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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 and Coxeter-Dynkin diagram . In the book '' Polyhedron Models'' by Magnus Wenninger, the polyhedron is misnamed ''great snub icosidodecahedron'', and vice versa. Cartesian coordinates Let \xi\approx -0.5055605785332548 be the largest (least negative) negative zero of the polynomial x^3+2x^2-\phi^, where \phi is the golden ratio. Let the point p be given by :p= \begin \xi \\ \phi^-\phi^\xi \\ -\phi^+\phi^\xi+2\phi^\xi^2 \end . Let the matrix M be given by :M= \begin 1/2 & -\phi/2 & 1/(2\phi) \\ \phi/2 & 1/(2\phi) & -1/2 \\ 1/(2\phi) & 1/2 & \phi/2 \end . M is the rotation around the axis (1, 0, \phi) by an angle of 2\pi/5, counterclockwise. Let the linear transformations T_0, \ldots, T_ be the transformations which send a point (x, y, z) to the ... [...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 and Coxeter-Dynkin diagram . In the book ''Polyhedron Models'' by Magnus Wenninger, the polyhedron is misnamed ''great snub icosidodecahedron'', and vice versa. Cartesian coordinates Let \xi\approx -0.5055605785332548 be the largest (least negative) negative zero of the polynomial x^3+2x^2-\phi^, where \phi is the golden ratio. Let the point p be given by :p= \begin \xi \\ \phi^-\phi^\xi \\ -\phi^+\phi^\xi+2\phi^\xi^2 \end . Let the matrix M be given by :M= \begin 1/2 & -\phi/2 & 1/(2\phi) \\ \phi/2 & 1/(2\phi) & -1/2 \\ 1/(2\phi) & 1/2 & \phi/2 \end . M is the rotation around the axis (1, 0, \phi) by an angle of 2\pi/5, counterclockwise. Let the linear transformations T_0, \ldots, T_ be the transformations which send a point (x, y, z) to the par ... [...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 Let \xi\approx -1.8934600671194555 be the smallest (most negative) zero of the polynomial x^3+2x^2-\phi^, where \phi is the golden ratio. Let the point p be given by :p= \begin \xi \\ \phi^-\phi^\xi \\ -\phi^+\phi^\xi+2\phi^\xi^2 \end . Let the matrix M be given by :M= \begin 1/2 & -\phi/2 & 1/(2\phi) \\ \phi/2 & 1/(2\phi) & -1/2 \\ 1/(2\phi) & 1/2 & \phi/2 \end . M is the rotation around the axis (1, 0, \phi) by an angle of 2\pi/5, counterclockwise. Let the linear transformations T_0, \ldots, T_ be the transformations which send a point (x, y, z) to the even permutations of (\pm x, \pm y, \pm z) with an even number of minus signs. The tr ... [...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 Let \xi\approx 0.3990206456527105 be the positive zero of the polynomial x^3+2x^2-\phi^, where \phi is the golden ratio. Let the point p be given by :p= \begin \xi \\ \phi^-\phi^\xi \\ -\phi^+\phi^\xi+2\phi^\xi^2 \end . Let the matrix M be given by :M= \begin 1/2 & -\phi/2 & 1/(2\phi) \\ \phi/2 & 1/(2\phi) & -1/2 \\ 1/(2\phi) & 1/2 & \phi/2 \end . M is the ro ... [...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 degene ... [...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 figure, isohedral polyhedron. It is the Dual polyhedron, 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 equals \arccos(\xi/(\xi+1))\approx 60.901\,133\,713\,21^. Part of each face lies inside the solid, hence is invisible in solid models. The other two zeroes of the polynomial P play a similar role in the description of the great pentagonal ... [...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 Father Magnus J. Wenninger OSB (October 31, 1919Banchoff (2002)– February 17, 2017) was an American mathematician who worked on constructing polyhedron models, and wrote the first book on their construction. Early life and education Born to ..., the polyhedron is misnamed ''great inverted snub icosidodecahedron'', and vice versa. Cartesian coordinates Let \xi\approx 0.3990206456527105 be the positive zero of the polynomial x^3+2x^2-\phi^, where \phi is the golden ratio. Let the point p be given by :p= \begin \xi ... [...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 ... [...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 pol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedron
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer either to a solid figure or to its boundary surface (mathematics), surface. The terms solid polyhedron and polyhedral surface are commonly used to distinguish the two concepts. Also, the term ''polyhedron'' is often used to refer implicitly to the whole structure (mathematics), structure formed by a solid polyhedron, its polyhedral surface, its faces, its edges, and its vertices. There are many definitions of polyhedron. Nevertheless, the polyhedron is typically understood as a generalization of a two-dimensional polygon and a three-dimensional specialization of a polytope, a more general concept in any number of dimensions. Polyhedra have several general characteristics that include the number of faces, topological classification by Eule ... [...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 Face (geometry), faces are the same. More specifically, all faces must be not merely Congruence (geometry), 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 Translation (geometry), translations, Rotation (mathematics), rotations, and/or Reflection (mathematics), reflections that maps onto . For this reason, Convex polytope, 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 Parity (mathematics), even number of faces. The Dual polyhedron, dual of an isohedral polyhedron is vertex-transitive, i.e. isogonal. The Catalan solids, the bipyramids, and the trapezo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
