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Ditrigonal
In geometry, there are seven uniform and uniform dual polyhedra named as ditrigonal. Ditrigonal vertex figures There are five uniform ditrigonal polyhedra, all with icosahedral symmetry.Har'El, 1993 The three uniform star polyhedron with Wythoff symbol of the form 3 , ''p'' ''q'' or , ''p'' ''q'' are ditrigonal, at least if ''p'' and ''q'' are not 2. Each polyhedron includes two types of faces, being of triangles, pentagons, or pentagrams. Their vertex configurations are of the form ''p''.''q''.''p''.''q''.''p''.''q'' or (''p''.''q'')3 with a symmetry of order 3. Here, term ditrigonal refers to a hexagon having a symmetry of order 3 (triangular symmetry) acting with 2 rotational orbits on the 6 angles of the vertex figure (the word ''ditrigonal'' means "having two sets of 3 angles"). Mathworld (retrieved 10 June 2016) [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ditrigonal Dodecadodecahedron Cd
In geometry, there are seven uniform and uniform dual polyhedra named as ditrigonal. Ditrigonal vertex figures There are five uniform ditrigonal polyhedra, all with icosahedral symmetry.Har'El, 1993 The three uniform star polyhedron with Wythoff symbol of the form 3 , ''p'' ''q'' or , ''p'' ''q'' are ditrigonal, at least if ''p'' and ''q'' are not 2. Each polyhedron includes two types of faces, being of triangles, pentagons, or pentagrams. Their vertex configurations are of the form ''p''.''q''.''p''.''q''.''p''.''q'' or (''p''.''q'')3 with a symmetry of order 3. Here, term ditrigonal refers to a hexagon having a symmetry of order 3 (triangular symmetry) acting with 2 rotational orbits on the 6 angles of the vertex figure (the word ''ditrigonal'' means "having two sets of 3 angles"). Mathworld (retrieved 10 June 2016) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ditrigonal Dodecadodecahedron Vertfig
In geometry, there are seven uniform and uniform dual polyhedra named as ditrigonal. Ditrigonal vertex figures There are five uniform ditrigonal polyhedra, all with icosahedral symmetry.Har'El, 1993 The three uniform star polyhedron with Wythoff symbol of the form 3 , ''p'' ''q'' or , ''p'' ''q'' are ditrigonal, at least if ''p'' and ''q'' are not 2. Each polyhedron includes two types of faces, being of triangles, pentagons, or pentagrams. Their vertex configurations are of the form ''p''.''q''.''p''.''q''.''p''.''q'' or (''p''.''q'')3 with a symmetry of order 3. Here, term ditrigonal refers to a hexagon having a symmetry of order 3 (triangular symmetry) acting with 2 rotational orbits on the 6 angles of the vertex figure (the word ''ditrigonal'' means "having two sets of 3 angles"). Mathworld (retrieved 10 June 2016) [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Ditrigonal Icosidodecahedron
In geometry, the small ditrigonal icosidodecahedron (or small ditrigonary icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U30. It has 32 faces (20 triangles and 12 pentagrams), 60 edges, and 20 vertices. It has extended Schläfli symbol a, as an ''altered dodecahedron'', and Coxeter diagram or . It is constructed from Schwarz triangle (3 3 ) with Wythoff symbol 3 , 3. Its hexagonal vertex figure alternates equilateral triangle and pentagram faces. Related polyhedra Its convex hull is a regular dodecahedron. It additionally shares its edge arrangement with the great ditrigonal icosidodecahedron (having the triangular faces in common), the ditrigonal dodecadodecahedron (having the pentagrammic faces in common), and the regular compound of five cubes. As a simple polyhedron, it is also a hexakis truncated icosahedron where the triangles touching the pentagons are made coplanar, making the others concave. See also * List of uniform polyhedra In geometry, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ditrigonal Dodecadodecahedron
In geometry, the ditrigonal dodecadodecahedron (or ditrigonary dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U41. It has 24 faces (12 pentagons and 12 pentagrams), 60 edges, and 20 vertices. It has extended Schläfli symbol b, as a ''blended great dodecahedron'', and Coxeter diagram . It has 4 Schwarz triangle equivalent constructions, for example Wythoff symbol 3 , 5, and Coxeter diagram . Related polyhedra Its convex hull is a regular dodecahedron. It additionally shares its edge arrangement with the small ditrigonal icosidodecahedron (having the pentagrammic faces in common), the great ditrigonal icosidodecahedron (having the pentagonal faces in common), and the regular compound of five cubes. Furthermore, it may be viewed as a facetted dodecahedron: the pentagrammic faces are inscribed in the dodecahedron's pentagons. Its dual, the medial triambic icosahedron, is a stellation of the icosahedron. It is topologically equivalent to a quotient sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Ditrigonal Icosidodecahedron
