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Great Retrosnub Icosidodecahedron
In geometry, the great retrosnub icosidodecahedron or great inverted retrosnub icosidodecahedron is a nonconvex uniform polyhedron, indexed as . It has 92 faces (80 triangles and 12 pentagrams), 150 edges, and 60 vertices. It is given a Schläfli symbol Cartesian coordinates Cartesian coordinates for the vertices of a great retrosnub icosidodecahedron are all the even permutations of : (±2α, ±2, ±2β), : (±(α−βτ−1/τ), ±(α/τ+β−τ), ±(−ατ−β/τ−1)), : (±(ατ−β/τ+1), ±(−α−βτ+1/τ), ±(−α/τ+β+τ)), : (±(ατ−β/τ−1), ±(α+βτ+1/τ), ±(−α/τ+β−τ)) and : (±(α−βτ+1/τ), ±(−α/τ−β−τ), ±(−ατ−β/τ+1)), with an even number of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Retrosnub Icosidodecahedron
In geometry, the great retrosnub icosidodecahedron or great inverted retrosnub icosidodecahedron is a nonconvex uniform polyhedron, indexed as . It has 92 faces (80 triangles and 12 pentagrams), 150 edges, and 60 vertices. It is given a Schläfli symbol Cartesian coordinates Cartesian coordinates for the vertices of a great retrosnub icosidodecahedron are all the even permutations of : (±2α, ±2, ±2β), : (±(α−βτ−1/τ), ±(α/τ+β−τ), ±(−ατ−β/τ−1)), : (±(ατ−β/τ+1), ±(−α−βτ+1/τ), ±(−α/τ+β+τ)), : (±(ατ−β/τ−1), ±(α+βτ+1/τ), ±(−α/τ+β−τ)) and : (±(α−βτ+1/τ), ±(−α/τ−β−τ), ±(−ατ−β/τ+1)), with an even number of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Odd Permutation
In mathematics, when ''X'' is a finite set with at least two elements, the permutations of ''X'' (i.e. the bijective functions from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of ''X'' is fixed, the parity (oddness or evenness) of a permutation \sigma of ''X'' can be defined as the parity of the number of inversions for ''σ'', i.e., of pairs of elements ''x'', ''y'' of ''X'' such that and . The sign, signature, or signum of a permutation ''σ'' is denoted sgn(''σ'') and defined as +1 if ''σ'' is even and −1 if ''σ'' is odd. The signature defines the alternating character of the symmetric group S''n''. Another notation for the sign of a permutation is given by the more general Levi-Civita symbol (''ε''''σ''), which is defined for all maps from ''X'' to ''X'', and has value zero for non-bijective maps. The sign of a permutation can be explicitly expressed as : where ''N''(''σ'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Snub Icosidodecahedron
In geometry, the great snub icosidodecahedron is a nonconvex uniform polyhedron, indexed as U57. It has 92 faces (80 triangles and 12 pentagrams), 150 edges, and 60 vertices. It can be represented by a Schläfli symbol sr, and Coxeter-Dynkin diagram . This polyhedron is the snub member of a family that includes the great icosahedron, the great stellated dodecahedron and the great icosidodecahedron. In the book '' Polyhedron Models'' by Magnus Wenninger, the polyhedron is misnamed ''great inverted snub icosidodecahedron'', and vice versa. Cartesian coordinates Cartesian coordinates for the vertices of a great snub icosidodecahedron are all the even permutations of : (±2α, ±2, ±2β), : (±(α−βτ−1/τ), ±(α/τ+β−τ), ±(−ατ−β/τ−1)), : (±(ατ−β/τ+1), ±(−α−βτ+1/τ), ±(−α/τ+β+τ)), : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Uniform Polyhedra
In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive ( transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational symmetry. Uniform polyhedra can be divided between convex forms with convex regular polygon faces and star forms. Star forms have either regular star polygon faces or vertex figures or both. This list includes these: * all 75 nonprismatic uniform polyhedra; * a few representatives of the infinite sets of prisms and antiprisms; * one degenerate polyhedron, Skilling's figure with overlapping edges. It was proven in that there are only 75 uniform polyhedra other than the infinite families of prisms and antiprisms. John Skilling discovered an overlooked degenerate example, by relaxing the condition that only two faces may meet at an edge. This is a degenerate u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Inverted Snub Icosidodecahedron
In geometry, the great inverted snub icosidodecahedron (or great vertisnub icosidodecahedron) is a uniform star polyhedron, indexed as U69. It is given a Schläfli symbol sr, and Coxeter-Dynkin diagram . In the book '' Polyhedron Models'' by Magnus Wenninger, the polyhedron is misnamed '' great snub icosidodecahedron'', and vice versa. Cartesian coordinates Cartesian coordinates for the vertices of a great inverted snub icosidodecahedron are all the even permutations of : (±2α, ±2, ±2β), : (±(α−βτ−1/τ), ±(α/τ+β−τ), ±(−ατ−β/τ−1)), : (±(ατ−β/τ+1), ±(−α−βτ+1/τ), ±(−α/τ+β+τ)), : (±(ατ−β/τ−1), ±(α+βτ+1/τ), ±(−α/τ+β−τ)) and : (±(α−βτ+1/τ), ±(−α/τ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Snub Icosidodecahedron
