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Hexagonal Bifrustum
The hexagonal bifrustum or truncated hexagonal bipyramid is the fourth in an infinite series of bifrustum polyhedra. It has 12 trapezoid and 2 hexagonal faces. This polyhedron can be constructed by taking a hexagonal dipyramid and truncating the polar axis vertices, making it into two end-to-end frustums. Several types of crystal take this shape. It has also been used in the design of 14-sided dice, which may be used to generate randomly chosen playing card A playing card is a piece of specially prepared card stock, heavy paper, thin cardboard, plastic-coated paper, cotton-paper blend, or thin plastic that is marked with distinguishing motifs. Often the front (face) and back of each card has a f ...s. Patent US 8074986 B1, Douglas A. Gebhart, filed September 30, 2008. It also has app ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hexagonal Bifrustum
The hexagonal bifrustum or truncated hexagonal bipyramid is the fourth in an infinite series of bifrustum polyhedra. It has 12 trapezoid and 2 hexagonal faces. This polyhedron can be constructed by taking a hexagonal dipyramid and truncating the polar axis vertices, making it into two end-to-end frustums. Several types of crystal take this shape. It has also been used in the design of 14-sided dice, which may be used to generate randomly chosen playing card A playing card is a piece of specially prepared card stock, heavy paper, thin cardboard, plastic-coated paper, cotton-paper blend, or thin plastic that is marked with distinguishing motifs. Often the front (face) and back of each card has a f ...s. Patent US 8074986 B1, Douglas A. Gebhart, filed September 30, 2008. It also has app ... [...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 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 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 from τετράς (''tetrás'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has Schläfli symbol and can also be constructed as a truncated equilateral triangle, t, which alternates two types of edges. A regular hexagon is defined as a hexagon that is both equilateral and equiangular. It is bicentric, meaning that it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle). The common length of the sides equals the radius of the circumscribed circle or circumcircle, which equals \tfrac times the apothem (radius of the inscribed circle). All internal angles are 120 degrees. A regular hexagon has six rotational symmetries (''rotational symmetry of order six'') and six reflection symmetries (''six lines of symmetry''), making up the dihedral group D6. The longest diagonals 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 fiel ... [...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 fiel ... [...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 ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elongated Hexagonal Dipyramid
In geometry, the elongated hexagonal bipyramid is constructed by elongating a hexagonal bipyramid (by inserting a hexagonal prism between its congruent halves). Related polyhedra This polyhedron is in the family of elongated bipyramids, of which the first three can be Johnson solids: J14, J15 and J16. The hexagonal form can be constructed by all regular faces, but is not a ''Johnson solid'' because 6 equilateral triangles would form six co-planar faces (in a regular hexagon). Uses * A quartz crystal is an example of an elongated hexagonal bipyramid. Because it has 18 faces, it can be called an octadecahedron. Other chemicals also have this shape. [...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] |
Hexagonal Dipyramid
A hexagonal bipyramid is a polyhedron formed from two hexagonal pyramids joined at their bases. The resulting solid has 12 triangular faces, 8 vertices and 18 edges. The 12 faces are identical isosceles triangles. Although it is face-transitive, it is not a Platonic solid because some vertices have four faces meeting and others have six faces, and it is not a Johnson solid because its faces cannot be equilateral triangles; 6 equilateral triangles would make a flat vertex. It is one of an infinite set of bipyramids. Having twelve faces, it is a type of dodecahedron, although that name is usually associated with the regular polyhedral form with pentagonal faces. The hexagonal bipyramid has a plane of symmetry (which is horizontal in the figure to the right) where the bases of the two pyramids are joined. This plane is a regular hexagon. There are also six planes of symmetry crossing through the two apices. These planes are rhombic and lie at 30° angles to each other, perp ... [...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 frust ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crystal
A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macroscopic single crystals are usually identifiable by their geometrical shape, consisting of flat faces with specific, characteristic orientations. The scientific study of crystals and crystal formation is known as crystallography. The process of crystal formation via mechanisms of crystal growth is called crystallization or solidification. The word ''crystal'' derives from the Ancient Greek word (), meaning both "ice" and "rock crystal", from (), "icy cold, frost". Examples of large crystals include snowflakes, diamonds, and table salt. Most inorganic solids are not crystals but polycrystals, i.e. many microscopic crystals fused together into a single solid. Polycrystals include most metals, rocks, ceramics, and ice. A third category ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
