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Dodecadodecahedron
In geometry, the dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U36. It is the Rectification (geometry), rectification of the great dodecahedron (and that of its dual, the small stellated dodecahedron). It was discovered independently by , and . The edges of this model form 10 central hexagons, and these, projected onto a sphere, become 10 great circles. These 10, along with the great circles from projections of two other polyhedra, form the 31 great circles of the spherical icosahedron used in construction of geodesic domes. Wythoff constructions It has four Wythoff constructions between four Schwarz triangle families: 2 , 5 5/2, 2 , 5 5/3, 2 , 5/2 5/4, 2 , 5/3 5/4, but represent identical results. Similarly it can be given four extended Schläfli symbols: r, r, r, and r or as Coxeter-Dynkin diagrams: , , , and . Net A shape with the same exterior appearance as the dodecadodecahedron can be constructed by folding up these nets: 12 pentagrams and 20 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonconvex Uniform Polyhedron
In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, or both. The complete set of 57 nonprismatic uniform star polyhedra includes the 4 regular ones, called the Kepler–Poinsot polyhedron, Kepler–Poinsot polyhedra, 14 Quasiregular polyhedron#Nonconvex examples, quasiregular ones, and 39 semiregular ones. There are also two infinite sets of Uniform_polyhedron#.28p_2_2.29_Prismatic_.5Bp.2C2.5D.2C_I2.28p.29_family_.28Dph_dihedral_symmetry.29, ''uniform star prisms'' and ''uniform star antiprisms''. Just as (nondegenerate) star polygons (which have density (polytope), polygon density greater than 1) correspond to circular polygons with overlapping Tessellation, tiles, star polyhedra that do not pass through the center have polytope density greater than 1, and correspond to spherical polyhe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dodecadodecahedron (fixed Geometry)
In geometry, the dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U36. It is the rectification of the great dodecahedron (and that of its dual, the small stellated dodecahedron). It was discovered independently by , and . The edges of this model form 10 central hexagons, and these, projected onto a sphere, become 10 great circles. These 10, along with the great circles from projections of two other polyhedra, form the 31 great circles of the spherical icosahedron used in construction of geodesic domes. Wythoff constructions It has four Wythoff constructions between four Schwarz triangle families: 2 , 5 5/2, 2 , 5 5/3, 2 , 5/2 5/4, 2 , 5/3 5/4, but represent identical results. Similarly it can be given four extended Schläfli symbols: r, r, r, and r or as Coxeter-Dynkin diagrams: , , , and . Net A shape with the same exterior appearance as the dodecadodecahedron can be constructed by folding up these nets: 12 pentagrams and 20 rhombic clusters are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dodecadodecahedron
In geometry, the dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U36. It is the Rectification (geometry), rectification of the great dodecahedron (and that of its dual, the small stellated dodecahedron). It was discovered independently by , and . The edges of this model form 10 central hexagons, and these, projected onto a sphere, become 10 great circles. These 10, along with the great circles from projections of two other polyhedra, form the 31 great circles of the spherical icosahedron used in construction of geodesic domes. Wythoff constructions It has four Wythoff constructions between four Schwarz triangle families: 2 , 5 5/2, 2 , 5 5/3, 2 , 5/2 5/4, 2 , 5/3 5/4, but represent identical results. Similarly it can be given four extended Schläfli symbols: r, r, r, and r or as Coxeter-Dynkin diagrams: , , , and . Net A shape with the same exterior appearance as the dodecadodecahedron can be constructed by folding up these nets: 12 pentagrams and 20 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Dodecahemicosahedron
In geometry, the great dodecahemicosahedron (or great dodecahemiicosahedron) is a nonconvex uniform polyhedron, indexed as U65. It has 22 faces (12 pentagons and 10 hexagons), 60 edges, and 30 vertices. Its vertex figure is a crossed quadrilateral. It is a hemipolyhedron with ten hexagonal faces passing through the model center. Related polyhedra Its convex hull is the icosidodecahedron. It also shares its edge arrangement with the dodecadodecahedron (having the pentagonal faces in common), and with the small dodecahemicosahedron (having the hexagonal faces in common). Great dodecahemicosacron The great dodecahemicosacron is the dual of the great dodecahemicosahedron, and is one of nine dual hemipolyhedra. It appears visually indistinct from the small dodecahemicosacron. Since the hemipolyhedra have faces passing through the center, the dual figures have corresponding vertices at infinity; properly, on the real projective plane at infinity. In Magnus Wenninger ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Dodecahemicosahedron
