|
Triangular Bifrustum
In geometry, the triangular bifrustum is the second in an infinite series of bifrustum polyhedra. It has 6 trapezoid and 2 triangle faces. It may also be called the truncated triangular bipyramid; however, that term is ambiguous, as it may also refer to polyhedra formed by truncating all five vertices of a triangular bipyramid. This polyhedron can be constructed by taking a triangular bipyramid and truncating the polar axis vertices, making it into two end-to-end frustums. It appears as the form of certain nanocrystals.. A truncated triangular bipyramid can be constructed by connecting two stacked regular octahedra with 3 pairs of tetrahedra around the sides. This represents a portion of the gyrated alternated cubic honeycomb The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2. Other names incl .... : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bifrustum
An ''n''-agonal bifrustum is a polyhedron composed of three parallel planes of ''n''-agons, with the middle plane largest and usually the top and bottom congruent. It can be constructed as two congruent frusta combined across a plane of symmetry, and also as a bipyramid with the two polar vertices truncated. They are duals to the family of elongated bipyramids. Formulae For a regular -gonal bifrustum with the equatorial polygon sides , bases sides and semi-height (half the distance between the planes of bases) , the lateral surface area , total area and volume are: :A_l = n (a+b) \sqrt\,, :A = A_l + n \frac\,, :V = n \frach\,. Forms Three bifrusta are duals ''Duals'' is a compilation album by the Irish rock band U2. It was released in April 2011 to u2.com subscribers. Track listing :* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose Bowl, P ... to three Johnson solids, J14-16. In general, a n-agonal bifru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trapezoid
A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium (). A trapezoid is necessarily a Convex polygon, convex quadrilateral in Euclidean geometry. The parallel sides are called the ''bases'' of the trapezoid. The other two sides are called the ''legs'' (or the ''lateral sides'') if they are not parallel; otherwise, the trapezoid is a parallelogram, and there are two pairs of bases). A ''scalene trapezoid'' is a trapezoid with no sides of equal measure, in contrast with the #Special cases, special cases below. Etymology and ''trapezium'' versus ''trapezoid'' Ancient Greek mathematician Euclid defined five types of quadrilateral, of which four had two sets of parallel sides (known in English as square, rectangle, rhombus and rhomboid) and the last did not have two sets of parallel sides – a τραπέζια (''trapezia'' literally "a table", itself fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-Collinearity, collinear, determine a unique triangle and simultaneously, a unique Plane (mathematics), plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted. Types of triangle The terminology for categorizing triangles is more than two thousand years old, having been defined on the very first page of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetry Group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object ''X'' is ''G'' = Sym(''X''). For an object in a metric space, its symmetries form a subgroup of the isometry group of the ambient space. This article mainly considers symmetry groups in Euclidean geometry, but the concept may also be studied for more general types of geometric structure. Introduction We consider the "objects" possessing symmetry to be geometric figures, images, and patterns, such as a wallpaper pattern. For symmetry of physical objects, one may also take their physical composition as part of the pattern. (A pattern may be specified formally as a scalar field, a funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elongated Triangular Bipyramid
In geometry, the elongated triangular bipyramid (or dipyramid) or triakis triangular prism is one of the Johnson solids (), convex polyhedra whose faces are regular polygons. As the name suggests, it can be constructed by elongating a triangular bipyramid () by inserting a triangular prism between its congruent halves. The nirrosula, an African musical instrument woven out of strips of plant leaves, is made in the form of a series of elongated bipyramids with non-equilateral triangles as the faces of their end caps.. Formulae The following formulae for volume (V), surface area (A) and height (H) can be used if all faces are regular, with edge length ''a'': :V=\left(\frac\left(2\sqrt+3\sqrt\right)\right)\cdot a^3\approx 0.668715...a^3Stephen Wolfram,Elongated triangular dipyramid from Wolfram Alpha WolframAlpha ( ) is an answer engine developed by Wolfram Research. It answers factual queries by computing answers from externally sourced data. WolframAlpha was rel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others''Mathematical Programming'', by Melvyn W. Jeter (1986) p. 68/ref> (including this article) allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. In the influential textbooks of Grünbaum and Ziegler on the subject, as well as in many other texts in discrete geometry, convex polytopes are often simply called "polytopes". Grünbaum points out that this is solely to avoi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bifrustum
