|
Nonconvex Great Rhombicosidodecahedron
In geometry, the nonconvex great rhombicosidodecahedron is a nonconvex uniform polyhedron, indexed as U67. It has 62 faces (20 triangles, 30 squares and 12 pentagrams), 120 edges, and 60 vertices. It is also called the quasirhombicosidodecahedron. It is given a Schläfli symbol rr. Its vertex figure is a crossed quadrilateral. This model shares the name with the convex ''great rhombicosidodecahedron'', also known as the truncated icosidodecahedron. Cartesian coordinates Cartesian coordinates for the vertices of a nonconvex great rhombicosidodecahedron are all the even permutations of : (±1/τ2, 0, ±(2−1/τ)) : (±1, ±1/τ3, ±1) : (±1/τ, ±1/τ2, ±2/τ) where τ = (1+)/2 is the golden ratio (sometimes written φ). Related polyhedra It shares its vertex arrangement with the truncated great dodecahedron, and with the uniform compounds of 6 or 12 pentagonal prisms. It additionally shares its edge arrangement with the great dodecicosidodecahedron (having the tri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonconvex Great Rhombicosidodecahedron
In geometry, the nonconvex great rhombicosidodecahedron is a nonconvex uniform polyhedron, indexed as U67. It has 62 faces (20 triangles, 30 squares and 12 pentagrams), 120 edges, and 60 vertices. It is also called the quasirhombicosidodecahedron. It is given a Schläfli symbol rr. Its vertex figure is a crossed quadrilateral. This model shares the name with the convex ''great rhombicosidodecahedron'', also known as the truncated icosidodecahedron. Cartesian coordinates Cartesian coordinates for the vertices of a nonconvex great rhombicosidodecahedron are all the even permutations of : (±1/τ2, 0, ±(2−1/τ)) : (±1, ±1/τ3, ±1) : (±1/τ, ±1/τ2, ±2/τ) where τ = (1+)/2 is the golden ratio (sometimes written φ). Related polyhedra It shares its vertex arrangement with the truncated great dodecahedron, and with the uniform compounds of 6 or 12 pentagonal prisms. It additionally shares its edge arrangement with the great dodecicosidodecahedron (having the tri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edge Arrangement
In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes. For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equal distance and angles from a center point. Two polytopes share the same ''vertex arrangement'' if they share the same 0-skeleton. A group of polytopes that shares a vertex arrangement is called an ''army''. Vertex arrangement The same set of vertices can be connected by edges in different ways. For example, the ''pentagon'' and ''pentagram'' have the same ''vertex arrangement'', while the second connects alternate vertices. A ''vertex arrangement'' is often described by the convex hull polytope which contains it. For example, the regular ''pentagram'' can be said to have a (regular) ''pentagonal vertex arrangement''. Infinite tilings can also share common ''vertex arrangements''. For example, this triangular lattice of points ... [...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] |
Great Deltoidal Hexecontahedron
In geometry, the great deltoidal hexecontahedron (or great sagittal ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the nonconvex great rhombicosidodecahedron. It is visually identical to the great rhombidodecacron. It has 60 intersecting cross quadrilateral faces, 120 edges, and 62 vertices. Its faces are darts. Part of each dart lies inside the solid, hence is invisible in solid models. It is also called a ''great strombic hexecontahedron.'' Proportions The darts have two angles of \arccos(\frac+\frac\sqrt)\approx 18.699\,407\,085\,15^, one of \arccos(-\frac+\frac\sqrt)\approx 91.512\,394\,720\,74^ and one of 360^-\arccos(-\frac-\frac\sqrt)\approx 231.088\,791\,108\,96^. 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 uni ... equals \ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Of Twelve Pentagonal Prisms
This uniform polyhedron compound is a symmetric arrangement of 12 pentagonal 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 pentagonal prisms. In doing so, the vertices of the two enantiomorphs coincide, with the result that the full compound has two pentagonal prisms incident on each of its vertices. Related polyhedra This compound shares its vertex arrangement with four uniform polyhedra In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also fa ... as follows: References *. Polyhedral compounds {{polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Of Six Pentagonal Prisms
