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Small Hexacronic Icositetrahedron
In geometry, the small hexacronic icositetrahedron is the dual of the small cubicuboctahedron. It is visually identical to the small rhombihexacron. A part of each dart lies inside the solid, hence is invisible in solid models. Proportions Its faces are darts, having two angles of \arccos(\frac+\frac\sqrt)\approx 16.842\,116\,236\,30^, one of \arccos(\frac-\frac\sqrt)\approx 81.578\,941\,881\,85^ and one of 360^-\arccos(-\frac-\frac\sqrt)\approx 244.736\,825\,645\,55^. 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 138.117\,959\,055\,51^. The ratio between the lengths of the long edges and the short ones equals 2-\frac\sqrt\approx 1.292\,893\,218\,81. References * External links * Dual uniform polyhedra {{polyhedron-stub ... [...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 geom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Cubicuboctahedron
In geometry, the small cubicuboctahedron is a uniform star polyhedron, indexed as U13. It has 20 faces (8 triangles, 6 squares, and 6 octagons), 48 edges, and 24 vertices. Its vertex figure is a crossed quadrilateral. The small cubicuboctahedron is a faceting of the rhombicuboctahedron. Its square faces and its octagonal faces are parallel to those of a cube, while its triangular faces are parallel to those of an octahedron: hence the name ''cubicuboctahedron''. The ''small'' suffix serves to distinguish it from the great cubicuboctahedron, which also has faces in the aforementioned directions. Related polyhedra It shares its vertex arrangement with the stellated truncated hexahedron. It additionally shares its edge arrangement with the rhombicuboctahedron (having the triangular faces and 6 square faces in common), and with the small rhombihexahedron (having the octagonal faces in common). Related tilings As the Euler characteristic suggests, the small cubicuboctah ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Rhombihexacron
In geometry, the small rhombihexacron (or small dipteral disdodecahedron) is the dual of the small rhombihexahedron. It is visually identical to the small hexacronic icositetrahedron. Its faces are antiparallelograms formed by pairs of coplanar triangles. Proportions Each antiparallelogram has two angles of \arccos(\frac+\frac\sqrt)\approx 16.842\,116\,236\,30^ and two angles of \arccos(-\frac+\frac\sqrt)\approx 98.421\,058\,118\,15^. The diagonals of each antiparallelogram intersect at an angle of \arccos(\frac+\frac\sqrt)\approx 64.736\,825\,645\,55^. The dihedral angle 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 un ... equals \arccos(\frac)\approx 138.117\,959\,055\,51^. The ratio between the lengths of the long edges and the short ones equals \sqrt. References * External li ... [...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 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 university textbooks, and English language teaching and learning publications. It also publishes Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Spo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
