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Great Rhombic Triacontahedron
In geometry, the great rhombic triacontahedron is a nonconvex isohedral, isotoxal polyhedron. It is the dual of the great icosidodecahedron (U54). Like the convex rhombic triacontahedron it has 30 rhombic faces, 60 edges and 32 vertices (also 20 on 3-fold and 12 on 5-fold axes). It can be constructed from the convex solid by expanding the faces by factor of \varphi^3 \approx 4.236, where \varphi\! is the golden ratio. This solid is to the compound of great icosahedron and great stellated dodecahedron what the convex one is to the compound of dodecahedron and icosahedron: The crossing edges in the dual compound are the diagonals of the rhombs. What resembles an "excavated" rhombic triacontahedron (compare excavated dodecahedron and excavated icosahedron) can be seen within the middle of this compound. The rest of the polyhedron strikingly resembles a rhombic hexecontahedron In geometry, a rhombic hexecontahedron is a stellation of the rhombic triacontahedron. It is noncon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Of Dodecahedron And Icosahedron
In geometry, this polyhedron can be seen as either a polyhedral stellation or a compound. As a compound It can be seen as the compound of an icosahedron and dodecahedron. It is one of four compounds constructed from a Platonic solid or Kepler-Poinsot solid, and its dual. It has icosahedral symmetry (I''h'') and the same vertex arrangement as a rhombic triacontahedron. This can be seen as the three-dimensional equivalent of the compound of two pentagons ( " decagram"); this series continues into the fourth dimension as the compound of 120-cell and 600-cell and into higher dimensions as compounds of hyperbolic tilings. As a stellation This polyhedron is the first stellation of the icosidodecahedron, and given as Wenninger model index 47. The stellation facets for construction are: : As a Faceting The compound of a Dodecahedron and an Icosahedron shares the same vertices as a list of other polyhedra, including the Rhombic triacontahedron and the Small tria ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pyritohedral Symmetry
150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation. The group of all (not necessarily orientation preserving) symmetries is isomorphic to the group S4, the symmetric group of permutations of four objects, since there is exactly one such symmetry for each permutation of the vertices of the tetrahedron. The set of orientation-preserving symmetries forms a group referred to as the alternating subgroup A4 of S4. Details Chiral and full (or achiral tetrahedral symmetry and pyritohedral symmetry) are discrete point symmetries (or equivalently, symmetries on the sphere). They are among the crystallographic point groups of the cubic crystal system. Seen in stereographic projection the edges of the tetrakis hexahedron form 6 circles (or centrally radial lines) in the plane. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Medial Rhombic Triacontahedron
In geometry, the medial rhombic triacontahedron (or midly rhombic triacontahedron) is a nonconvex isohedral polyhedron. It is a stellation of the rhombic triacontahedron, and can also be called small stellated triacontahedron. Its dual is the dodecadodecahedron. Its 24 vertices are all on the 12 axes with 5-fold symmetry (i.e. each corresponds to one of the 12 vertices of the icosahedron). This means that on each axis there is an inner and an outer vertex. The ratio of outer to inner vertex radius is \varphi \approx 1.618, the golden ratio. It has 30 intersecting rhombic faces, which correspond to the faces of the convex rhombic triacontahedron. The diagonals in the rhombs of the convex solid have a ratio of 1 to \varphi. The medial solid can be generated from the convex one by stretching the shorter diagonal from length 1 to \varphi^3 \approx 4.236. So the ratio of rhomb diagonals in the medial solid is 1 to \varphi^2 \approx 2.618. This solid is to the compound of small st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rhombic Hexecontahedron
