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Nonconvex Great Rhombicuboctahedron
In geometry, the nonconvex great rhombicuboctahedron is a nonconvex uniform polyhedron, indexed as U17. It has 26 faces (8 triangles and 18 squares), 48 edges, and 24 vertices. It is represented by the Schläfli symbol rr and Coxeter-Dynkin diagram of . Its vertex figure is a crossed quadrilateral. This model shares the name with the convex ''great rhombicuboctahedron'', also called the truncated cuboctahedron. An alternative name for this figure is quasirhombicuboctahedron. From that derives its Bowers acronym: querco. Orthographic projections Cartesian coordinates Cartesian coordinates for the vertices of a ''nonconvex great rhombicuboctahedron'' centered at the origin with edge length 1 are all the permutations of : (±''ξ'', ±1, ±1), where ''ξ'' = − 1. Related polyhedra It shares the vertex arrangement with the convex truncated cube. It additionally shares its edge arrangement In geometry, a vertex arrangement is a set of points in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonconvex Great Rhombicuboctahedron
In geometry, the nonconvex great rhombicuboctahedron is a nonconvex uniform polyhedron, indexed as U17. It has 26 faces (8 triangles and 18 squares), 48 edges, and 24 vertices. It is represented by the Schläfli symbol rr and Coxeter-Dynkin diagram of . Its vertex figure is a crossed quadrilateral. This model shares the name with the convex ''great rhombicuboctahedron'', also called the truncated cuboctahedron. An alternative name for this figure is quasirhombicuboctahedron. From that derives its Bowers acronym: querco. Orthographic projections Cartesian coordinates Cartesian coordinates for the vertices of a ''nonconvex great rhombicuboctahedron'' centered at the origin with edge length 1 are all the permutations of : (±''ξ'', ±1, ±1), where ''ξ'' = − 1. Related polyhedra It shares the vertex arrangement with the convex truncated cube. It additionally shares its edge arrangement In geometry, a vertex arrangement is a set of points in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Cubicuboctahedron
In geometry, the great cubicuboctahedron is a nonconvex uniform polyhedron, indexed as U14. It has 20 faces (8 triangles, 6 squares and 6 octagrams), 48 edges, and 24 vertices. Its square faces and its octagrammic faces are parallel to those of a cube, while its triangular faces are parallel to those of an octahedron: hence the name ''cubicuboctahedron''. The ''great'' suffix serves to distinguish it from the small cubicuboctahedron, which also has faces in the aforementioned directions. Orthographic projections Related polyhedra It shares the vertex arrangement with the convex truncated cube and two other nonconvex uniform polyhedra. It additionally shares its edge arrangement with the nonconvex great rhombicuboctahedron (having the triangular faces and 6 square faces in common), and with the great rhombihexahedron (having the octagrammic faces in common). Great hexacronic icositetrahedron The great hexacronic icositetrahedron (or great lanceal disdodecahedron) is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Deltoidal Icositetrahedron
In geometry, the great deltoidal icositetrahedron (or great sagittal disdodecahedron) is the dual of the nonconvex great rhombicuboctahedron. Its faces are darts. Part of each dart lies inside the solid, hence is invisible in solid models. One of its halves can be rotated by 45 degrees to form the pseudo great deltoidal icositetrahedron, analogous to the pseudo-deltoidal icositetrahedron. Proportions Faces have three angles of \arccos(\frac+\frac\sqrt)\approx 31.399\,714\,809\,92^ and one of 360^-\arccos(-\frac+\frac\sqrt)\approx 265.800\,855\,570\,24^. Its 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 uni ... equal \arccos()\approx 94.531\,580\,798\,20^. The ratio between the lengths of the long edges and the short ones equals 2+\frac\sqrt\approx 2.707\,106\,781\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo Great Rhombicuboctahedron
