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Medial Pentagonal Hexecontahedron
In geometry, the medial pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the snub dodecadodecahedron. It has 60 intersecting irregular pentagonal faces. Proportions Denote the golden ratio by \phi, and let \xi\approx -0.409\,037\,788\,014\,42 be the smallest (most negative) real zero of the polynomial P=8x^4-12x^3+5x+1. Then each face has three equal angles of \arccos(\xi)\approx 114.144\,404\,470\,43^, one of \arccos(\phi^2\xi+\phi)\approx 56.827\,663\,280\,94^ and one of \arccos(\phi^\xi-\phi^)\approx 140.739\,123\,307\,76^. Each face has one medium length edge, two short and two long ones. If the medium length is 2, then the short edges have length :1+\sqrt\approx 1.550\,761\,427\,20, and the long edges have length :1+\sqrt\approx 3.854\,145\,870\,08. The dihedral angle equals \arccos(\xi/(\xi+1))\approx 133.800\,984\,233\,53^. The other real zero of the polynomial P plays a similar role for the medial inverted pentagonal hexecontahedron In ... [...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] |
Isohedral Figure
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent but must be ''transitive'', i.e. must lie within the same '' symmetry orbit''. In other words, for any two faces and , there must be a symmetry of the ''entire'' figure by translations, rotations, and/or reflections that maps onto . For this reason, convex isohedral polyhedra are the shapes that will make fair dice. Isohedral polyhedra are called isohedra. They can be described by their face configuration. An isohedron has an even number of faces. The dual of an isohedral polyhedron is vertex-transitive, i.e. isogonal. The Catalan solids, the bipyramids, and the trapezohedra are all isohedral. They are the duals of the (isogonal) Archimedean solids, prisms, and antiprisms, respectively. The Platonic solids, which are either self-du ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on the same plane. Cubes and pyramids are examples of convex polyhedra. A polyhedron is a 3-dimensional example of a polytope, a more general concept in any number of dimensions. Definition Convex polyhedra are well-defined, with several equivalent standard definitions. However, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of "polyhedron" have been given within particular contexts,. some more rigorous than others, and there is not universal agreement over which of these to choose. Some of these definitions exclude shapes that have often been counted as polyhedra (such as the self-crossing polyhedra) or include shapes that are often not considered as valid polyhedr ... [...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] |
Snub Dodecadodecahedron
In geometry, the snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as . It has 84 faces (60 triangles, 12 pentagons, and 12 pentagrams), 150 edges, and 60 vertices. It is given a Schläfli symbol as a snub great dodecahedron. Cartesian coordinates Cartesian coordinates for the vertices of a snub dodecadodecahedron are all the even permutations of : (±2α, ±2, ±2β), : (±(α+β/τ+τ), ±(-ατ+β+1/τ), ±(α/τ+βτ-1)), : (±(-α/τ+βτ+1), ±(-α+β/τ-τ), ±(ατ+β-1/τ)), : (±(-α/τ+βτ-1), ±(α-β/τ-τ), ±(ατ+β+1/τ)) and : (±(α+β/τ-τ), ±(ατ-β+1/τ), ±(α/τ+βτ+1)), with an even number of plus signs, where : β = (α2/τ+τ)/(ατ−1/τ), where τ = (1+)/2 is the golden mean and α is the positive real root of τα4−α3+2α2−α−1/τ, or approximately 0.7964421. Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one. Related polyhedr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golden Ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( or \phi) denotes the golden ratio. The constant \varphi satisfies the quadratic equation \varphi^2 = \varphi + 1 and is an irrational number with a value of The golden ratio was called the extreme and mean ratio by Euclid, and the divine proportion by Luca Pacioli, and also goes by several other names. Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the construction of the dodecahedron and icosahedron. A golden rectangle—that is, a rectangle with an aspect ratio of \varphi—may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has been used to analyze the proportions of natural object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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] |
Medial Inverted Pentagonal Hexecontahedron
In geometry, the inverted snub dodecadodecahedron (or vertisnub dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U60. It is given a Schläfli symbol sr. Cartesian coordinates Cartesian coordinates for the vertices of an inverted snub dodecadodecahedron are all the even permutations of : (±2α, ±2, ±2β), : (±(α+β/τ+τ), ±(-ατ+β+1/τ), ±(α/τ+βτ-1)), : (±(-α/τ+βτ+1), ±(-α+β/τ-τ), ±(ατ+β-1/τ)), : (±(-α/τ+βτ-1), ±(α-β/τ-τ), ±(ατ+β+1/τ)) and : (±(α+β/τ-τ), ±(ατ-β+1/τ), ±(α/τ+βτ+1)), with an even number of plus signs, where : β = (α2/τ+τ)/(ατ−1/τ), where τ = (1+)/2 is the golden mean and α is the negative real root of τα4−α3+2α2−α−1/τ, or approximately −0.3352090. Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one. Related polyhedra Medial inverted pentagonal hexecontahedron The medial i ... [...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] |
