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Dodecadodecahedron
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 nec ... [...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 polyhedra, 5 quasiregular ones, and 48 semiregular ones. There are also two infinite sets of ''uniform star prisms'' and ''uniform star antiprisms''. Just as (nondegenerate) star polygons (which have polygon density greater than 1) correspond to circular polygons with overlapping tiles, star polyhedra that do not pass through the center have polytope density greater than 1, and correspond to spherical polyhedra with overlapping tiles; there are 47 nonprismatic such uniform star polyhedra. The remaining 10 nonprismatic uniform star polyhedra, those that pass through the center, are the hemipolyhedra as well ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated Dodecadodecahedron
In geometry, the truncated dodecadodecahedron (or stellatruncated dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U59. It is given a Schläfli symbol t0,1,2. It has 54 faces (30 squares, 12 decagons, and 12 decagrams), 180 edges, and 120 vertices. The central region of the polyhedron is connected to the exterior via 20 small triangular holes. The name ''truncated dodecadodecahedron'' is somewhat misleading: truncation of the dodecadodecahedron would produce rectangular faces rather than squares, and the pentagram faces of the dodecadodecahedron would turn into truncated pentagrams rather than decagrams. However, it is the quasitruncation of the dodecadodecahedron, as defined by . For this reason, it is also known as the quasitruncated dodecadodecahedron. Coxeter et al. credit its discovery to a paper published in 1881 by Austrian mathematician Johann Pitsch. Cartesian coordinates Cartesian coordinates for the vertices of a truncated dodecadodecahedron are all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dodecadodecahedron Net
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 ne ... [...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 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 nec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Dodecahedron
In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol and Coxeter–Dynkin diagram of . It is one of four nonconvex regular 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. The discovery of the great dodecahedron is sometimes credited to Louis Poinsot in 1810, though there is a drawing of something very similar to a great dodecahedron in the 1568 book ''Perspectiva Corporum Regularium'' by Wenzel Jamnitzer. The great dodecahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the -pentagonal polytope faces of the core -polytope (pentagons for the great dodecahedron, and line segments for the pentagram) until the figure again closes. Images Related polyhedra It shares the same edge arrangement as the convex regular icosahedron; the compoun ... [...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 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 a degenerate uniform compound figure. It is the 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. Topology If the pentagrammic faces are considered as 5 triangular faces, it shares the same surface topology as the pentakis dodecahedron, but with much taller isosceles triangle faces, with the ... [...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] |
Great Dodecahemicosahedron
In geometry, the great dodecahemicosahedron (or small dodecahemiicosahedron) is a nonconvex uniform polyhedron, indexed as U65. It has 22 faces (12 Pentagon, pentagons and 10 Hexagon, hexagons), 60 edges, and 30 vertices. Its vertex figure is a quadrilateral#More quadrilaterals, 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 Face (geometry), faces passing through the center, the Dual polyhedron, dual figures have corresponding Vertex (geome ... [...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. It also shares its edge arrangement with the dodecadodecahedron (having the pentagrammic faces in common), and with the great dodecahemicosahedron In geometry, the great dodecahemicosahedron (or small dodecahemiicosahedron) is a nonconvex uniform polyhedron, indexed as U65. It has 22 faces (12 Pentagon, pentagons and 10 Hexagon, hexagons), 60 edges, and 30 vertices. Its vertex figure is a q ... (having the hexagonal faces in common). Gallery See also * List of uniform polyhedra References External links * Uniform polyhedra and duals Uniform polyhedra {{Polyhe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Dodecahemicosahedron
In geometry, the great dodecahemicosahedron (or small dodecahemiicosahedron) is a nonconvex uniform polyhedron, indexed as U65. It has 22 faces (12 Pentagon, pentagons and 10 Hexagon, hexagons), 60 edges, and 30 vertices. Its vertex figure is a quadrilateral#More quadrilaterals, 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 Face (geometry), faces passing through the center, the Dual polyhedron, dual figures have corresponding Vertex (geome ... [...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. It also shares its edge arrangement with the dodecadodecahedron (having the pentagrammic faces in common), and with the great dodecahemicosahedron In geometry, the great dodecahemicosahedron (or small dodecahemiicosahedron) is a nonconvex uniform polyhedron, indexed as U65. It has 22 faces (12 Pentagon, pentagons and 10 Hexagon, hexagons), 60 edges, and 30 vertices. Its vertex figure is a q ... (having the hexagonal faces in common). Gallery See also * List of uniform polyhedra References External links * Uniform polyhedra and duals Uniform polyhedra {{Polyhe ... [...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 tw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
