the altitude dropped ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Gyroelongated Triangular Cupola
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 number, a number system used in automatic differentiation * Dual (grammatical number), a grammatical category used in some languages * Dual county, a Gaelic games county which competes 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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolfram Alpha
WolframAlpha ( ) is an answer engine developed by Wolfram Research. It is offered as an online service that answers factual queries by computing answers from externally sourced data. History Launch preparations for WolframAlpha began on May 15, 2009, at 7:00 pm CDT with a live broadcast on Justin.tv. The plan was to publicly launch the service a few hours later. However, there were issues due to extreme load. The service officially launched on May 18, 2009, receiving mixed reviews. The engine is based on Wolfram's earlier product Wolfram Mathematica, a technical computing platform. The coding is written in Wolfram Language, a general multi-paradigm programming language, and implemented in Mathematica. WolframAlpha gathers data from academic and commercial websites such as the CIA's ''The World Factbook'', the United States Geological Survey, a Cornell University Library publication called ''All About Birds'', '' Chambers Biographical Dictionary'', Dow Jones, the ''Cat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stephen Wolfram
Stephen Wolfram ( ; born 29 August 1959) is a British-American computer scientist, physicist, and businessman. He is known for his work in computer algebra and theoretical physics. In 2012, he was named a fellow of the American Mathematical Society. As a businessman, he is the founder and CEO of the software company Wolfram Research, where he works as chief designer of Mathematica and the Wolfram Alpha answer engine. Early life Family Stephen Wolfram was born in London in 1959 to Hugo and Sybil Wolfram, both German Jewish refugees to the United Kingdom. His maternal grandmother was British psychoanalyst Kate Friedlander. Wolfram's father, Hugo Wolfram, was a textile manufacturer and served as managing director of the Lurex Company—makers of the fabric Lurex. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex polygon, convex'' or ''star polygon, star''. In the limit (mathematics), limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon (effectively a Line (geometry), straight line), if the edge length is fixed. General properties These properties apply to all regular polygons, whether convex or star polygon, star: *A regular ''n''-sided polygon has rotational symmetry of order ''n''. *All vertices of a regular polygon lie on a common circle (the circumscribed circle); i.e., they are concyclic points. That is, a regular polygon is a cyclic polygon. *Together with the property of equal-length sides, this implies that every regular polygon also h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Face (geometry)
In solid geometry, a face is a flat surface (a Plane (geometry), planar region (mathematics), region) that forms part of the boundary of a solid object. For example, a cube has six faces in this sense. In more modern treatments of the geometry of polyhedra and higher-dimensional polytopes, a "face" is defined in such a way that it may have any dimension. The vertices, edges, and (2-dimensional) faces of a polyhedron are all faces in this more general sense. Polygonal face In elementary geometry, a face is a polygon on the boundary of a polyhedron. (Here a "polygon" should be viewed as including the 2-dimensional region inside it.) Other names for a polygonal face include polyhedron side and Euclidean plane ''tessellation, tile''. For example, any of the six square (geometry), squares that bound a cube is a face of the cube. Sometimes "face" is also used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface Area
The surface area (symbol ''A'') of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with flat polygonal faces), for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces. This definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration. A general definition of surface area was sought by Henri Lebesgue and Hermann Minkowski at the turn of the twentieth century. Their work led to the development of geometric measure theory, which studies various notions of surface area for irregular objects of any dimension. An important example is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volume
Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The definition of length and height (cubed) is interrelated with volume. The volume of a container is generally understood to be the capacity of the container; i.e., the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. By metonymy, the term "volume" sometimes is used to refer to the corresponding region (e.g., bounding volume). In ancient times, volume was measured using similar-shaped natural containers. Later on, standardized containers were used. Some simple three-dimensional shapes can have their volume easily calculated using arithmetic formulas. Volumes of more complicated shapes can be calculated with integral calculus if a formula exists for the shape ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formula
In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship between given quantities. The plural of ''formula'' can be either ''formulas'' (from the most common English plural noun form) or, under the influence of scientific Latin, ''formulae'' (from the original Latin). In mathematics In mathematics, a formula generally refers to an equation or inequality relating one mathematical expression to another, with the most important ones being mathematical theorems. For example, determining the volume of a sphere requires a significant amount of integral calculus or its geometrical analogue, the method of exhaustion. However, having done this once in terms of some parameter (the radius for example), mathematicians have produced a formula to describe the volume of a sphere in terms of its radius: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygon
In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon's '' vertices'' or ''corners''. An ''n''-gon is a polygon with ''n'' sides; for example, a triangle is a 3-gon. A simple polygon is one which does not intersect itself. More precisely, the only allowed intersections among the line segments that make up the polygon are the shared endpoints of consecutive segments in the polygonal chain. A simple polygon is the boundary of a region of the plane that is called a ''solid polygon''. The interior of a solid polygon is its ''body'', also known as a ''polygonal region'' or ''polygonal area''. In contexts where one is concerned only with simple and solid polygons, a ''polygon'' may refer only to a simple polygon or to a solid polygon. A polygonal chain may cross over itself, creating star polyg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cupola
In architecture, a cupola () is a relatively small, usually dome-like structure on top of a building often crowning a larger roof or dome. Cupolas often serve as a roof lantern to admit light and air or as a lookout. The word derives, via Italian language, Italian, from lower Latin ''cupula'' (classical Latin ''cupella''), (Latin ''cupa''), indicating a vault resembling an upside-down cup. The cylindrical drum underneath a larger cupola is called a tholobate. Background The cupola evolved during the Renaissance from the older Oculus (architecture), oculus. Being weatherproof, the cupola was better suited to the wetter climates of northern Europe. The chhatri, seen in Architecture of India, Indian architecture, fits the definition of a cupola when it is used atop a larger structure. Cupolas often serve as a Bell tower, belfry, Belvedere (structure), belvedere, or roof lantern above a main roof. In other cases they may crown a spire, tower, or Turret (architecture), turret. B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
