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Great Deltoidal Icositetrahedron
In geometry, the great deltoidal icositetrahedron (or great sagittal disdodecahedron) is the dual of the nonconvex great rhombicuboctahedron. Its faces are darts. Part of each dart lies inside the solid, hence is invisible in solid models. One of its halves can be rotated by 45 degrees to form the pseudo great deltoidal icositetrahedron, analogous to the pseudo-deltoidal icositetrahedron. Proportions Faces have three angles of \arccos(\frac+\frac\sqrt)\approx 31.399\,714\,809\,92^ and one of 360^-\arccos(-\frac+\frac\sqrt)\approx 265.800\,855\,570\,24^. Its dihedral angles A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the uni ... equal \arccos()\approx 94.531\,580\,798\,20^. The ratio between the lengths of the long edges and the short ones equals 2+\frac\sqrt\approx 2.707\,106\,781\ ... [...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] |
Nonconvex Great Rhombicuboctahedron
In geometry, the nonconvex great rhombicuboctahedron is a nonconvex uniform polyhedron, indexed as U17. It has 26 faces (8 triangles and 18 squares), 48 edges, and 24 vertices. It is represented by the Schläfli symbol rr and Coxeter-Dynkin diagram of . Its vertex figure is a crossed quadrilateral. This model shares the name with the convex ''great rhombicuboctahedron'', also called the truncated cuboctahedron. An alternative name for this figure is quasirhombicuboctahedron. From that derives its Bowers acronym: querco. Orthographic projections Cartesian coordinates Cartesian coordinates for the vertices of a ''nonconvex great rhombicuboctahedron'' centered at the origin with edge length 1 are all the permutations of : (±''ξ'', ±1, ±1), where ''ξ'' = − 1. Related polyhedra It shares the vertex arrangement with the convex truncated cube. It additionally shares its edge arrangement In geometry, a vertex arrangement is a set of points in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo Great Deltoidal Icositetrahedron
The prefix pseudo- (from Greek ψευδής, ''pseudes'', "false") is used to mark something that superficially appears to be (or behaves like) one thing, but is something else. Subject to context, ''pseudo'' may connote coincidence, imitation, intentional deception, or a combination thereof. * In scholarship and studies, pseudo-scholarship refers to material that is presented as, but is not, the product of rigorous and objective study or research. Examples: ** Pseudoarchaeology ** Pseudohistory ** Pseudolinguistics *** Pseudoscientific language comparison *** Folk linguistics ** Pseudomathematics ** Pseudophilosophy ** Pseudonym ** Pseudoscience **Pseudoculture * In biology and botany, the prefix 'pseudo' is used to indicate a species with a coincidental visual similarity to another genus. For example, '' Iris pseudacorus'' is known as '''pseud''acorus' for having leaves similar to those of '' Acorus calamus''. In biology, coincidental similarity is not the same as mimicry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo-deltoidal Icositetrahedron
The pseudo-deltoidal icositetrahedron is a convex polyhedron with congruent kites as its faces. It is the dual of the elongated square gyrobicupola, also known as the pseudorhombicuboctahedron. As the pseudorhombicuboctahedron is tightly related to the rhombicuboctahedron, but has a twist along an equatorial belt of faces (and edges), the pseudo-deltoidal icositetrahedron is tightly related to the deltoidal icositetrahedron, but has a twist along an equator of (vertices and) edges. Properties Vertices As the faces of the pseudorhombicuboctahedron are regular, the vertices of the pseudo-deltoidal icositetrahedron are regular. But due to the twist, these vertices are of four different kinds: *eight vertices connecting three short edges (yellow vertices in 1st figure below), *two apices connecting four long edges (top and bottom vertices, light red in 1st figure below), *eight vertices connecting four alternate edges: short-long-short-long (dark red vertices in 1st figure below), * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dihedral Angles
A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the union of a line and two half-planes that have this line as a common edge. In higher dimensions, a dihedral angle represents the angle between two hyperplanes. The planes of a flying machine are said to be at positive dihedral angle when both starboard and port main planes (commonly called wings) are upwardly inclined to the lateral axis. When downwardly inclined they are said to be at a negative dihedral angle. Mathematical background When the two intersecting planes are described in terms of Cartesian coordinates by the two equations : a_1 x + b_1 y + c_1 z + d_1 = 0 :a_2 x + b_2 y + c_2 z + d_2 = 0 the dihedral angle, \varphi between them is given by: :\cos \varphi = \frac and satisfies 0\le \varphi \le \pi/2. Alternatively, if an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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