In geometry, the great ditrigonal icosidodecahedron (or great ditrigonary icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U47. It has 32 faces (20 triangles and 12 pentagons), 60 edges, and 20 vertices. It has 4 Schwarz triangle equivalent constructions, for example Wythoff symbol 3 , 3 gives Coxeter diagram = . It has extended Schläfli symbol a or c, as an ''altered great stellated dodecahedron'' or ''converted great icosahedron''. Its circumradius is \frac times the length of its edge, a value it shares with the cube. Related polyhedra Its convex hull is a regular dodecahedron. It additionally shares its edge arrangement with the small ditrigonal icosidodecahedron (having the triangular faces in common), the ditrigonal dodecadodecahedron (having the pentagonal faces in common), and the regular compound of five cubes The compound of five cubes is one of the five regular polyhedral compounds. It was first described by Edmund Hess in 1876. It is one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Star 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 as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Ditrigonal Dodecicosidodecahedron
In geometry, the great ditrigonal dodecicosidodecahedron (or great dodekified icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U42. It has 44 faces (20 triangles, 12 pentagons, and 12 decagrams), 120 edges, and 60 vertices. Its vertex figure is an isosceles trapezoid. Related polyhedra It shares its vertex arrangement with the truncated dodecahedron. It additionally shares its edge arrangement with the great icosicosidodecahedron (having the triangular and pentagonal faces in common) and the great dodecicosahedron (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] |
Small Ditrigonal Dodecicosidodecahedron
In geometry, the small ditrigonal dodecicosidodecahedron (or small dodekified icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U43. It has 44 faces (20 triangles, 12 pentagrams and 12 decagons), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral. Related polyhedra It shares its vertex arrangement with the great stellated truncated dodecahedron. It additionally shares its edges with the small icosicosidodecahedron (having the triangular and pentagrammic faces in common) and the small dodecicosahedron (having the decagonal 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] |
Great Ditrigonal Dodecacronic Hexecontahedron
In geometry, the great ditrigonal dodecacronic hexecontahedron (or great lanceal trisicosahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform great ditrigonal dodecicosidodecahedron. Its faces are kites. Part of each kite lies inside the solid, hence is invisible in solid models. Proportions Kite faces have two angles of \arccos(\frac-\frac\sqrt)\approx 98.183\,872\,491\,81^, one of \arccos(-\frac+\frac\sqrt)\approx 112.296\,452\,073\,54^ and one of \arccos(-\frac+\frac\sqrt)\approx 51.335\,802\,942\,83^. 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 127.686\,523\,427\,48^. The ratio between the lengths of the long edges and the short ones equals \frac\approx 1.917\,288\,176\,70. References * p. 62 E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Ditrigonal Dodecacronic Hexecontahedron
In geometry, the small ditrigonal dodecacronic hexecontahedron (or fat star) is a nonconvex isohedral polyhedron. It is the dual of the uniform small ditrigonal dodecicosidodecahedron. It is visually identical to the small dodecicosacron. Its faces are darts. A part of each dart lies inside the solid, hence is invisible in solid models. Proportions Faces have two angles of \arccos(\frac+\frac\sqrt)\approx 12.661\,078\,804\,43^, one of \arccos(-\frac-\frac\sqrt)\approx 116.996\,396\,851\,70^ and one of 360^-\arccos(-\frac-\frac\sqrt)\approx 217.681\,445\,539\,45^. 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 146.230\,659\,755\,53^. The ratio between the lengths of the long and short edges is \frac\approx 1.110\,008\,944\,41. Re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagram
A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle around the five points creates a similar symbol referred to as the pentacle, which is used widely by Wiccans and in paganism, or as a sign of life and connections. The word "pentagram" refers only to the five-pointed star, not the surrounding circle of a pentacle. Pentagrams were used symbolically in ancient Greece and Babylonia. Christians once commonly used the pentagram to represent the Five Holy Wounds, five wounds of Jesus. Today the symbol is widely used by the Wiccans, witches, and pagans. The pentagram has Magic (supernatural), magical associations. Many people who practice neopaganism wear jewelry incorporating the symbol. The word ''pentagram'' comes from the Greek language, Greek word πεντάγραμμον (''pentagrammon''), fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