In geometry, the pseudo great rhombicuboctahedron is one of the two pseudo uniform polyhedra, the other being the convex elongated square gyrobicupola or pseudo rhombicuboctahedron. It has the same vertex figure as the nonconvex great rhombicuboctahedron (a uniform polyhedron), but is not a uniform polyhedron (due to not being isogonal), and has a smaller symmetry group. It can be obtained from the great rhombicuboctahedron by taking a square face and the 8 faces with a common vertex to it (forming a crossed square cupola) and rotating them by an angle of . It is related to the nonconvex great rhombicuboctahedron In geometry, the nonconvex great rhombicuboctahedron is a nonconvex uniform polyhedron, indexed as U17. It has 26 faces (8 triangles and 18 squares), 48 edges, and 24 vertices. It is represented by the Schläfli symbol rr and Coxeter-Dynkin dia ... in the same way that the pseudo rhombicuboctahedron is related to the rhombicuboctahedron. Related polyhedra The pseud ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo-great Rhombicuboctahedron
In geometry, the pseudo great rhombicuboctahedron is one of the two pseudo uniform polyhedra, the other being the convex elongated square gyrobicupola or pseudo rhombicuboctahedron. It has the same vertex figure as the nonconvex great rhombicuboctahedron (a uniform polyhedron), but is not a uniform polyhedron (due to not being isogonal), and has a smaller symmetry group. It can be obtained from the great rhombicuboctahedron by taking a square face and the 8 faces with a common vertex to it (forming a crossed square cupola) and rotating them by an angle of . It is related to the nonconvex great rhombicuboctahedron In geometry, the nonconvex great rhombicuboctahedron is a nonconvex uniform polyhedron, indexed as U17. It has 26 faces (8 triangles and 18 squares), 48 edges, and 24 vertices. It is represented by the Schläfli symbol rr and Coxeter-Dynkin dia ... in the same way that the pseudo rhombicuboctahedron is related to the rhombicuboctahedron. Related polyhedra The pseud ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Rhombihexahedron
In geometry, the great rhombihexahedron (or great rhombicube) is a nonconvex uniform polyhedron, indexed as U21. It has 18 faces (12 squares and 6 octagrams), 48 edges, and 24 vertices. Its dual is the great rhombihexacron. Its vertex figure is a crossed quadrilateral. Orthogonal projections Gallery Related polyhedra It shares the vertex arrangement with the convex truncated cube. It additionally shares its edge arrangement with the nonconvex great rhombicuboctahedron (having 12 square faces in common), and with the great cubicuboctahedron (having the octagrammic faces in common). It may be constructed as the exclusive or (blend) of three octagrammic prisms. Similarly, the small rhombihexahedron may be constructed as the exclusive or of three octagonal prisms. Great rhombihexacron The great rhombihexacron is a nonconvex isohedral polyhedron. It is the dual of the uniform great rhombihexahedron (U21). It has 24 identical bow-tie-shaped faces, 18 vertices, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Cubicuboctahedron
In geometry, the great cubicuboctahedron is a nonconvex uniform polyhedron, indexed as U14. It has 20 faces (8 triangles, 6 squares and 6 octagrams), 48 edges, and 24 vertices. Its square faces and its octagrammic faces are parallel to those of a cube, while its triangular faces are parallel to those of an octahedron: hence the name ''cubicuboctahedron''. The ''great'' suffix serves to distinguish it from the small cubicuboctahedron, which also has faces in the aforementioned directions. Orthographic projections Related polyhedra It shares the vertex arrangement with the convex truncated cube and two other nonconvex uniform polyhedra. It additionally shares its edge arrangement with the nonconvex great rhombicuboctahedron (having the triangular faces and 6 square faces in common), and with the great rhombihexahedron (having the octagrammic faces in common). Great hexacronic icositetrahedron The great hexacronic icositetrahedron (or great lanceal disdodecahedron) is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Great Rhombicuboctahedron
