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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] |
Dual Elongated Pentagonal Dipyramid
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 t ... [...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] |
Quasicrystal
A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. While crystals, according to the classical crystallographic restriction theorem, can possess only two-, three-, four-, and six-fold rotational symmetries, the Bragg diffraction pattern of quasicrystals shows sharp peaks with other symmetry orders—for instance, five-fold. Aperiodic tilings were discovered by mathematicians in the early 1960s, and, some twenty years later, they were found to apply to the study of natural quasicrystals. The discovery of these aperiodic forms in nature has produced a paradigm shift in the field of crystallography. In crystallography the quasicrystals were predicted in 1981 by a five-fold symmetry study of Alan Lindsay Mackay,—that also brought in 1982, with the crystallographic Fourier transform of a Penrose tiling,Alan L. Mackay, "Crystallogra ... [...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] |
George W
George Walker Bush (born July 6, 1946) is an American politician who served as the 43rd president of the United States from 2001 to 2009. A member of the Republican Party, Bush family, and son of the 41st president George H. W. Bush, he previously served as the 46th governor of Texas from 1995 to 2000. While in his twenties, Bush flew warplanes in the Texas Air National Guard. After graduating from Harvard Business School in 1975, he worked in the oil industry. In 1978, Bush unsuccessfully ran for the House of Representatives. He later co-owned the Texas Rangers of Major League Baseball before he was elected governor of Texas in 1994. As governor, Bush successfully sponsored legislation for tort reform, increased education funding, set higher standards for schools, and reformed the criminal justice system. He also helped make Texas the leading producer of wind powered electricity in the nation. In the 2000 presidential election, Bush defeated Democratic incum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagonal Prism
In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with seven faces, fifteen edges, and ten vertices. As a semiregular (or uniform) polyhedron If faces are all regular, the pentagonal prism is a semiregular polyhedron, more generally, a uniform polyhedron, and the third in an infinite set of prisms formed by square sides and two regular polygon caps. It can be seen as a '' truncated pentagonal hosohedron'', represented by Schläfli symbol t. Alternately it can be seen as the Cartesian product of a regular pentagon and a line segment, and represented by the product ×. The dual of a pentagonal prism is a pentagonal bipyramid. The symmetry group of a right pentagonal prism is ''D5h'' of order 20. The rotation group is ''D5'' of order 10. Volume The volume, as for all prisms, is the product of the area of the pentagonal base times the height or distance along any edge perpendicular to the base. For a uniform pentagonal prism ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches of recreational mathematics, most notably the invention of the cellular automaton called the Game of Life. Born and raised in Liverpool, Conway spent the first half of his career at the University of Cambridge before moving to the United States, where he held the John von Neumann Professorship at Princeton University for the rest of his career. On 11 April 2020, at age 82, he died of complications from COVID-19. Early life and education Conway was born on 26 December 1937 in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at a very early age. By the time he was 11, his ambition was to become a mathematician. After leaving sixth form, he studied mathematics at Gonville and Caius College, Ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagonal Bipyramid
In geometry, the pentagonal bipyramid (or dipyramid) is third of the infinite set of face-transitive bipyramids, and the 13th Johnson solid (). Each bipyramid is the dual of a uniform prism. Although it is face-transitive, it is not a Platonic solid because some vertices have four faces meeting and others have five faces. Properties If the faces are equilateral triangles, it is a deltahedron and a Johnson solid (''J''13). It can be seen as two pentagonal pyramids (''J''2) connected by their bases. : The pentagonal dipyramid is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size. The other three polyhedra with this property are the regular octahedron, the snub disphenoid, and an irregular polyhedron with 12 vertices and 20 triangular faces.. Formulae The fol ... [...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] |
Johnson Solid
In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson solid is the square-based pyramid with equilateral sides ( ); it has 1 square face and 4 triangular faces. Some authors require that the solid not be uniform (i.e., not Platonic solid, Archimedean solid, uniform prism, or uniform antiprism) before they refer to it as a “Johnson solid”. As in any strictly convex solid, at least three faces meet at every vertex, and the total of their angles is less than 360 degrees. Since a regular polygon has angles at least 60 degrees, it follows that at most five faces meet at any vertex. The pentagonal pyramid () is an example that has a degree-5 vertex. Although there is no obvious restriction that any given regular polygon cannot be a face of a Johnson solid, it turns out that the faces of ... [...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] |
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] |