In geometry, a rhombic hexecontahedron is a stellation of the rhombic triacontahedron. It is nonconvex with 60 golden rhombus, golden rhombic faces with icosahedral symmetry. It was described mathematically in 1940 by Helmut Unkelbach. It is topologically identical to the convex deltoidal hexecontahedron which has Kite (geometry), kite faces. Dissection The rhombic hexecontahedron can be dissected into 20 acute golden rhombohedra meeting at a central point. This gives the volume of a hexecontahedron of side length ''a'' to be V = (10 + 2\sqrt 5)a^3 and the area to be A = (24\sqrt 5)a^2. : Construction A rhombic hexecontahedron can be constructed from a regular dodecahedron, by taking its vertices, its face centers and its edge centers and scaling them in or out from the body center to different extents. Thus, if the 20 vertices of a dodecahedron are pulled out to increase the circumradius by a factor of ≈ 1.309, the 12 face centers are pushed in to decrease the inradius to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Dodecahedron
In geometry, the great dodecahedron is one of four Kepler–Poinsot polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagons), intersecting each other making a pentagrammic path, with five pentagons meeting at each vertex. Construction One way to construct a great dodecahedron is by faceting the regular icosahedron. In other words, it is constructed from the regular icosahedron by removing its polygonal faces without changing or creating new vertices. For each vertex of the icosahedron, the five neighboring vertices become those of a regular pentagon face of the great dodecahedron. The resulting shape has a pentagram as its vertex figure, so its Schläfli symbol is \ . The great dodecahedron may also be interpreted as the ''second stellation of dodecahedron''. The construction started from a regular dodecahedron by attaching 12 pentagonal pyramids onto each of its faces, known as the ''first stellation''. The second stellation appears when 30 wed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Excavated Dodecahedron
In geometry, the excavated dodecahedron is a star polyhedron that looks like a regular dodecahedron, dodecahedron with concave pentagonal pyramids in place of its faces. Its exterior surface represents the The Fifty Nine Icosahedra, Ef1g1 stellation of the icosahedron. It appears in Magnus Wenninger's book ''Polyhedron Models'' as model 28, the ''third stellation of icosahedron''. Description All 20 vertices and 30 of its 60 edges belong to its regular dodecahedron, dodecahedral hull. The 30 other internal edges are longer and belong to a great stellated dodecahedron. (Each contains one of the 30 edges of the regular icosahedron, icosahedral core.) Each face is a List of self-intersecting polygons, self-intersecting hexagon with alternating long and short edges and 60° angles. The equilateral triangles touching a short edge are part of the face. (The smaller one between the long edges is a face of the icosahedral core.) Faceting of the dodecahedron It has the same external for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Compound
In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram. The outer vertices of a compound can be connected to form a convex polyhedron called its convex hull. A compound is a faceting of its convex hull. Another convex polyhedron is formed by the small central space common to all members of the compound. This polyhedron can be used as the core for a set of stellations. Regular compounds A regular polyhedral compound can be defined as a compound which, like a regular polyhedron, is vertex-transitive, edge-transitive, and face-transitive. Unlike the case of polyhedra, this is not equivalent to the symmetry group acting transitively on its flags; the compound of two tetrahedra is the only regular compound with that property. There are five regular compounds of polyhedra: Best known is the regular compound of two tetrahedra, often called t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Of Great Icosahedron And Great Stellated Dodecahedron
There are two different compounds of great icosahedron and great stellated dodecahedron: one is a dual compound and a stellation of the great icosidodecahedron, the other is a stellation of the icosidodecahedron. Dual compound It can be seen as a polyhedron compound of a great icosahedron and great stellated dodecahedron. It is one of five compounds constructed from a Platonic solid or Kepler-Poinsot solid, and its dual. It is a stellation of the great icosidodecahedron. It has icosahedral symmetry (I''h'') and it has the same vertex arrangement as a great rhombic triacontahedron. This can be seen as one of the two three-dimensional equivalents of the compound of two pentagrams ( " decagram"); this series continues into the fourth dimension as compounds of star 4-polytopes. Stellation of the icosidodecahedron This polyhedron is a stellation of the icosidodecahedron In geometry, an icosidodecahedron or pentagonal gyrobirotunda is a polyhedron with twent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
