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Great Inverted Pentagonal Hexecontahedron
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 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 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] |
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 Pentagrammic Hexecontahedron
In geometry, the great pentagrammic hexecontahedron (or great dentoid ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the great retrosnub icosidodecahedron. Its 60 faces are irregular pentagrams. Proportions Denote the golden ratio by \phi. Let \xi\approx 0.946\,730\,033\,56 be the largest positive zero of the polynomial P = 8x^3-8x^2+\phi^. Then each pentagrammic face has four equal angles of \arccos(\xi)\approx 18.785\,633\,958\,24^ and one angle of \arccos(-\phi^+\phi^\xi)\approx 104.857\,464\,167\,03^. Each face has three long and two short edges. The ratio l between the lengths of the long and the short edges is given by :l = \frac\approx 1.774\,215\,864\,94. 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(\xi/ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Pentagonal Hexecontahedron
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] |
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] |
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] |
Concave Polygon
A simple polygon that is not convex is called concave, non-convex or reentrant. A concave polygon will always have at least one reflex interior angle—that is, an angle with a measure that is between 180 degrees and 360 degrees exclusive. Polygon Some lines containing interior points of a concave polygon intersect its boundary at more than two points. Some diagonals of a concave polygon lie partly or wholly outside the polygon. Some sidelines of a concave polygon fail to divide the plane into two half-planes one of which entirely contains the polygon. None of these three statements holds for a convex polygon. As with any simple polygon, the sum of the internal angles of a concave polygon is ×(''n'' − 2) radians, equivalently 180×(''n'' − 2) degrees (°), where ''n'' is the number of sides. It is always possible to partition a concave polygon into a set of convex polygons. A polynomial-time algorithm for finding a decomposition into as few convex polyg ... [...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] |
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] |
