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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] |
Elongated Bipyramid
In geometry, the elongated bipyramids are an infinite set of polyhedra, constructed by elongating an bipyramid (by inserting an prism between its congruent halves). There are three ''elongated bipyramids'' that are Johnson solids: * Elongated triangular bipyramid (), * Elongated square bipyramid (), and * Elongated pentagonal bipyramid (). Higher forms can be constructed with isosceles triangles. Forms See also * Gyroelongated bipyramid * Gyroelongated pyramid * Elongated pyramid * Diminished trapezohedron In geometry, a diminished trapezohedron is a polyhedron in an infinite set of polyhedra, constructed by removing one of the polar vertices of a trapezohedron and replacing it by a new face (diminishment). It has one regular base face, triangle ... References * Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elongated Bipyramid
In geometry, the elongated bipyramids are an infinite set of polyhedra, constructed by elongating an bipyramid (by inserting an prism between its congruent halves). There are three ''elongated bipyramids'' that are Johnson solids: * Elongated triangular bipyramid (), * Elongated square bipyramid (), and * Elongated pentagonal bipyramid (). Higher forms can be constructed with isosceles triangles. Forms See also * Gyroelongated bipyramid * Gyroelongated pyramid * Elongated pyramid * Diminished trapezohedron In geometry, a diminished trapezohedron is a polyhedron in an infinite set of polyhedra, constructed by removing one of the polar vertices of a trapezohedron and replacing it by a new face (diminishment). It has one regular base face, triangle ... References * Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elongated Dipyramid
In geometry, the elongated bipyramids are an infinite set of polyhedra, constructed by elongating an bipyramid (by inserting an prism between its congruent halves). There are three ''elongated bipyramids'' that are Johnson solids: * Elongated triangular bipyramid (), * Elongated square bipyramid (), and * Elongated pentagonal bipyramid (). Higher forms can be constructed with isosceles triangles. Forms See also * Gyroelongated bipyramid * Gyroelongated pyramid * Elongated pyramid * Diminished trapezohedron References *Norman W. Johnson Norman Woodason Johnson () was a mathematician at Wheaton College, Norton, Massachusetts. Early life and education Norman Johnson was born on in Chicago. His father had a bookstore and published a local newspaper. Johnson earned his unde ..., "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no ... [...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] |
Square Bifrustum
The square bifrustum or ''square truncated bipyramid'' is the second in an infinite series of bifrustum polyhedra. It has 4 trapezoidal and 2 square faces. This polyhedron can be constructed by taking a square bipyramid (octahedron) and truncating the polar axis vertices, making it into two end-to-end frustums. It is dual 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 ... to the elongated square dipyramid. Polyhedra {{Polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagonal Bifrustum
The pentagonal bifrustum or truncated pentagonal bipyramid is the third in an infinite series of bifrustum polyhedra. It has 10 trapezoid and 2 pentagonal faces. Constructions The pentagonal bifrustum is the dual polyhedron of a Johnson solid, the elongated pentagonal bipyramid. This polyhedron can be constructed by taking a pentagonal bipyramid and truncating the polar axis vertices. In Conway's notation for polyhedra, it can be represented as the polyhedron "t5dP5", meaning the truncation of the degree-five vertices of the dual of a pentagonal prism. Alternatively, it can be constructed by gluing together two end-to-end pentagonal frustums, or (if coplanar faces are allowed) by gluing together two pentagonal prisms on their pentagonal faces. Application In the formation of quasicrystals, a 15-site truncated pentagonal bipyramid structure may form the nucleus Nucleus ( : nuclei) is a Latin word for the seed inside a fruit. It most often refers to: *Atomic nucleus, the v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Elongated Square Bipyramid
In geometry, the elongated square bipyramid (or elongated octahedron) is one of the Johnson solids (). As the name suggests, it can be constructed by elongating an octahedron by inserting a cube between its congruent halves. It has been named the pencil cube or 12-faced pencil cube due to its shape.Order in Space: A design source book, Keith Critchlow, p.46-47 A zircon crystal is an example of an elongated square bipyramid. Formulae The following formulae for volume (V), surface area (A) and height (H) can be used if all faces are regular, with edge length L: :V = L^3\cdot \left( 1 + \frac\right) \approx L^3\cdot 1.471404521 :A = L^2\cdot \left(4 + 2\sqrt\right) \approx L^2\cdot 7.464101615 :H = L\cdot \left( 1 + \sqrt\right) \approx L\cdot 2.414213562 Dual polyhedron The dual of the elongated square bipyramid is called a square bifrustum and has 10 faces: 8 trapezoidal and 2 square. Related polyhedra and honeycombs A special kind of elongated square bipyramid ... [...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] |
Elongated Pentagonal Bipyramid
In geometry, the elongated pentagonal bipyramid or pentakis pentagonal prism is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a pentagonal bipyramid () by inserting a pentagonal prism between its congruent halves. Dual polyhedron The dual of the elongated square bipyramid is a pentagonal bifrustum The pentagonal bifrustum or truncated pentagonal bipyramid is the third in an infinite series of bifrustum polyhedra. It has 10 trapezoid and 2 pentagonal faces. Constructions The pentagonal bifrustum is the dual polyhedron of a Johnson solid, .... See also * Elongated pentagonal pyramid External links * Johnson solids Pyramids and bipyramids {{Polyhedron-stub ... [...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] |
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] |