In geometry, the great snub icosidodecahedron is a nonconvex uniform polyhedron, indexed as U57. It has 92 faces (80 triangles and 12 pentagrams), 150 edges, and 60 vertices. It can be represented by a Schläfli symbol sr, and Coxeter-Dynkin diagram . This polyhedron is the snub member of a family that includes the great icosahedron, the great stellated dodecahedron and the great icosidodecahedron. In the book '' Polyhedron Models'' by Magnus Wenninger, the polyhedron is misnamed ''great inverted snub icosidodecahedron'', and vice versa. Cartesian coordinates Cartesian coordinates for the vertices of a great snub icosidodecahedron are all the even permutations of : (±2α, ±2, ±2β), : (±(α−βτ−1/τ), ±(α/τ+β−τ), ±(−ατ−β/τ−1)), : (±(ατ−β/τ+1), ±(−α−βτ+1/τ), ±(−α/τ+β+τ)), : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Snub Dodecahedron
In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces. The snub dodecahedron has 92 faces (the most of the 13 Archimedean solids): 12 are pentagons and the other 80 are equilateral triangles. It also has 150 edges, and 60 vertices. It has two distinct forms, which are mirror images (or " enantiomorphs") of each other. The union of both forms is a compound of two snub dodecahedra, and the convex hull of both forms is a truncated icosidodecahedron. Kepler first named it in Latin as dodecahedron simum in 1619 in his Harmonices Mundi. H. S. M. Coxeter, noting it could be derived equally from either the dodecahedron or the icosahedron, called it snub icosidodecahedron, with a vertical extended Schläfli symbol s \scriptstyle\begin 5 \\ 3 \end and flat Schläfli symbol sr. Cartesian coordinates Let ''ξ'' ≈ be the real zero of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sextic Equation
In algebra, a sextic (or hexic) polynomial is a polynomial of degree six. A sextic equation is a polynomial equation of degree six—that is, an equation whose left hand side is a sextic polynomial and whose right hand side is zero. More precisely, it has the form: :ax^6+bx^5+cx^4+dx^3+ex^2+fx+g=0,\, where and the ''coefficients'' may be integers, rational numbers, real numbers, complex numbers or, more generally, members of any field. A sextic function is a function defined by a sextic polynomial. Because they have an even degree, sextic functions appear similar to quartic functions when graphed, except they may possess an additional local maximum and local minimum each. The derivative of a sextic function is a quintic function. Since a sextic function is defined by a polynomial with even degree, it has the same infinite limit when the argument goes to positive or negative infinity. If the leading coefficient is positive, then the function increases to positive infinity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chirality (mathematics)
In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. An object that is not chiral is said to be ''achiral''. A chiral object and its mirror image are said to be enantiomorphs. The word ''chirality'' is derived from the Greek (cheir), the hand, the most familiar chiral object; the word ''enantiomorph'' stems from the Greek (enantios) 'opposite' + (morphe) 'form'. Examples Some chiral three-dimensional objects, such as the helix, can be assigned a right or left handedness, according to the right-hand rule. Many other familiar objects exhibit the same chiral symmetry of the human body, such as gloves and shoes. Right shoes differ from left shoes only by being mirror images of each other. In contrast thin gloves may not be considered chiral if you can wear them inside-out. The J, L, S and Z-shaped ''tetrominoes'' of the popul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero Of A Function
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or equivalently, x is the solution to the equation f(x) = 0. A "zero" of a function is thus an input value that produces an output of 0. A root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial f of degree two, defined by f(x)=x^2-5x+6 has the two roots (or zeros) that are 2 and 3. f(2)=2^2-5\times 2+6= 0\textf(3)=3^2-5\times 3+6=0. If the function maps real numbers to real numbers, then it ... [...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] |
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] |
