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Small Dodecicosidodecahedron
In geometry, the small dodecicosidodecahedron (or small dodekicosidodecahedron) is a nonconvex uniform polyhedron, indexed as U33. It has 44 faces (20 triangles, 12 pentagons, and 12 decagons), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral. Related polyhedra It shares its vertex arrangement with the small stellated truncated dodecahedron and the uniform compounds of 6 or 12 pentagrammic prisms. It additionally shares its edge arrangement with the rhombicosidodecahedron (having the triangular and pentagonal faces in common), and with the small rhombidodecahedron (having the decagonal faces in common). Dual The dual polyhedron to the small dodecicosidodecahedron is the small dodecacronic hexecontahedron (or small sagittal ditriacontahedron). It is visually identical to the small rhombidodecacron. Its faces are darts. A part of each dart lies inside the solid, hence is invisible in solid models. Proportions Faces have two angles of \arccos(\f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Dodecicosidodecahedron
In geometry, the small dodecicosidodecahedron (or small dodekicosidodecahedron) is a nonconvex uniform polyhedron, indexed as U33. It has 44 faces (20 triangles, 12 pentagons, and 12 decagons), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral. Related polyhedra It shares its vertex arrangement with the small stellated truncated dodecahedron and the uniform compounds of 6 or 12 pentagrammic prisms. It additionally shares its edge arrangement with the rhombicosidodecahedron (having the triangular and pentagonal faces in common), and with the small rhombidodecahedron (having the decagonal faces in common). Dual The dual polyhedron to the small dodecicosidodecahedron is the small dodecacronic hexecontahedron (or small sagittal ditriacontahedron). It is visually identical to the small rhombidodecacron. Its faces are darts. A part of each dart lies inside the solid, hence is invisible in solid models. Proportions Faces have two angles of \arccos(\f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Rhombidodecahedron
In geometry, the small rhombidodecahedron is a nonconvex uniform polyhedron, indexed as U39. It has 42 faces (30 squares and 12 decagons), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral. Related polyhedra It shares its vertex arrangement with the small stellated truncated dodecahedron and the uniform compounds of 6 or 12 pentagrammic prisms. It additionally shares its edge arrangement with the rhombicosidodecahedron (having the square faces in common), and with the small dodecicosidodecahedron (having the decagonal faces in common). Small rhombidodecacron The small rhombidodecacron (or small dipteral ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the small rhombidodecahedron. It is visually identical to the Small dodecacronic hexecontahedron. It has 60 intersecting antiparallelogram In geometry, an antiparallelogram is a type of self-crossing quadrilateral. Like a parallelogram, an antiparallelogram has two op ... [...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] |
Small Rhombidodecacron
In geometry, the small rhombidodecacron is a nonconvex isohedral polyhedron. It is the dual of the small rhombidodecahedron. It is visually identical to the Small dodecacronic hexecontahedron. It has 60 intersecting antiparallelogram faces. Proportions Each face has two angles of \arccos(\frac+\frac\sqrt)\approx 25.242\,832\,961\,52^ and two angles of \arccos(-\frac+\frac\sqrt)\approx 93.025\,844\,508\,96^. The diagonals of each antiparallelogram intersect at an angle of \arccos(\frac+\frac\sqrt)\approx 61.731\,322\,529\,52^. The ratio between the lengths of the long edges and the short ones equals \frac+\frac\sqrt, which is the golden ratio. 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 154.121\,363\,125\,78^. References * ... [...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] |
Small Dodecacronic Hexecontahedron
In geometry, the small dodecicosidodecahedron (or small dodekicosidodecahedron) is a nonconvex uniform polyhedron, indexed as U33. It has 44 faces (20 triangles, 12 pentagons, and 12 decagons), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral. Related polyhedra It shares its vertex arrangement with the small stellated truncated dodecahedron and the uniform compounds of 6 or 12 pentagrammic prisms. It additionally shares its edge arrangement with the rhombicosidodecahedron (having the triangular and pentagonal faces in common), and with the small rhombidodecahedron (having the decagonal faces in common). Dual The dual polyhedron to the small dodecicosidodecahedron is the small dodecacronic hexecontahedron (or small sagittal ditriacontahedron). It is visually identical to the small rhombidodecacron. Its faces are darts. A part of each dart lies inside the solid, hence is invisible in solid models. Proportions Faces have two angles of \arccos(\fra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Of Twelve Pentagrammic Prisms
This uniform polyhedron compound is a symmetric arrangement of 12 pentagrammic prisms, aligned in pairs with the axes of fivefold rotational symmetry of a dodecahedron. It results from composing the two enantiomorphs of the compound of six pentagrammic prisms This uniform polyhedron compound is a chiral symmetric arrangement of 6 pentagrammic prism In geometry, the pentagrammic prism is one of an infinite set of nonconvex prisms formed by square sides and two regular star polygon caps, in this cas .... In doing so, the vertices of the two enantiomorphs coincide, with the result that the full compound has two pentagrammic prisms incident on each of its vertices. Related polyhedra This compound shares its vertex arrangement with four uniform polyhedra as follows: References *. Polyhedral compounds {{polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Of Six Pentagrammic Prisms
This uniform polyhedron compound is a chiral symmetric arrangement of 6 pentagrammic prism In geometry, the pentagrammic prism is one of an infinite set of nonconvex prisms formed by square sides and two regular star polygon caps, in this case two pentagrams. It is a special case of a right prism with a pentagram as base, which in ge ...s, aligned with the axes of fivefold rotational symmetry of a dodecahedron. Related polyhedra This compound shares its vertex arrangement with four uniform polyhedra as follows: References *. Polyhedral compounds {{polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Stellated Truncated Dodecahedron
In geometry, the small stellated truncated dodecahedron (or quasitruncated small stellated dodecahedron or small stellatruncated dodecahedron) is a nonconvex uniform polyhedron, indexed as U58. It has 24 faces (12 pentagons and 12 decagrams), 90 edges, and 60 vertices. It is given a Schläfli symbol t, and Coxeter diagram . Related polyhedra It shares its vertex arrangement with three other uniform polyhedra: the convex rhombicosidodecahedron, the small dodecicosidodecahedron and the small rhombidodecahedron. It also has the same vertex arrangement as the uniform compounds of 6 or 12 pentagrammic prisms. See also * 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 c ... References External links * Uniform polyhedra {{Polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Rhombidodecahedron
In geometry, the small rhombidodecahedron is a nonconvex uniform polyhedron, indexed as U39. It has 42 faces (30 squares and 12 decagons), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral. Related polyhedra It shares its vertex arrangement with the small stellated truncated dodecahedron and the uniform compounds of 6 or 12 pentagrammic prisms. It additionally shares its edge arrangement with the rhombicosidodecahedron (having the square faces in common), and with the small dodecicosidodecahedron (having the decagonal faces in common). Small rhombidodecacron The small rhombidodecacron (or small dipteral ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the small rhombidodecahedron. It is visually identical to the Small dodecacronic hexecontahedron. It has 60 intersecting antiparallelogram In geometry, an antiparallelogram is a type of self-crossing quadrilateral. Like a parallelogram, an antiparallelogram has two op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