In geometry, the small dodecahemicosahedron (or great dodecahemiicosahedron) is a nonconvex uniform polyhedron, indexed as U62. It has 22 faces (12 pentagrams and 10 hexagons), 60 edges, and 30 vertices. Its vertex figure is a crossed quadrilateral. It is a hemipolyhedron with ten hexagonal faces passing through the model center. Related polyhedra Its convex hull is the icosidodecahedron In geometry, an icosidodecahedron or pentagonal gyrobirotunda is a polyhedron with twenty (''icosi-'') triangular faces and twelve (''dodeca-'') pentagonal faces. An icosidodecahedron has 30 identical Vertex (geometry), vertices, with two triang .... It also shares its edge arrangement with the dodecadodecahedron (having the pentagrammic faces in common), and with the great dodecahemicosahedron (having the hexagonal faces in common). Gallery See also * List of uniform polyhedra References External links * Uniform polyhedra and duals Uniform polyhedra {{Polyh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
31 Great Circles Of The Spherical Icosahedron
In geometry, the 31 great circles of the spherical icosahedron is an arrangement of 31 great circles in icosahedral symmetry. It was first identified by Buckminster Fuller and is used in construction of geodesic domes. Construction The 31 great circles can be seen in 3 sets: 15, 10, and 6, each representing edges of a polyhedron projected onto a sphere. Fifteen great circles represent the edges of a disdyakis triacontahedron, the dual of a truncated icosidodecahedron. Six more great circles represent the edges of an icosidodecahedron, and the last ten great circles come from the edges of the uniform star polyhedron, uniform star dodecadodecahedron, making pentagrams with vertices at the edge centers of the icosahedron. There are 62 points of intersection, positioned at the 12 vertices, and center of the 30 edges, and 20 faces of a regular icosahedron. Images The 31 great circles are shown here in 3 directions, with 5-fold, 3-fold, and 2-fold symmetry. There are 4 types of righ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Stellated Dodecahedron
In geometry, the small stellated dodecahedron is a Kepler–Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol . It is one of four nonconvex List of regular polytopes#Non-convex 2, regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex. It shares the same vertex arrangement as the convex regular icosahedron. It also shares the same edge arrangement with the great icosahedron, with which it forms Great complex icosidodecahedron, a degenerate uniform compound figure. It is the List of Wenninger polyhedron models#Stellations of dodecahedron, second of four stellations of the dodecahedron (including the original dodecahedron itself). The small stellated dodecahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the edges (1-faces) of the core polytope until a point is reached where they intersect. Construction and properties The small stellated dodecahedron is ... [...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] |
Small Dodecahemicosahedron
In geometry, the small dodecahemicosahedron (or great dodecahemiicosahedron) is a nonconvex uniform polyhedron, indexed as U62. It has 22 faces (12 pentagrams and 10 hexagons), 60 edges, and 30 vertices. Its vertex figure is a crossed quadrilateral. It is a hemipolyhedron with ten hexagonal faces passing through the model center. Related polyhedra Its convex hull is the icosidodecahedron In geometry, an icosidodecahedron or pentagonal gyrobirotunda is a polyhedron with twenty (''icosi-'') triangular faces and twelve (''dodeca-'') pentagonal faces. An icosidodecahedron has 30 identical Vertex (geometry), vertices, with two triang .... It also shares its edge arrangement with the dodecadodecahedron (having the pentagrammic faces in common), and with the great dodecahemicosahedron (having the hexagonal faces in common). Gallery See also * List of uniform polyhedra References External links * Uniform polyhedra and duals Uniform polyhedra {{Polyh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncation (geometry)
In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new Facet (geometry), facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids. Uniform truncation In general any polyhedron (or polytope) can also be truncated with a degree of freedom as to how deep the cut is, as shown in Conway polyhedron notation truncation operation. A special kind of truncation, usually implied, is a uniform truncation, a truncation operator applied to a regular polyhedron (or regular polytope) which creates a resulting uniform polyhedron (uniform polytope) with equal edge lengths. There are no degrees of freedom, and it represents a fixed geometric, just like the regular polyhedra. In general all single ringed uniform polytopes have a uniform truncation. For example, the icosidodecahedron, represented as Schläfli symbols r or \begin 5 \\ 3 \end, and Coxeter-Dynkin diagram or has a uniform truncation, the truncate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