An ''n''-agonal bifrustum is a polyhedron composed of three parallel planes of ''n''-agons, with the middle plane largest and usually the top and bottom congruent. It can be constructed as two congruent frusta combined across a plane of symmetry, and also as a bipyramid with the two polar vertices truncated. They are duals to the family of elongated bipyramids. Formulae For a regular -gonal bifrustum with the equatorial polygon sides , bases sides and semi-height (half the distance between the planes of bases) , the lateral surface area , total area and volume are: :A_l = n (a+b) \sqrt\,, :A = A_l + n \frac\,, :V = n \frach\,. Forms Three bifrusta are duals ''Duals'' is a compilation album by the Irish rock band U2. It was released in April 2011 to u2.com subscribers. Track listing :* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose Bowl, P ... to three Johnson solids, J14-16. In general, a n-agonal bifru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncation (geometry)
In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new Facet (geometry), facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids. Uniform truncation In general any polyhedron (or polytope) can also be truncated with a degree of freedom as to how deep the cut is, as shown in Conway polyhedron notation truncation operation. A special kind of truncation, usually implied, is a uniform truncation, a truncation operator applied to a regular polyhedron (or regular polytope) which creates a resulting uniform polyhedron (uniform polytope) with equal edge lengths. There are no degrees of freedom, and it represents a fixed geometric, just like the regular polyhedra. In general all single ringed uniform polytopes have a uniform truncation. For example, the icosidodecahedron, represented as Schläfli symbols r or \begin 5 \\ 3 \end, and Coxeter-Dynkin diagram or has a uniform truncation, the truncate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Bipyramid
In geometry, the triangular bipyramid (or dipyramid) is a type of hexahedron, being the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces. As the name suggests, it can be constructed by joining two tetrahedra along one face. Although all its faces are congruent and the solid is face-transitive, it is not a Platonic solid because some vertices adjoin three faces and others adjoin four. The bipyramid whose six faces are all equilateral triangles is one of the Johnson solids, (). As a Johnson solid with all faces equilateral triangles, it is also a deltahedron. Formulae The following formulae for the height (H), surface area (A) and volume (V) can be used if all faces are regular, with edge length L: :H = L\cdot \frac \approx L\cdot 1.632993162 :A = L^2 \cdot \frac \approx L^2\cdot 2.598076211 :V = L^3 \cdot \frac \approx L^3\cdot 0.235702260 Dual polyhedron The dual polyhedron of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frustum
In geometry, a (from the Latin for "morsel"; plural: ''frusta'' or ''frustums'') is the portion of a solid (normally a pyramid or a cone) that lies between two parallel planes cutting this solid. In the case of a pyramid, the base faces are polygonal, the side faces are trapezoidal. A right frustum is a right pyramid or a right cone truncated perpendicularly to its axis; otherwise it is an oblique frustum. If all its edges are forced to become of the same length, then a frustum becomes a prism (possibly oblique or/and with irregular bases). In computer graphics, the viewing frustum is the three-dimensional region which is visible on the screen. It is formed by a clipped pyramid; in particular, ''frustum culling'' is a method of hidden surface determination. In the aerospace industry, a frustum is the fairing between two stages of a multistage rocket (such as the Saturn V), which is shaped like a truncated cone. Elements, special cases, and related concepts A frustu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nanocrystal
A ''nanocrystal'' is a material particle having at least one dimension smaller than 100 nanometres, based on quantum dots (a nanoparticle) and composed of atoms in either a single- or poly-crystalline arrangement. The size of nanocrystals distinguishes them from larger crystals. For example, silicon nanocrystals can provide efficient light emission while bulk silicon does not and may be used for memory components. When embedded in solids, nanocrystals may exhibit much more complex melting behaviour than conventional solids and may form the basis of a special class of solids. They can behave as single-domain systems (a volume within the system having the same atomic or molecular arrangement throughout) that can help explain the behaviour of macroscopic samples of a similar material without the complicating presence of grain boundaries and other defects. Semiconductor nanocrystals having dimensions smaller than 10 nm are also described as quantum dots. Synthesis The t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