This uniform polyhedron compound is a chiral symmetric arrangement of six pentagonal prisms, aligned with the axes of fivefold rotational symmetry of a dodecahedron. Related polyhedra This compound shares its vertex arrangement with four uniform polyhedra In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also fa ... as follows: References *. Polyhedral compounds {{polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated Great Dodecahedron
In geometry, the truncated great dodecahedron is a nonconvex uniform polyhedron, indexed as U37. It has 24 faces (12 pentagrams and 12 decagons), 90 edges, and 60 vertices. It is given a Schläfli symbol t. Related polyhedra It shares its vertex arrangement with three other uniform polyhedra: the nonconvex great rhombicosidodecahedron, the great dodecicosidodecahedron, and the great rhombidodecahedron; and with the uniform compounds of 6 or 12 pentagonal prisms. This polyhedron is the truncation of the great dodecahedron: The truncated small stellated dodecahedron looks like a dodecahedron on the surface, but it has 24 faces, 12 pentagons from the truncated vertices and 12 overlapping as (truncated pentagrams). Small stellapentakis dodecahedron The small stellapentakis dodecahedron (or small astropentakis dodecahedron) is a nonconvex isohedral polyhedron. It is the dual of the truncated great dodecahedron. It has 60 intersecting triangular faces. See al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Truncated Dodecahedron
In geometry, the truncated great dodecahedron is a nonconvex uniform polyhedron, indexed as U37. It has 24 faces (12 pentagrams and 12 decagons), 90 edges, and 60 vertices. It is given a Schläfli symbol t. Related polyhedra It shares its vertex arrangement with three other uniform polyhedra: the nonconvex great rhombicosidodecahedron, the great dodecicosidodecahedron, and the great rhombidodecahedron; and with the uniform compounds of 6 or 12 pentagonal prisms. This polyhedron is the truncation of the great dodecahedron: The truncated small stellated dodecahedron looks like a dodecahedron on the surface, but it has 24 faces, 12 pentagons from the truncated vertices and 12 overlapping as (truncated pentagrams). Small stellapentakis dodecahedron The small stellapentakis dodecahedron (or small astropentakis dodecahedron) is a nonconvex isohedral polyhedron. It is the dual of the truncated great dodecahedron. It has 60 intersecting triangular faces. See also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Rhombidodecahedron
In geometry, the great rhombidodecahedron is a nonconvex uniform polyhedron, indexed as U73. It has 42 faces (30 squares, 12 decagram (geometry), decagrams), 120 edges and 60 vertices. Its vertex figure is a antiparallelogram, crossed quadrilateral. Related polyhedra It shares its vertex arrangement with the truncated great dodecahedron and the Polyhedron compound#Uniform compounds, uniform compounds of compound of six pentagonal prisms, 6 or compound of twelve pentagonal prisms, 12 pentagonal prisms. It additionally shares its edge arrangement with the nonconvex great rhombicosidodecahedron (having the square faces in common), and with the great dodecicosidodecahedron (having the decagrammic faces in common). Gallery See also * List of uniform polyhedra References External links * {{Polyhedron-stub Uniform polyhedra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Dodecicosidodecahedron
In geometry, the great dodecicosidodecahedron (or great dodekicosidodecahedron) is a nonconvex uniform polyhedron, indexed as U61. It has 44 faces (20 triangles, 12 pentagrams and 12 decagrams), 120 edges and 60 vertices. Related polyhedra It shares its vertex arrangement with the truncated great dodecahedron and the uniform compounds of 6 or 12 pentagonal prisms. It additionally shares its edge arrangement with the nonconvex great rhombicosidodecahedron (having the triangular and pentagrammic faces in common), and with the great rhombidodecahedron (having the decagrammic faces in common). 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] |