In geometry, the nonconvex great rhombicuboctahedron is a nonconvex uniform polyhedron, indexed as U17. It has 26 faces (8 triangles and 18 squares), 48 edges, and 24 vertices. It is represented by the Schläfli symbol rr and Coxeter-Dynkin diagram of . Its vertex figure is a crossed quadrilateral. This model shares the name with the convex ''great rhombicuboctahedron'', also called the truncated cuboctahedron. An alternative name for this figure is quasirhombicuboctahedron. From that derives its Bowers acronym: querco. Orthographic projections Cartesian coordinates Cartesian coordinates for the vertices of a ''nonconvex great rhombicuboctahedron'' centered at the origin with edge length 1 are all the permutations of : (±''ξ'', ±1, ±1), where ''ξ'' = − 1. Related polyhedra It shares the vertex arrangement with the convex truncated cube. It additionally shares its edge arrangement with the great cubicuboctahedron (having the triangula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated Cube
In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices. If the truncated cube has unit edge length, its dual triakis octahedron has edges of lengths 2 and 2 + . Area and volume The area ''A'' and the volume ''V'' of a truncated cube of edge length ''a'' are: :\begin A &= 2\left(6+6\sqrt+\sqrt\right)a^2 &&\approx 32.434\,6644a^2 \\ V &= \fraca^3 &&\approx 13.599\,6633a^3. \end Orthogonal projections The ''truncated cube'' has five special orthogonal projections, centered, on a vertex, on two types of edges, and two types of faces: triangles, and octagons. The last two correspond to the B2 and A2 Coxeter planes. Spherical tiling The truncated cube can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated Hexahedron
In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices. If the truncated cube has unit edge length, its dual triakis octahedron has edges of lengths 2 and 2 + . Area and volume The area ''A'' and the volume ''V'' of a truncated cube of edge length ''a'' are: :\begin A &= 2\left(6+6\sqrt+\sqrt\right)a^2 &&\approx 32.434\,6644a^2 \\ V &= \fraca^3 &&\approx 13.599\,6633a^3. \end Orthogonal projections The ''truncated cube'' has five special orthogonal projections, centered, on a vertex, on two types of edges, and two types of faces: triangles, and octagons. The last two correspond to the B2 and A2 Coxeter planes. Spherical tiling The truncated cube can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo Great Rhombicuboctahedron
In geometry, the pseudo great rhombicuboctahedron is one of the two pseudo uniform polyhedra, the other being the convex elongated square gyrobicupola or pseudo rhombicuboctahedron. It has the same vertex figure as the nonconvex great rhombicuboctahedron (a uniform polyhedron), but is not a uniform polyhedron (due to not being isogonal), and has a smaller symmetry group. It can be obtained from the great rhombicuboctahedron by taking a square face and the 8 faces with a common vertex to it (forming a crossed square cupola) and rotating them by an angle of . It is related to the nonconvex great rhombicuboctahedron In geometry, the nonconvex great rhombicuboctahedron is a nonconvex uniform polyhedron, indexed as U17. It has 26 faces (8 triangles and 18 squares), 48 edges, and 24 vertices. It is represented by the Schläfli symbol rr and Coxeter-Dynkin dia ... in the same way that the pseudo rhombicuboctahedron is related to the rhombicuboctahedron. Related polyhedra The pseud ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Rhombihexahedron
In geometry, the great rhombihexahedron (or great rhombicube) is a nonconvex uniform polyhedron, indexed as U21. It has 18 faces (12 squares and 6 octagrams), 48 edges, and 24 vertices. Its dual is the great rhombihexacron. Its vertex figure is a crossed quadrilateral. Orthogonal projections Gallery Related polyhedra It shares the vertex arrangement with the convex truncated cube. It additionally shares its edge arrangement with the nonconvex great rhombicuboctahedron (having 12 square faces in common), and with the great cubicuboctahedron (having the octagrammic faces in common). It may be constructed as the exclusive or (blend) of three octagrammic prisms. Similarly, the small rhombihexahedron may be constructed as the exclusive or of three octagonal prisms. Great rhombihexacron The great rhombihexacron is a nonconvex isohedral polyhedron. It is the dual of the uniform great rhombihexahedron (U21). It has 24 identical bow-tie-shaped faces, 18 vertices, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
